gmm/0000755000176200001440000000000014020351121011017 5ustar liggesusersgmm/NAMESPACE0000644000176200001440000000654213467070535012272 0ustar liggesusersuseDynLib(gmm, .registration = TRUE, .fixes="F_") import(stats) importFrom(sandwich, estfun, bread, kernHAC, weightsAndrews, vcovHAC, bwAndrews, meatHC) importFrom(methods, is) importFrom(graphics, abline, legend, lines, panel.smooth, par, plot, points) importFrom(grDevices, dev.interactive, devAskNewPage, extendrange) importFrom(utils, tail) export(gmm,summary.gmm,smoothG,getDat,summary.gel,getLamb,gel, estfun.gmmFct, estfun.gmm, estfun.gel, bread.gel, bread.gmm, print.gmm,coef.gmm,vcov.gmm,print.summary.gmm, confint.gel, print.gel, print.summary.gel, vcov.gel, coef.gel, fitted.gmm, residuals.gmm, fitted.gel, residuals.gel, plot.gmm, plot.gel,formula.gmm, formula.gel, charStable, specTest, specTest.gmm, specTest.gel, print.specTest, momentEstim.baseGmm.twoStep, momentEstim.baseGmm.twoStep.formula, momentEstim.baseGmm.iterative.formula, momentEstim.baseGmm.iterative, momentEstim.baseGmm.cue.formula, momentEstim.baseGmm.cue, getModel.baseGmm, getModel.baseGel, getModel.constGmm, getModel.constGel, FinRes.baseGmm.res, momentEstim.baseGel.mod, momentEstim.baseGel.modFormula,tsls,summary.tsls, print.summary.tsls, KTest, print.gmmTests, gmmWithConst, estfun.tsls, model.matrix.tsls,vcov.tsls, bread.tsls, evalGmm, momentEstim.baseGmm.eval, momentEstim.baseGel.eval, evalGel, confint.gmm, print.confint, sysGmm, getModel.sysGmm, momentEstim.sysGmm.twoStep.formula, momentEstim.tsls.twoStep.formula, getModel.tsls, summary.sysGmm, print.sysGmm, print.summary.sysGmm, sur, threeSLS, randEffect, five, bwWilhelm, getModel.ateGel, ATEgel, summary.ategel, vcov.ategel, confint.ategel, marginal, marginal.ategel,checkConv,getImpProb, getImpProb.gel) S3method(marginal, ategel) S3method(summary, gmm) S3method(summary, sysGmm) S3method(summary, tsls) S3method(summary, gel) S3method(summary, ategel) S3method(print, gmm) S3method(print, sysGmm) S3method(print, summary.gmm) S3method(print, summary.sysGmm) S3method(coef, gmm) S3method(vcov, gmm) S3method(vcov, tsls) S3method(confint, gmm) S3method(confint, ategel) S3method(fitted, gmm) S3method(residuals, gmm) S3method(plot, gmm) S3method(formula, gmm) S3method(print, gel) S3method(print, summary.gel) S3method(print, summary.tsls) S3method(print, gmmTests) S3method(coef, gel) S3method(vcov, gel) S3method(vcov, ategel) S3method(confint, gel) S3method(fitted, gel) S3method(residuals, gel) S3method(plot, gel) S3method(formula, gel) S3method(specTest, gmm) S3method(specTest, gel) S3method(getImpProb, gel) S3method(print, specTest) S3method(print, confint) S3method(FinRes, baseGmm.res) S3method(getModel, baseGmm) S3method(getModel, baseGel) S3method(getModel, sysGmm) S3method(getModel, constGmm) S3method(getModel, constGel) S3method(getModel, tsls) S3method(getModel, ateGel) S3method(momentEstim, baseGmm.twoStep) S3method(momentEstim, baseGmm.eval) S3method(momentEstim, baseGmm.twoStep.formula) S3method(momentEstim, tsls.twoStep.formula) S3method(momentEstim, sysGmm.twoStep.formula) S3method(momentEstim, baseGmm.iterative.formula) S3method(momentEstim, baseGmm.iterative) S3method(momentEstim, baseGmm.cue.formula) S3method(momentEstim, baseGmm.cue) S3method(momentEstim, baseGel.mod) S3method(momentEstim, baseGel.modFormula) S3method(momentEstim, baseGel.eval) S3method(estfun, gmmFct) S3method(estfun, gmm) S3method(estfun, tsls) S3method(model.matrix, tsls) S3method(estfun, gel) S3method(bread, gmm) S3method(bread, gel) S3method(bread, tsls) gmm/data/0000755000176200001440000000000014017531111011735 5ustar liggesusersgmm/data/wage.rda0000644000176200001440000002720014017531162013357 0ustar liggesusers7zXZi"6!X{.D])TW"nRʟF X&\LHv`Dr19L{ynmZJ~{ g,{o^"g|μd% zcDnfiId/2BW^RȦe]UG"DQ54=*Q'G~/KXEBN.~2P1flNn"ʼ RA0%+#({oLaL~ [+CF>B۱$39@T%; 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For example, if \code{g} is a formula and \code{type} is set to "cue", it creates an object of class \code{baseGmm.cue.formula}. It this case, \code{momentEstim}, applied to this object, computes the continuously updated GMM of a linear model. It allows more flexibility this way. For example, it could be easy to add a GMM method which is robust in presence of weak identification simply by creating a new class of model and the associated \code{momentEstime} method. } gmm/man/gmm.Rd0000644000176200001440000004160114017527033012660 0ustar liggesusers\name{gmm} \alias{gmm} \alias{gmmWithConst} \alias{evalGmm} \title{Generalized method of moment estimation} \description{ Function to estimate a vector of parameters based on moment conditions using the GMM method of Hansen(82). } \usage{ gmm(g,x,t0=NULL,gradv=NULL, type=c("twoStep","cue","iterative"), wmatrix = c("optimal","ident"), vcov=c("HAC","MDS","iid","TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"), crit=10e-7,bw = bwAndrews, prewhite = 1, ar.method = "ols", approx="AR(1)", tol = 1e-7, itermax=100,optfct=c("optim","optimize","nlminb", "constrOptim"), model=TRUE, X=FALSE, Y=FALSE, TypeGmm = "baseGmm", centeredVcov = TRUE, weightsMatrix = NULL, traceIter = FALSE, data, eqConst = NULL, eqConstFullVcov = FALSE, mustar = NULL, onlyCoefficients=FALSE, ...) evalGmm(g, x, t0, tetw=NULL, gradv=NULL, wmatrix = c("optimal","ident"), vcov=c("HAC","iid","TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"),crit=10e-7,bw = bwAndrews, prewhite = FALSE, ar.method = "ols", approx="AR(1)",tol = 1e-7, model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE, weightsMatrix = NULL, data, mustar = NULL) gmmWithConst(obj, which, value) } \arguments{ \item{g}{A function of the form \eqn{g(\theta,x)} and which returns a \eqn{n \times q} matrix with typical element \eqn{g_i(\theta,x_t)} for \eqn{i=1,...q} and \eqn{t=1,...,n}. This matrix is then used to build the q sample moment conditions. It can also be a formula if the model is linear (see details below).} \item{x}{The matrix or vector of data from which the function \eqn{g(\theta,x)} is computed. If "g" is a formula, it is an \eqn{n \times Nh} matrix of instruments or a formula (see details below).} \item{t0}{A \eqn{k \times 1} vector of starting values. It is required only when "g" is a function because only then a numerical algorithm is used to minimize the objective function. If the dimension of \eqn{\theta} is one, see the argument "optfct".} \item{tetw}{A \eqn{k \times 1} vector to compute the weighting matrix.} \item{gradv}{A function of the form \eqn{G(\theta,x)} which returns a \eqn{q\times k} matrix of derivatives of \eqn{\bar{g}(\theta)} with respect to \eqn{\theta}. By default, the numerical algorithm \code{numericDeriv} is used. It is of course strongly suggested to provide this function when it is possible. This gradient is used to compute the asymptotic covariance matrix of \eqn{\hat{\theta}} and to obtain the analytical gradient of the objective function if the method is set to "CG" or "BFGS" in \code{\link{optim}} and if "type" is not set to "cue". If "g" is a formula, the gradiant is not required (see the details below).} \item{type}{The GMM method: "twostep" is the two step GMM proposed by Hansen(1982) and the "cue" and "iterative" are respectively the continuous updated and the iterative GMM proposed by Hansen, Eaton et Yaron (1996)} \item{wmatrix}{Which weighting matrix should be used in the objective function. By default, it is the inverse of the covariance matrix of \eqn{g(\theta,x)}. The other choice is the identity matrix which is usually used to obtain a first step estimate of \eqn{\theta} } \item{vcov}{Assumption on the properties of the random vector x. By default, x is a weakly dependant process. The "iid" option will avoid using the HAC matrix which will accelerate the estimation if one is ready to make that assumption. The option "TrueFixed" is used only when the matrix of weights is provided and it is the optimal one.} \item{kernel}{type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see \code{\link{kernHAC}} for more details)} \item{crit}{The stopping rule for the iterative GMM. It can be reduce to increase the precision.} \item{bw}{The method to compute the bandwidth parameter in the HAC weighting matrix. The default is \code{link{bwAndrews}} (as proposed in Andrews (1991)), which minimizes the MSE of the weighting matrix. Alternatives are \code{link{bwWilhelm}} (as proposed in Wilhelm (2015)), which minimizes the mean-square error (MSE) of the resulting GMM estimator, and \code{link{bwNeweyWest}} (as proposed in Newey-West(1994)).} \item{prewhite}{logical or integer. Should the estimating functions be prewhitened? If \code{TRUE} or greater than 0 a VAR model of order \code{as.integer(prewhite)} is fitted via \code{ar} with method \code{"ols"} and \code{demean = FALSE}.} \item{ar.method}{character. The \code{method} argument passed to \code{\link{ar}} for prewhitening.} \item{approx}{A character specifying the approximation method if the bandwidth has to be chosen by \code{bwAndrews}.} \item{tol}{Weights that exceed \code{tol} are used for computing the covariance matrix, all other weights are treated as 0.} \item{itermax}{The maximum number of iterations for the iterative GMM. It is unlikely that the algorithm does not converge but we keep it as a safety.} \item{optfct}{Only when the dimension of \eqn{\theta} is 1, you can choose between the algorithm \code{\link{optim}} or \code{\link{optimize}}. In that case, the former is unreliable. If \code{\link{optimize}} is chosen, "t0" must be \eqn{1\times 2} which represents the interval in which the algorithm seeks the solution. It is also possible to choose the \code{\link{nlminb}} algorithm. In that case, boundaries for the coefficients can be set by the options \code{upper=} and \code{lower=}. The \code{\link{constrOptim}} is only available for nonlinear models for now. The standard errors may have to be corrected if the estimtes reach the boundary set by ui and ci.} \item{model, X, Y}{logical. If \code{TRUE} the corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula.} \item{TypeGmm}{The name of the class object created by the method \code{getModel}. It allows developers to extend the package and create other GMM methods.} \item{centeredVcov}{Should the moment function be centered when computing its covariance matrix. Doing so may improve inference.} \item{weightsMatrix}{It allows users to provide \code{gmm} with a fixed weighting matrix. This matrix must be \eqn{q \times q}, symmetric and strictly positive definite. When provided, the \code{type} option becomes irrelevant. } \item{traceIter}{Tracing information for GMM of type "iter"} \item{data}{A data.frame or a matrix with column names (Optional). } \item{eqConst}{Either a named vector (if "g" is a function), a simple vector for the nonlinear case indicating which of the \eqn{\theta_0} is restricted, or a qx2 vector defining equality constraints of the form \eqn{\theta_i=c_i}. See below for an example.} \item{which, value}{The equality constraint is of the form which=value. "which" can be a vector of type characters with the names of the coefficients being constrained, or a vector of type numeric with the position of the coefficient in the whole vector.} \item{obj}{Object of class "gmm"} \item{eqConstFullVcov}{If FALSE, the constrained coefficients are assumed to be fixed and only the covariance of the unconstrained coefficients is computed. If TRUE, the covariance matrix of the full set of coefficients is computed.} \item{mustar}{If not null, it must be a vector with the number of elements being equal to the number of moment conditions. In that case, the vector is subtracted from the sample moment vector before minimizing the objective function. It is useful to do a bootstrap procedure. } \item{onlyCoefficients}{If set to \code{TRUE}, the function only returns the coefficient estimates. It may be of interest when the standard errors are not needed} \item{...}{More options to give to \code{\link{optim}}.} } \details{ If we want to estimate a model like \eqn{Y_t = \theta_1 + X_{2t} \theta_2 + \cdots + X_{k}\theta_k + \epsilon_t} using the moment conditions \eqn{Cov(\epsilon_tH_t)=0}, where \eqn{H_t} is a vector of \eqn{Nh} instruments, than we can define "g" like we do for \code{\link{lm}}. We would have \eqn{g = y ~\tilde{}~ x2+x3+ \cdots +xk} and the argument "x" above would become the matrix H of instruments. As for \code{\link{lm}}, \eqn{Y_t} can be a \eqn{Ny \times 1} vector which would imply that \eqn{k=Nh \times Ny}. The intercept is included by default so you do not have to add a column of ones to the matrix \eqn{H}. You do not need to provide the gradiant in that case since in that case it is embedded in \code{\link{gmm}}. The intercept can be removed by adding -1 to the formula. In that case, the column of ones need to be added manually to H. It is also possible to express "x" as a formula. For example, if the instruments are \eqn{\{1,z_1,z_2,z_3\}}, we can set "x" to \eqn{\tilde{} z1+z2+z3}. By default, a column of ones is added. To remove it, set "x" to \eqn{\tilde{}z1+z2+z3-1}. The following explains the last example bellow. Thanks to Dieter Rozenich, a student from the Vienna University of Economics and Business Administration. He suggested that it would help to understand the implementation of the Jacobian. For the two parameters of a normal distribution \eqn{(\mu,\sigma)} we have the following three moment conditions: \deqn{ m_{1} = \mu - x_{i} } \deqn{ m_{2} = \sigma^2 - (x_{i}-\mu)^2 } \deqn{ m_{3} = x_{i}^{3} - \mu (\mu^2+3\sigma^{2}) } \eqn{m_{1},m_{2}} can be directly obtained by the definition of \eqn{(\mu,\sigma)}. The third moment condition comes from the third derivative of the moment generating function (MGF) \deqn{ M_{X}(t) = exp\Big(\mu t + \frac{\sigma^{2}t^{2}}{2}\Big) } evaluated at \eqn{(t=0)}. Note that we have more equations (3) than unknown parameters (2). The Jacobian of these two conditions is (it should be an array but I can't make it work): \deqn{ 1~~~~~~~~~~ 0 } \deqn{ -2\mu+2x ~~~~~ 2\sigma } \deqn{-3\mu^{2}-3\sigma^{2} ~~~~ -6\mu\sigma} \code{gmmWithConst()} re-estimates an unrestricted model by adding an equality constraint. \code{evalGmm()} creates an object of class '"gmm"' for a given parameter vector. If no vector "tetw" is provided and the weighting matrix needs to be computed, "t0" is used., } \value{ 'gmm' returns an object of 'class' '"gmm"' The functions 'summary' is used to obtain and print a summary of the results. It also compute the J-test of overidentying restriction The object of class "gmm" is a list containing at least: \item{coefficients}{\eqn{k\times 1} vector of coefficients} \item{residuals}{the residuals, that is response minus fitted values if "g" is a formula.} \item{fitted.values}{the fitted mean values if "g" is a formula.} \item{vcov}{the covariance matrix of the coefficients} \item{objective}{the value of the objective function \eqn{\| var(\bar{g})^{-1/2}\bar{g}\|^2}} \item{terms}{the \code{\link{terms}} object used when g is a formula.} \item{call}{the matched call.} \item{y}{if requested, the response used (if "g" is a formula).} \item{x}{if requested, the model matrix used if "g" is a formula or the data if "g" is a function.} \item{model}{if requested (the default), the model frame used if "g" is a formula.} \item{algoInfo}{Information produced by either \code{\link{optim}} or \code{\link{nlminb}} related to the convergence if "g" is a function. It is printed by the \code{summary.gmm} method.} } \references{ Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. Pierre Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likelihood with R. \emph{Journal of Statistical Software}, \bold{34}(11), 1--35. URL \url{https://www.jstatsoft.org/v34/i11/}. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. \emph{Econometrica}, \bold{59}, 817--858. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. \emph{Econometrica}, \bold{55}, 703--708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation. \emph{Review of Economic Studies}, \bold{61}, 631-653. Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finite-Sample Properties of Some Alternative GMM Estimators. \emph{Journal of Business and Economic Statistics}, \bold{14} 262-280. } \examples{ ## CAPM test with GMM data(Finance) r <- Finance[1:300, 1:10] rm <- Finance[1:300, "rm"] rf <- Finance[1:300, "rf"] z <- as.matrix(r-rf) t <- nrow(z) zm <- rm-rf h <- matrix(zm, t, 1) res <- gmm(z ~ zm, x = h) summary(res) ## linear tests can be performed using linearHypothesis from the car package ## The CAPM can be tested as follows: library(car) linearHypothesis(res,cbind(diag(10),matrix(0,10,10)),rep(0,10)) # The CAPM of Black g <- function(theta, x) { e <- x[,2:11] - theta[1] - (x[,1] - theta[1]) \%*\% matrix(theta[2:11], 1, 10) gmat <- cbind(e, e*c(x[,1])) return(gmat) } x <- as.matrix(cbind(rm, r)) res_black <- gmm(g, x = x, t0 = rep(0, 11)) summary(res_black)$coefficients ## APT test with Fama-French factors and GMM f1 <- zm f2 <- Finance[1:300, "hml"] f3 <- Finance[1:300, "smb"] h <- cbind(f1, f2, f3) res2 <- gmm(z ~ f1 + f2 + f3, x = h) coef(res2) summary(res2)$coefficients ## Same result with x defined as a formula: res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3) coef(res2) ## The following example has been provided by Dieter Rozenich (see details). # It generates normal random numbers and uses the GMM to estimate # mean and sd. #------------------------------------------------------------------------------- # Random numbers of a normal distribution # First we generate normally distributed random numbers and compute the two parameters: n <- 1000 x <- rnorm(n, mean = 4, sd = 2) # Implementing the 3 moment conditions g <- function(tet, x) { m1 <- (tet[1] - x) m2 <- (tet[2]^2 - (x - tet[1])^2) m3 <- x^3 - tet[1]*(tet[1]^2 + 3*tet[2]^2) f <- cbind(m1, m2, m3) return(f) } # Implementing the jacobian Dg <- function(tet, x) { jacobian <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2], -6*tet[1]*tet[2]), nrow=3,ncol=2) return(jacobian) } # Now we want to estimate the two parameters using the GMM. gmm(g, x, c(0, 0), grad = Dg) # Two-stage-least-squares (2SLS), or IV with iid errors. # The model is: # Y(t) = b[0] + b[1]C(t) + b[2]Y(t-1) + e(t) # e(t) is an MA(1) # The instruments are Z(t)={1 C(t) y(t-2) y(t-3) y(t-4)} getdat <- function(n) { e <- arima.sim(n,model=list(ma=.9)) C <- runif(n,0,5) Y <- rep(0,n) Y[1] = 1 + 2*C[1] + e[1] for (i in 2:n){ Y[i] = 1 + 2*C[i] + 0.9*Y[i-1] + e[i] } Yt <- Y[5:n] X <- cbind(1,C[5:n],Y[4:(n-1)]) Z <- cbind(1,C[5:n],Y[3:(n-2)],Y[2:(n-3)],Y[1:(n-4)]) return(list(Y=Yt,X=X,Z=Z)) } d <- getdat(5000) res4 <- gmm(d$Y~d$X-1,~d$Z-1,vcov="iid") res4 ### Examples with equality constraint ###################################### # Random numbers of a normal distribution ## Not run: # The following works but produces warning message because the dimension of coef is 1 # Brent should be used # without named vector # Method Brent is used because the problem is now one-dimensional gmm(g, x, c(4, 0), grad = Dg, eqConst=1, method="Brent", lower=-10,upper=10) # with named vector gmm(g, x, c(mu=4, sig=2), grad = Dg, eqConst="sig", method="Brent", lower=-10,upper=10) ## End(Not run) gmm(g, x, c(4, 0), grad = Dg, eqConst=1,method="Brent",lower=0,upper=6) gmm(g, x, c(mu=4, sig=2), grad = Dg, eqConst="sig",method="Brent",lower=0,upper=6) # Example with formula # first coef = 0 and second coef = 1 # Only available for one dimensional yt z <- z[,1] res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, eqConst = matrix(c(1,2,0,1),2,2)) res2 # CUE with starting t0 requires eqConst to be a vector res3 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, t0=c(0,1,.5,.5), type="cue", eqConst = c(1,2)) res3 ### Examples with equality constraints, where the constrained coefficients is used to compute ### the covariance matrix. ### Useful when some coefficients have been estimated before, they are just identified in GMM ### and don't need to be re-estimated. ### To use with caution because the covariance won't be valid if the coefficients do not solve ### the GMM FOC. ###################################### res4 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, t0=c(0,1,.5,.5), eqConst = c(1,2), eqConstFullVcov=TRUE) summary(res4) ### Examples with equality constraint using gmmWithConst ########################################################### res2 <- gmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3) gmmWithConst(res2,c("f2","f3"),c(.5,.5)) gmmWithConst(res2,c(2,3),c(.5,.5)) ## Creating an object without estimation for a fixed parameter vector ################################################################### res2_2 <- evalGmm(z ~ f1 + f2 + f3, ~ f1 + f2 + f3, t0=res2$coefficients, tetw=res2$coefficients) summary(res2_2) } gmm/man/getLamb.Rd0000644000176200001440000000751713464357722013476 0ustar liggesusers\name{getLamb} \alias{getLamb} \title{Solving for the Lagrange multipliers of Generalized Empirical Likelihood (GEL) } \description{ It computes the vector of Lagrange multipliers, which maximizes the GEL objective function, using an iterative Newton method. } \usage{ getLamb(gt, l0, type = c("EL","ET","CUE", "ETEL", "HD","ETHD","RCUE"), tol_lam = 1e-7, maxiterlam = 100, tol_obj = 1e-7, k = 1, method = c("nlminb", "optim", "iter", "Wu"), control = list()) } \arguments{ \item{gt}{A \eqn{n \times q} matrix with typical element \eqn{g_i(\theta,x_t)}} \item{l0}{Vector of starting values for lambda} \item{type}{"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continuous updated estimator, and "HD" for Hellinger Distance. See details for "ETEL" and "ETHD". "RCUE" is a restricted version of "CUE" in which the probabilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution.} \item{tol_lam}{Tolerance for \eqn{\lambda} between two iterations. The algorithm stops when \eqn{\|\lambda_i -\lambda_{i-1}\|} reaches \code{tol_lam} } \item{maxiterlam}{The algorithm stops if there is no convergence after "maxiterlam" iterations.} \item{tol_obj}{Tolerance for the gradiant of the objective function. The algorithm returns a non-convergence message if \eqn{\max(|gradiant|)} does not reach \code{tol_obj}. It helps the \code{gel} algorithm to select the right space to look for \eqn{\theta}} \item{k}{It represents the ratio k1/k2, where \eqn{k1=\int_{-\infty}^{\infty} k(s)ds} and \eqn{k2=\int_{-\infty}^{\infty} k(s)^2 ds}. See Smith(2004).} \item{method}{The iterative procedure uses a Newton method for solving the FOC. It i however recommended to use \code{optim} or \code{nlminb}. If type is set to "EL" and method to "optim", \code{\link{constrOptim}} is called to prevent \eqn{log(1-gt'\lambda)} from producing NA. The gradient and hessian is provided to \code{nlminb} which speed up the convergence. The latter is therefore the default value. "Wu" is for "EL" only. It uses the algorithm of Wu (2005). The value of \code{method} is ignored for "CUE" because in that case, the analytical solution exists.} \item{control}{Controls to send to \code{\link{optim}}, \code{\link{nlminb}} or \code{\link{constrOptim}}} } \details{ It solves the problem \eqn{\max_{\lambda} \frac{1}{n}\sum_{t=1}^n \rho(gt'\lambda)}. For the type "ETEL", it is only used by \code{\link{gel}}. In that case \eqn{\lambda} is obtained by maximizing \eqn{\frac{1}{n}\sum_{t=1}^n \rho(gt'\lambda)}, using \eqn{\rho(v)=-\exp{v}} (so ET) and \eqn{\theta} by minimizing the same equation but with \eqn{\rho(v)-\log{(1-v)}}. To avoid NA's, \code{\link{constrOptim}} is used with the restriction \eqn{\lambda'g_t < 1}. The type "ETHD" is experimental and proposed by Antoine-Dovonon (2015). The paper is not yet available. } \value{ lambda: A \eqn{q\times 1} vector of Lagrange multipliers which solve the system of equations given above. \code{conv}: Details on the type of convergence. } \references{ Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. \emph{Econometrica}, \bold{72}, 219-255. Smith, R.J. (2004), GEL Criteria for Moment Condition Models. \emph{Working paper, CEMMAP}. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling. \emph{Survey Methodology}, \bold{31}(2), page 239. } \examples{ g <- function(tet,x) { n <- nrow(x) u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)]) f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)]) return(f) } n = 500 phi<-c(.2, .7) thet <- 0.2 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1) gt <- g(c(0,phi),x) getLamb(gt, type = "EL",method="optim") } gmm/man/specTest.Rd0000644000176200001440000000244511305655400013672 0ustar liggesusers\name{specTest} \alias{specTest} \alias{specTest.gel} \alias{specTest.gmm} \alias{print.specTest} \title{Compute tests of specification} \description{ Generic function for testing the specification of estimated models. It computes the J-test from \code{gmm} objects and J-test, LR-test and LM-test from \code{gel} objects. } \usage{ \method{specTest}{gmm}(x, ...) \method{specTest}{gel}(x, ...) \method{print}{specTest}(x, digits = 5, ...) specTest(x, \dots) } \arguments{ \item{x}{A fitted model object.} \item{digits}{The number of digits to be printed.} \item{\dots}{Arguments passed to methods.} } \value{ Tests and p-values } \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Smith, R. J. (2004), GEL Criteria for Moment Condition Models. \emph{CeMMAP working papers, Institute for Fiscal Studies} } \examples{ ################# n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H t0 <- c(0,.5,.5) res <- gel(g, x, t0) specTest(res) ################### res <- gmm(g, x) specTest(res) } gmm/man/sysGmm.Rd0000644000176200001440000001333214017527033013357 0ustar liggesusers\name{sysGmm} \alias{sysGmm} \alias{five} \alias{sur} \alias{randEffect} \alias{threeSLS} \title{Generalized method of moment estimation for system of equations} \description{ Functions to estimate a system of equations based on GMM. } \usage{ sysGmm(g, h, wmatrix = c("optimal","ident"), vcov=c("MDS", "HAC", "CondHom", "TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"), crit=10e-7,bw = bwAndrews, prewhite = FALSE, ar.method = "ols", approx="AR(1)", tol = 1e-7, model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE, weightsMatrix = NULL, data, crossEquConst = NULL, commonCoef = FALSE) five(g, h, commonCoef = FALSE, data = NULL) threeSLS(g, h, commonCoef = FALSE, data = NULL) sur(g, commonCoef = FALSE, data = NULL) randEffect(g, data = NULL) } \arguments{ \item{g}{A possibly named list of formulas} \item{h}{A formula if the same instruments are used in each equation or a list of formulas.} \item{wmatrix}{Which weighting matrix should be used in the objective function. By default, it is the inverse of the covariance matrix of \eqn{g(\theta,x)}. The other choice is the identity matrix.} \item{vcov}{Assumption on the properties of the moment vector. By default, it is a martingale difference sequence. "HAC" is for weakly dependent processes and "CondHom" implies conditional homoscedasticity. The option "TrueFixed" is used only when the matrix of weights is provided and it is the optimal one.} \item{kernel}{type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see \code{\link{kernHAC}} for more details)} \item{crit}{The stopping rule for the iterative GMM. It can be reduce to increase the precision.} \item{bw}{The method to compute the bandwidth parameter. By default it is \code{\link{bwAndrews}} which is proposed by Andrews (1991). The alternative is \code{\link{bwNeweyWest}} of Newey-West(1994).} \item{prewhite}{logical or integer. Should the estimating functions be prewhitened? If \code{TRUE} or greater than 0 a VAR model of order \code{as.integer(prewhite)} is fitted via \code{ar} with method \code{"ols"} and \code{demean = FALSE}.} \item{ar.method}{character. The \code{method} argument passed to \code{\link{ar}} for prewhitening.} \item{approx}{A character specifying the approximation method if the bandwidth has to be chosen by \code{bwAndrews}.} \item{tol}{Weights that exceed \code{tol} are used for computing the covariance matrix, all other weights are treated as 0.} \item{model, X, Y}{logical. If \code{TRUE} the corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula.} \item{centeredVcov}{Should the moment function be centered when computing its covariance matrix. Doing so may improve inference.} \item{weightsMatrix}{It allows users to provide \code{gmm} with a fixed weighting matrix. This matrix must be \eqn{q \times q}, symmetric and strictly positive definite. When provided, the \code{type} option becomes irrelevant. } \item{data}{A data.frame or a matrix with column names (Optional). } \item{commonCoef}{If true, coefficients accross equations are the same} \item{crossEquConst}{Only used if the number of regressors are the same in each equation. It is a vector which indicates which coefficient are constant across equations. The order is 1 for Intercept and 2 to k as it is formulated in the formulas \code{g}. Setting it to \code{1:k} is equivalent to setting commonCoef to TRUE.} } \details{ This set of functions implement the estimation of system of equations as presented in Hayashi (2000) } \value{ 'sysGmm' returns an object of 'class' '"sysGmm"' The functions 'summary' is used to obtain and print a summary of the results. It also compute the J-test of overidentying restriction The object of class "sysGmm" is a list containing at least: \item{coefficients}{list of vectors of coefficients for each equation} \item{residuals}{list of the residuals for each equation.} \item{fitted.values}{list of the fitted values for each equation.} \item{vcov}{the covariance matrix of the stacked coefficients} \item{objective}{the value of the objective function \eqn{\| var(\bar{g})^{-1/2}\bar{g}\|^2}} \item{terms}{The list of \code{\link{terms}} objects for each equation} \item{call}{the matched call.} \item{y}{If requested, a list of response variables.} \item{x}{if requested, a list of the model matrices.} \item{model}{if requested (the default), a list of the model frames.} } \references{ Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. \emph{Econometrica}, \bold{59}, 817--858. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. \emph{Econometrica}, \bold{55}, 703--708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation. \emph{Review of Economic Studies}, \bold{61}, 631-653. Hayashi, F. (2000), Econometrics. \emph{Princeton University Press}. } \examples{ data(wage) eq1 <- LW~S+IQ+EXPR eq2 <- LW80~S80+IQ+EXPR80 g2 <- list(Wage69=eq1, WAGE80=eq2) h2 <- list(~S+EXPR+MED+KWW, ~S80+EXPR80+MED+KWW) res <- sysGmm(g2, h2, data=wage, commonCoef=TRUE) summary(res) res2 <- sysGmm(g2, h2, data=wage) summary(res2) five(g2, h2, data=wage) threeSLS(g2, h2[[1]], data=wage) sur(g2, data=wage) randEffect(g2, data=wage) ## Cross-Equation restrictions ## All but the intercept are assumed to be the same res <- sysGmm(g2, h2, data=wage, crossEquConst = 2:4) summary(res) } gmm/man/gel.Rd0000644000176200001440000003272314017527033012654 0ustar liggesusers\name{gel} \alias{gel} \alias{evalGel} \title{Generalized Empirical Likelihood estimation} \description{ Function to estimate a vector of parameters based on moment conditions using the GEL method as presented by Newey-Smith(2004) and Anatolyev(2005). } \usage{ gel(g, x, tet0 = NULL, gradv = NULL, smooth = FALSE, type = c("EL","ET","CUE","ETEL","HD","ETHD","RCUE"), kernel = c("Truncated", "Bartlett"), bw = bwAndrews, approx = c("AR(1)", "ARMA(1,1)"), prewhite = 1, ar.method = "ols", tol_weights = 1e-7, tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, constraint = FALSE, optfct = c("optim", "optimize", "nlminb"), optlam = c("nlminb", "optim", "iter", "Wu"), data, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, TypeGel = "baseGel", alpha = NULL, eqConst = NULL, eqConstFullVcov = FALSE, onlyCoefficients=FALSE, ...) evalGel(g, x, tet0, gradv = NULL, smooth = FALSE, type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD","RCUE"), kernel = c("Truncated", "Bartlett"), bw = bwAndrews, approx = c("AR(1)", "ARMA(1,1)"), prewhite = 1, ar.method = "ols", tol_weights = 1e-7, tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, optlam = c("nlminb", "optim", "iter", "Wu"), data, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, alpha = NULL, ...) } \arguments{ \item{g}{A function of the form \eqn{g(\theta,x)} and which returns a \eqn{n \times q} matrix with typical element \eqn{g_i(\theta,x_t)} for \eqn{i=1,...q} and \eqn{t=1,...,n}. This matrix is then used to build the q sample moment conditions. It can also be a formula if the model is linear (see details below). } \item{tet0}{A \eqn{k \times 1} vector of starting values. If the dimension of \eqn{\theta} is one, see the argument "optfct". In the linear case, if tet0=NULL, the 2-step gmm estimator is used as starting value. However, it has to be provided when eqConst is not NULL} \item{x}{The matrix or vector of data from which the function \eqn{g(\theta,x)} is computed. If "g" is a formula, it is an \eqn{n \times Nh} matrix of instruments (see details below).} \item{gradv}{A function of the form \eqn{G(\theta,x)} which returns a \eqn{q\times k} matrix of derivatives of \eqn{\bar{g}(\theta)} with respect to \eqn{\theta}. By default, the numerical algorithm \code{numericDeriv} is used. It is of course strongly suggested to provide this function when it is possible. This gradiant is used compute the asymptotic covariance matrix of \eqn{\hat{\theta}}. If "g" is a formula, the gradiant is not required (see the details below).} \item{smooth}{If set to TRUE, the moment function is smoothed as proposed by Kitamura(1997)} \item{type}{"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continuous updated estimator, "ETEL" for exponentially tilted empirical likelihood of Schennach(2007), "HD" for Hellinger Distance of Kitamura-Otsu-Evdokimov (2013), and "ETHD" for the exponentially tilted Hellinger distance of Antoine-Dovonon (2015). "RCUE" is a restricted version of "CUE" in which the probabilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution.} \item{kernel}{type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see \code{\link{kernHAC}} for more details) and to smooth the moment conditions if "smooth" is set to TRUE. Only two types of kernel are available. The truncated implies a Bartlett kernel for the HAC matrix and the Bartlett implies a Parzen kernel (see Smith 2004).} \item{bw}{The method to compute the bandwidth parameter. By default it is \code{\link{bwAndrews}} which is proposed by Andrews (1991). The alternative is \code{\link{bwNeweyWest}} of Newey-West(1994).} \item{prewhite}{logical or integer. Should the estimating functions be prewhitened? If \code{TRUE} or greater than 0 a VAR model of order \code{as.integer(prewhite)} is fitted via \code{ar} with method \code{"ols"} and \code{demean = FALSE}.} \item{ar.method}{character. The \code{method} argument passed to \code{\link{ar}} for prewhitening.} \item{approx}{a character specifying the approximation method if the bandwidth has to be chosen by \code{bwAndrews}.} \item{tol_weights}{numeric. Weights that exceed \code{tol} are used for computing the covariance matrix, all other weights are treated as 0.} \item{tol_lam}{Tolerance for \eqn{\lambda} between two iterations. The algorithm stops when \eqn{\|\lambda_i -\lambda_{i-1}\|} reaches \code{tol_lamb} (see \code{\link{getLamb}}) } \item{maxiterlam}{The algorithm to compute \eqn{\lambda} stops if there is no convergence after "maxiterlam" iterations (see \code{\link{getLamb}}).} \item{tol_obj}{Tolerance for the gradiant of the objective function to compute \eqn{\lambda} (see \code{\link{getLamb}}).} \item{optfct}{Only when the dimension of \eqn{\theta} is 1, you can choose between the algorithm \code{\link{optim}} or \code{\link{optimize}}. In that case, the former is unreliable. If \code{\link{optimize}} is chosen, "t0" must be \eqn{1\times 2} which represents the interval in which the algorithm seeks the solution.It is also possible to choose the \code{\link{nlminb}} algorithm. In that case, borns for the coefficients can be set by the options \code{upper=} and \code{lower=}.} \item{constraint}{If set to TRUE, the constraint optimization algorithm is used. See \code{\link{constrOptim}} to learn how it works. In particular, if you choose to use it, you need to provide "ui" and "ci" in order to impose the constraint \eqn{ui \theta - ci \geq 0}.} \item{tol_mom}{It is the tolerance for the moment condition \eqn{\sum_{t=1}^n p_t g(\theta(x_t)=0}, where \eqn{p_t=\frac{1}{n}D\rho()} is the implied probability. It adds a penalty if the solution diverges from its goal.} \item{optlam}{Algorithm used to solve for the lagrange multiplier in \code{\link{getLamb}}. The algorithm Wu is only for \code{type="EL"}. The value of \code{optlam} is ignored for "CUE" because in that case, the analytical solution exists.} \item{data}{A data.frame or a matrix with column names (Optional). } \item{Lambdacontrol}{Controls for the optimization of the vector of Lagrange multipliers used by either \code{\link{optim}}, \code{\link{nlminb}} or \code{\link{constrOptim}}} \item{model, X, Y}{logicals. If \code{TRUE} the corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula.} \item{TypeGel}{The name of the class object created by the method \code{getModel}. It allows developers to extand the package and create other GEL methods.} \item{alpha}{Regularization coefficient for discrete CGEL estimation (experimental). By setting alpha to any value, the model is estimated by CGEL of type specified by the option \code{type}. See Chausse (2011)} \item{eqConst}{Either a named vector (if "g" is a function), a simple vector for the nonlinear case indicating which of the \eqn{\theta_0} is restricted, or a qx2 vector defining equality constraints of the form \eqn{\theta_i=c_i}. See \code{\link{gmm}} for an example.} \item{eqConstFullVcov}{If FALSE, the constrained coefficients are assumed to be fixed and only the covariance of the unconstrained coefficients is computed. If TRUE, the covariance matrix of the full set of coefficients is computed.} \item{onlyCoefficients}{If \code{TRUE}, only the vector of coefficients and Lagrange multipliers are returned} \item{...}{More options to give to \code{\link{optim}}, \code{\link{optimize}} or \code{\link{constrOptim}}.} } \details{ If we want to estimate a model like \eqn{Y_t = \theta_1 + X_{2t}\theta_2 + ... + X_{k}\theta_k + \epsilon_t} using the moment conditions \eqn{Cov(\epsilon_tH_t)=0}, where \eqn{H_t} is a vector of \eqn{Nh} instruments, than we can define "g" like we do for \code{\link{lm}}. We would have \code{g = y~x2+x3+...+xk} and the argument "x" above would become the matrix H of instruments. As for \code{\link{lm}}, \eqn{Y_t} can be a \eqn{Ny \times 1} vector which would imply that \eqn{k=Nh \times Ny}. The intercept is included by default so you do not have to add a column of ones to the matrix \eqn{H}. You do not need to provide the gradiant in that case since in that case it is embedded in \code{\link{gel}}. The intercept can be removed by adding -1 to the formula. In that case, the column of ones need to be added manually to H. If "smooth" is set to TRUE, the sample moment conditions \eqn{\sum_{t=1}^n g(\theta,x_t)} is replaced by: \eqn{\sum_{t=1}^n g^k(\theta,x_t)}, where \eqn{g^k(\theta,x_t)=\sum_{i=-r}^r k(i) g(\theta,x_{t+i})}, where \eqn{r} is a truncated parameter that depends on the bandwidth and \eqn{k(i)} are normalized weights so that they sum to 1. The method solves \eqn{\hat{\theta} = \arg\min \left[\arg\max_\lambda \frac{1}{n}\sum_{t=1}^n \rho() - \rho(0) \right]} \code{\link{evalGel}} generates the object of class "gel" for a fixed vector of parameters. There is no estimation for \eqn{\theta}, but the optimal vector of Lagrange multipliers \eqn{\lambda} is computed. The objective function is then the profiled likelihood for a given \eqn{\theta}. It can be used to construct a confidence interval by inverting the likelihood ratio test. } \value{ 'gel' returns an object of 'class' '"gel"' The functions 'summary' is used to obtain and print a summary of the results. The object of class "gel" is a list containing at least the following: \item{coefficients}{\eqn{k\times 1} vector of parameters} \item{residuals}{the residuals, that is response minus fitted values if "g" is a formula.} \item{fitted.values}{the fitted mean values if "g" is a formula.} \item{lambda}{\eqn{q \times 1} vector of Lagrange multipliers.} \item{vcov_par}{the covariance matrix of "coefficients"} \item{vcov_lambda}{the covariance matrix of "lambda"} \item{pt}{The implied probabilities} \item{objective}{the value of the objective function} \item{conv_lambda}{Convergence code for "lambda" (see \code{\link{getLamb}})} \item{conv_mes}{Convergence message for "lambda" (see \code{\link{getLamb}})} \item{conv_par}{Convergence code for "coefficients" (see \code{\link{optim}}, \code{\link{optimize}} or \code{\link{constrOptim}})} \item{terms}{the \code{\link{terms}} object used when g is a formula.} \item{call}{the matched call.} \item{y}{if requested, the response used (if "g" is a formula).} \item{x}{if requested, the model matrix used if "g" is a formula or the data if "g" is a function.} \item{model}{if requested (the default), the model frame used if "g" is a formula.} } \references{ Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias. \emph{Econometrica}, \bold{73}, 983-1002. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. \emph{Econometrica}, \bold{59}, 817--858. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes. \emph{The Annals of Statistics}, \bold{25}, 2084-2102. Kitamura, Y. and Otsu, T. and Evdokimov, K. (2013), Robustness, Infinitesimal Neighborhoods and Moment Restrictions. \emph{Econometrica}, \bold{81}, 1185-1201. Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. \emph{Econometrica}, \bold{72}, 219-255. Smith, R.J. (2004), GEL Criteria for Moment Condition Models. \emph{Working paper, CEMMAP}. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. \emph{Econometrica}, \bold{55}, 703--708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation. \emph{Review of Economic Studies}, \bold{61}, 631-653. Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood. \emph{Econometrica}, \bold{35}, 634-672. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling. \emph{Survey Methodology}, \bold{31}(2), page 239. Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likelihood with R. \emph{Journal of Statistical Software}, \bold{34}(11), 1--35. URL \url{https://www.jstatsoft.org/v34/i11/}. Chausse (2011), Generalized Empirical likelihood for a continumm of moment conditions. \emph{Working Paper}, \emph{Department of Economics}, \emph{University of Waterloo}. } \examples{ # First, an exemple with the fonction g() g <- function(tet, x) { n <- nrow(x) u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)]) f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)]) return(f) } Dg <- function(tet,x) { n <- nrow(x) xx <- cbind(rep(1, (n-6)), x[6:(n-1)], x[5:(n-2)]) H <- cbind(rep(1, (n-6)), x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) f <- -crossprod(H, xx)/(n-6) return(f) } n = 200 phi<-c(.2, .7) thet <- 0.2 sd <- .2 set.seed(123) x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1) res <- gel(g, x, c(0, .3, .6), grad = Dg) summary(res) # The same model but with g as a formula.... much simpler in that case y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gel(g, x, c(0, .3, .6)) summary(res) # Using evalGel to create the object without estimation res <- evalGel(g, x, res$coefficients) } gmm/man/KTest.Rd0000644000176200001440000000345011743071151013130 0ustar liggesusers\name{KTest} \alias{KTest} \alias{print.gmmTests} \title{Compute the K statistics of Kleibergen} \description{The test is proposed by Kleibergen (2005). It is robust to weak identification. } \usage{ KTest(obj, theta0 = NULL, alphaK = 0.04, alphaJ = 0.01) \method{print}{gmmTests}(x, digits = 5, ...) } \arguments{ \item{obj}{Object of class "gmm" returned by \code{\link{gmm}}} \item{theta0}{The null hypothesis being tested. See details.} \item{alphaK, alphaJ}{The size of the J and K tests when combining the two. The overall size is alphaK+alphaJ.} \item{x}{An object of class \code{gmmTests} returned by \code{KTest}} \item{digits}{The number of digits to be printed} \item{...}{Other arguments when \code{print} is applied to another class object} } \details{ The function produces the J-test and K-statistics which are robust to weak identification. The test is either \eqn{H0:\theta=theta_0}, in which case theta0 must be provided, or \eqn{\beta=\beta_0}, where \eqn{\theta=(\alpha', \beta')'}, and \eqn{\alpha} is assumed to be identified. In the latter case, theta0 is NULL and obj is a restricted estimation in which \eqn{\beta} is fixed to \eqn{\beta_0}. See \code{\link{gmm}} and the option "eqConst" for more details. } \value{ Tests and p-values } \references{ Keibergen, F. (2005), Testing Parameters in GMM without assuming that they are identified. \emph{Econometrica}, \bold{73}, 1103-1123, } \examples{ library(mvtnorm) sig <- matrix(c(1,.5,.5,1),2,2) n <- 400 e <- rmvnorm(n,sigma=sig) x4 <- rnorm(n) w <- exp(-x4^2) + e[,1] y <- 0.1*w + e[,2] h <- cbind(x4, x4^2, x4^3, x4^6) g3 <- y~w res <- gmm(g3,h) # Testing the whole vector: KTest(res,theta0=c(0,.1)) # Testing a subset of the vector (See \code{\link{gmm}}) res2 <- gmm(g3, h, eqConst=matrix(c(2,.1),1,2)) res2 KTest(res2) } gmm/man/marginal.Rd0000644000176200001440000000277013107655275013707 0ustar liggesusers\name{marginal} \alias{marginal} \alias{marginal.ategel} \title{Marginal effects Summary} \description{ It produces the summary table of marginal effects for GLM estimation with GEL. Only implemented for ATEgel. } \usage{ \method{marginal}{ategel}(object, ...) } \arguments{ \item{object}{An object of class \code{ategel} returned by the function \code{\link{ATEgel}}} \item{...}{Other arguments for other methods} } \value{ It returns a matrix with the marginal effects, the standard errors based on the Delta method when the link is nonlinear, the t-ratios, and the pvalues.} \references{ Owen, A.B. (2001), Empirical Likelihood. \emph{Monographs on Statistics and Applied Probability 92, Chapman and Hall/CRC} } \examples{ ## We create some artificial data with unbalanced groups and binary outcome genDat <- function(n) { eta=c(-1, .5, -.25, -.1) Z <- matrix(rnorm(n*4),ncol=4) b <- c(27.4, 13.7, 13.7, 13.7) bZ <- c(Z\%*\%b) Y1 <- as.numeric(rnorm(n, mean=210+bZ)>220) Y0 <- as.numeric(rnorm(n, mean=200-.5*bZ)>220) etaZ <- c(Z\%*\%eta) pZ <- exp(etaZ)/(1+exp(etaZ)) T <- rbinom(n, 1, pZ) Y <- T*Y1+(1-T)*Y0 X1 <- exp(Z[,1]/2) X2 <- Z[,2]/(1+exp(Z[,1])) X3 <- (Z[,1]*Z[,3]/25+0.6)^3 X4 <- (Z[,2]+Z[,4]+20)^2 data.frame(Y=Y, cbind(X1,X2,X3,X4), T=T) } dat <- genDat(200) res <- ATEgel(Y~T, ~X1+X2+X3+X4, data=dat, type="ET", family="logit") summary(res) marginal(res) } gmm/man/summary.Rd0000644000176200001440000000565413107655275013616 0ustar liggesusers\name{summary} \alias{summary.gmm} \alias{summary.sysGmm} \alias{summary.gel} \alias{summary.ategel} \alias{summary.tsls} \alias{print.summary.gmm} \alias{print.summary.sysGmm} \alias{print.summary.gel} \alias{print.summary.tsls} \title{Method for object of class gmm or gel} \description{ It presents the results from the \code{gmm} or \code{gel} estimation in the same fashion as \code{summary} does for the \code{lm} class objects for example. It also compute the tests for overidentifying restrictions. } \usage{ \method{summary}{gmm}(object, ...) \method{summary}{sysGmm}(object, ...) \method{summary}{gel}(object, ...) \method{summary}{ategel}(object, robToMiss = TRUE, ...) \method{summary}{tsls}(object, vcov = NULL, ...) \method{print}{summary.gmm}(x, digits = 5, ...) \method{print}{summary.sysGmm}(x, digits = 5, ...) \method{print}{summary.gel}(x, digits = 5, ...) \method{print}{summary.tsls}(x, digits = 5, ...) } \arguments{ \item{object}{An object of class \code{gmm} or \code{gel} returned by the function \code{\link{gmm}} or \code{\link{gel}}} \item{x}{An object of class \code{summary.gmm} or \code{summary.gel} returned by the function \code{\link{summary.gmm}} \code{\link{summary.gel}}} \item{digits}{The number of digits to be printed} \item{vcov}{An alternative covariance matrix computed with \code{vcov.tsls}} \item{robToMiss}{If \code{TRUE}, it computes the robust to misspecification covariance matrix} \item{...}{Other arguments when summary is applied to another class object} } \value{ It returns a list with the parameter estimates and their standard deviations, t-stat and p-values. It also returns the J-test and p-value for the null hypothesis that \eqn{E(g(\theta,X)=0} } \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. \emph{Journal of Business and Economic Statistics}, \bold{14} 262-280. Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias. \emph{Econometrica}, \bold{73}, 983-1002. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes. \emph{The Annals of Statistics}, \bold{25}, 2084-2102. Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. \emph{Econometrica}, \bold{72}, 219-255. } \examples{ # GMM # set.seed(444) n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] ym3 <- x[4:(n-3)] ym4 <- x[3:(n-4)] ym5 <- x[2:(n-5)] ym6 <- x[1:(n-6)] g <- y ~ ym1 + ym2 x <- ~ym3+ym4+ym5+ym6 res <- gmm(g, x) summary(res) # GEL # t0 <- res$coef res <- gel(g, x, t0) summary(res) # tsls # res <- tsls(y ~ ym1 + ym2,~ym3+ym4+ym5+ym6) summary(res) } gmm/man/residuals.Rd0000644000176200001440000000261711305632271014075 0ustar liggesusers\name{residuals} \alias{residuals.gel} \alias{residuals.gmm} \title{Residuals of GEL or GMM} \description{ Method to extract the residuals of the model estimated by \code{gmm} or \code{gel}. } \usage{ \method{residuals}{gel}(object, ...) \method{residuals}{gmm}(object, ...) } \arguments{ \item{object}{An object of class \code{gmm} or \code{gel} returned by the function \code{\link{gmm}} or \code{\link{gel}}} \item{...}{Other arguments when \code{residuals} is applied to an other classe object} } \value{ It returns the matrix of residuals \eqn{(y-\hat{y})} in \code{g=y~x} as it is done by \code{residuals.lm}. } \examples{ # GEL can deal with endogeneity problems n = 200 phi<-c(.2,.7) thet <- 0.2 sd <- .2 set.seed(123) x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gel(g, x, c(0,.3,.6)) e <- residuals(res) plot(e, type = 'l', main = "Residuals from an ARMA fit using GEL") # GMM is like GLS for linear models without endogeneity problems set.seed(345) n = 200 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- 10 + 5*rnorm(n) + x res <- gmm(y ~ x, x) plot(x, residuals(res), main = "Residuals of an estimated model with GMM") } gmm/man/charStable.Rd0000644000176200001440000000324014010606726014145 0ustar liggesusers\name{charStable} \alias{charStable} \title{The characteristic function of a stable distribution} \description{ It computes the theoretical characteristic function of a stable distribution for two different parametrizations. It is used in the vignette to illustrate the estimation of the parameters using GMM.} \usage{ charStable(theta, tau, pm = 0) } \arguments{ \item{theta}{Vector of parameters of the stable distribution. See details.} \item{tau}{A vector of numbers at which the function is evaluated.} \item{pm}{The type of parametization. It takes the values 0 or 1.} } \value{ It returns a vector of complex numbers with the dimension equals to \code{length(tau)}. } \details{ The function returns the vector \eqn{\Psi(\theta,\tau,pm)} defined as \eqn{E(e^{ix\tau}}, where \eqn{\tau} is a vector of real numbers, \eqn{i} is the imaginary number, \eqn{x} is a stable random variable with parameters \eqn{\theta} = \eqn{(\alpha,\beta,\gamma,\delta)} and \code{pm} is the type of parametrization. The vector of parameters are the characteristic exponent, the skewness, the scale and the location parameters, respectively. The restrictions on the parameters are: \eqn{\alpha \in (0,2]}, \eqn{\beta\in [-1,1]} and \eqn{\gamma>0}. For mode details see Nolan(2009). } \references{ Nolan J. P. (2020), Univariate Stable Distributions - Models for Heavy Tailed Data. \emph{Springer Series in Operations Research and Financial Engineering}. URL \url{https://edspace.american.edu/jpnolan/stable/}. } \examples{ # GMM is like GLS for linear models without endogeneity problems pm <- 0 theta <- c(1.5,.5,1,0) tau <- seq(-3, 3, length.out = 20) char_fct <- charStable(theta, tau, pm) } gmm/man/formula.Rd0000644000176200001440000000206211305632271013541 0ustar liggesusers\name{formula} \alias{formula.gel} \alias{formula.gmm} \title{Formula method for gel and gmm objects} \description{ Method to extract the formula from \code{gel} or \code{gmm} objects. } \usage{ \method{formula}{gel}(x, ...) \method{formula}{gmm}(x, ...) } \arguments{ \item{x}{An object of class \code{gel} or \code{gmm} returned by the function \code{\link{gel}} or \code{\link{gmm}}} \item{...}{Other arguments to pass to other methods} } \examples{ ## GEL ## n = 200 phi<-c(.2,.7) thet <- 0.2 sd <- .2 set.seed(123) x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gel(g, x, c(0,.3,.6)) formula(res) # GMM is like GLS for linear models without endogeneity problems set.seed(345) n = 200 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- 10 + 5*rnorm(n) + x res <- gmm(y ~ x, x) formula(res) } gmm/man/vcov.Rd0000644000176200001440000000312313107655275013063 0ustar liggesusers\name{vcov} \alias{vcov.gmm} \alias{vcov.gel} \alias{vcov.tsls} \alias{vcov.ategel} \title{Variance-covariance matrix of GMM or GEL} \description{ It extracts the matrix of variances and covariances from \code{gmm} or \code{gel} objects. } \usage{ \method{vcov}{gmm}(object, ...) \method{vcov}{gel}(object, lambda = FALSE, ...) \method{vcov}{tsls}(object, type=c("Classical","HC0","HC1","HAC"), hacProp = list(), ...) \method{vcov}{ategel}(object, lambda = FALSE, robToMiss = TRUE, ...) } \arguments{ \item{object}{An object of class \code{gmm} or \code{gmm} returned by the function \code{\link{gmm}} or \code{\link{gel}}} \item{lambda}{If set to TRUE, the covariance matrix of the Lagrange multipliers is produced.} \item{type}{Type of covariance matrix for the meat} \item{hacProp}{A list of arguments to pass to \code{\link{kernHAC}}} \item{robToMiss}{If \code{TRUE}, it computes the robust to misspecification covariance matrix} \item{...}{Other arguments when \code{vcov} is applied to another class object} } \details{ For tsls(), if vcov is set to a different value thand "Classical", a sandwich covariance matrix is computed. } \value{ A matrix of variances and covariances } \examples{ # GMM # n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n,list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gmm(g, x) vcov(res) ## GEL ## t0 <- c(0,.5,.5) res <- gel(g, x, t0) vcov(res) vcov(res, lambda = TRUE) } gmm/man/tsls.Rd0000644000176200001440000000475411665451374013107 0ustar liggesusers\name{tsls} \alias{tsls} \title{Two stage least squares estimation} \description{ Function to estimate a linear model by the two stage least squares method. } \usage{ tsls(g,x,data) } \arguments{ \item{g}{A formula describing the linear regression model (see details below).} \item{x}{The matrix of instruments (see details below).} \item{data}{A data.frame or a matrix with column names (Optionnal). } } \details{ The function just calls \code{\link{gmm}} with the option vcov="iid". It just simplifies the the implementation of 2SLS. The users don't have to worry about all the options offered in \code{\link{gmm}}. The model is \deqn{ Y_i = X_i\beta + u_i } In the first step, \code{\link{lm}} is used to regress \eqn{X_i} on the set of instruments \eqn{Z_i}. The second step also uses \code{\link{lm}} to regress \eqn{Y_i} on the fitted values of the first step. } \value{ 'tsls' returns an object of 'class' '"tsls"' which inherits from class '"gmm"'. The functions 'summary' is used to obtain and print a summary of the results. It also compute the J-test of overidentying restriction The object of class "gmm" is a list containing at least: \item{coefficients}{\eqn{k\times 1} vector of coefficients} \item{residuals}{the residuals, that is response minus fitted values if "g" is a formula.} \item{fitted.values}{the fitted mean values if "g" is a formula.} \item{vcov}{the covariance matrix of the coefficients} \item{objective}{the value of the objective function \eqn{\| var(\bar{g})^{-1/2}\bar{g}\|^2}} \item{terms}{the \code{\link{terms}} object used when g is a formula.} \item{call}{the matched call.} \item{y}{if requested, the response used (if "g" is a formula).} \item{x}{if requested, the model matrix used if "g" is a formula or the data if "g" is a function.} \item{model}{if requested (the default), the model frame used if "g" is a formula.} \item{algoInfo}{Information produced by either \code{\link{optim}} or \code{\link{nlminb}} related to the convergence if "g" is a function. It is printed by the \code{summary.gmm} method.} } \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, } \examples{ n <- 1000 e <- arima.sim(n,model=list(ma=.9)) C <- runif(n,0,5) Y <- rep(0,n) Y[1] = 1 + 2*C[1] + e[1] for (i in 2:n){ Y[i] = 1 + 2*C[i] + 0.9*Y[i-1] + e[i] } Yt <- Y[5:n] X <- cbind(C[5:n],Y[4:(n-1)]) Z <- cbind(C[5:n],Y[3:(n-2)],Y[2:(n-3)],Y[1:(n-4)]) res <- tsls(Yt~X,~Z) res } gmm/man/Finance.Rd0000644000176200001440000000275514017527033013452 0ustar liggesusers\name{Finance} \docType{data} \alias{Finance} \title{Returns on selected stocks} \description{ Daily returns on selected stocks, the Market portfolio and factors of Fama and French from 1993-01-05 to 2009-01-30 for CAPM and APT analysis } \usage{data(Finance)} \format{ A data frame containing 24 time series. Dates are reported as rownames(). In the following description, company symboles are used. \describe{ \item{WMK}{Returns of WEIS MARKETS INC} \item{UIS}{Returns of UNISYS CP NEW} \item{ORB}{Returns of ORBITAL SCIENCES CP} \item{MAT}{Returns of Mattel, Inc.} \item{ABAX}{Returns of ABAXIS, Inc.} \item{T}{Returns of AT&T INC.} \item{EMR}{Returns of EMERSON ELEC CO} \item{JCS}{Returns of Communications Systems Inc.} \item{VOXX}{Returns of Audiovox Corp.} \item{ZOOM}{Returns of ZOOM Technologies Inc.} \item{TDW}{Returns of TIDEWATER INC} \item{ROG}{Returns of Rogers Corporation} \item{GGG}{Returns of Graco Inc.} \item{PC}{Returns of Panasonic Corporation} \item{GCO}{Returns of Genesco Inc.} \item{EBF}{Returns of ENNIS, INC} \item{F}{Returns of FORD MOTOR CO} \item{FNM}{Returns of FANNIE MAE} \item{NHP}{Returns of NATIONWIDE HLTH PROP} \item{AA}{Returns of ALCOA INC} \item{rf}{Risk-free rate of Fama-French} \item{rm}{Return of the market portfolio of Fama-French} \item{hml}{Factor High-Minus-Low of Fama-French} \item{smb}{Factor Small-Minus-Big of Fama-French} } } \source{Yahoo Finance and \url{https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/}} \keyword{datasets} gmm/man/growth.Rd0000644000176200001440000000162712634331531013415 0ustar liggesusers\name{Growth} \docType{data} \alias{Growth} \title{Growth Data} \description{ Panel of Macroeconomic data for 125 countries from 1960 to 1985 constructed by Summers and Heston (1991)) } \usage{data(Growth)} \format{ A data frame containing 9 vectors. \describe{ \item{Country_ID}{Country identification number} \item{COM}{1 if the country is in a communist regime, 0 otherwise} \item{OPEC}{1 if the country is part of the OPEC, 0 otherwise} \item{Year}{Year} \item{GDP}{Per capita GDP (in thousands) in 1985 U.S. dollars.} \item{LagGDP}{GDP of the previous period} \item{SavRate}{Saving rate measured as the ratio of real investment to real GDP} \item{LagSavRate}{SavRate of the previous period} \item{Country}{Country names} \item{Pop}{Population in thousands} \item{LagPop}{Population of the previous period} } } \source{\url{http://fhayashi.fc2web.com/datasets.htm}} \keyword{datasets} gmm/man/momentEstim.Rd0000644000176200001440000000412512620733247014405 0ustar liggesusers\name{momentEstim} \alias{momentEstim.baseGmm.twoStep} \alias{momentEstim.baseGmm.twoStep.formula} \alias{momentEstim.sysGmm.twoStep.formula} \alias{momentEstim.tsls.twoStep.formula} \alias{momentEstim.baseGmm.iterative.formula} \alias{momentEstim.baseGmm.iterative} \alias{momentEstim.baseGmm.cue.formula} \alias{momentEstim.baseGmm.cue} \alias{momentEstim.baseGmm.eval} \alias{momentEstim.baseGel.mod} \alias{momentEstim.baseGel.modFormula} \alias{momentEstim.baseGel.eval} \title{Method for estimating models based on moment conditions} \description{ It estimates a model which is caracterized by the method \code{getModel} (see details). } \usage{ \method{momentEstim}{baseGmm.twoStep}(object, ...) \method{momentEstim}{baseGmm.twoStep.formula}(object, ...) \method{momentEstim}{sysGmm.twoStep.formula}(object, ...) \method{momentEstim}{tsls.twoStep.formula}(object, ...) \method{momentEstim}{baseGmm.iterative.formula}(object, ...) \method{momentEstim}{baseGmm.iterative}(object, ...) \method{momentEstim}{baseGmm.cue.formula}(object, ...) \method{momentEstim}{baseGmm.cue}(object, ...) \method{momentEstim}{baseGmm.eval}(object, ...) \method{momentEstim}{baseGel.mod}(object, ...) \method{momentEstim}{baseGel.modFormula}(object, ...) \method{momentEstim}{baseGel.eval}(object, ...) } \arguments{ \item{object}{An object created by the method \code{getModel}} \item{...}{Other arguments when \code{momentEstim} is applied to an other class object} } \value{ It returns an object of class determined by the argument "TypeGMM" of \code{\link{gmm}}. By default, it is of class \code{baseGmm.res}. It estimates the model and organize the results that will be finalized by the method \code{FinRes}. More methods can be created in order to use other GMM methods not yet included in the package. } \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. \emph{Journal of Business and Economic Statistics}, \bold{14} 262-280. } gmm/man/smoothG.Rd0000644000176200001440000000672114017527033013524 0ustar liggesusers\name{smoothG} \alias{smoothG} \title{Kernel smoothing of a matrix of time series} \description{ It applies the required kernel smoothing to the moment function in order for the GEL estimator to be valid. It is used by the \code{gel} function.} \usage{ smoothG(x, bw = bwAndrews, prewhite = 1, ar.method = "ols", weights = weightsAndrews, kernel = c("Bartlett", "Parzen", "Truncated", "Tukey-Hanning"), approx = c("AR(1)", "ARMA(1,1)"), tol = 1e-7) } \arguments{ \item{x}{a \eqn{n\times q} matrix of time series, where n is the sample size.} \item{bw}{The method to compute the bandwidth parameter. By default, it uses the bandwidth proposed by Andrews(1991). As an alternative, we can choose bw=bwNeweyWest (without "") which is proposed by Newey-West(1996).} \item{prewhite}{logical or integer. Should the estimating functions be prewhitened? If \code{TRUE} or greater than 0 a VAR model of order \code{as.integer(prewhite)} is fitted via \code{ar} with method \code{"ols"} and \code{demean = FALSE}.} \item{ar.method}{character. The \code{method} argument passed to \code{\link{ar}} for prewhitening.} \item{weights}{The smoothing weights can be computed by \code{\link{weightsAndrews}} of it can be provided manually. If provided, it has to be a \eqn{r\times 1}vector (see details). } \item{approx}{a character specifying the approximation method if the bandwidth has to be chosen by \code{bwAndrews}.} \item{tol}{numeric. Weights that exceed \code{tol} are used for computing the covariance matrix, all other weights are treated as 0.} \item{kernel}{The choice of kernel} } \details{ The sample moment conditions \eqn{\sum_{t=1}^n g(\theta,x_t)} is replaced by: \eqn{\sum_{t=1}^n g^k(\theta,x_t)}, where \eqn{g^k(\theta,x_t)=\sum_{i=-r}^r k(i) g(\theta,x_{t+i})}, where \eqn{r} is a truncated parameter that depends on the bandwidth and \eqn{k(i)} are normalized weights so that they sum to 1. If the vector of weights is provided, it gives only one side weights. For exemple, if you provide the vector (1,.5,.25), \eqn{k(i)} will become \eqn{(.25,.5,1,.5,.25)/(.25+.5+1+.5+.25) = (.1,.2,.4,.2,.1)} } \value{ smoothx: A \eqn{q \times q} matrix containing an estimator of the asymptotic variance of \eqn{\sqrt{n} \bar{x}}, where \eqn{\bar{x}} is \eqn{q\times 1}vector with typical element \eqn{\bar{x}_i = \frac{1}{n}\sum_{j=1}^nx_{ji}}. This function is called by \code{\link{gel}} but can also be used by itself. \code{kern_weights}: Vector of weights used for the smoothing. } \references{ Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias. \emph{Econometrica}, \bold{73}, 983-1002. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. \emph{Econometrica}, \bold{59}, 817--858. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes. \emph{The Annals of Statistics}, \bold{25}, 2084-2102. Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. } \examples{ g <- function(tet, x) { n <- nrow(x) u <- (x[7:n] - tet[1] - tet[2]*x[6:(n-1)] - tet[3]*x[5:(n-2)]) f <- cbind(u, u*x[4:(n-3)], u*x[3:(n-4)], u*x[2:(n-5)], u*x[1:(n-6)]) return(f) } n = 500 phi<-c(.2, .7) thet <- 0.2 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1) gt <- g(c(0, phi), x) sgt <- smoothG(gt)$smoothx plot(gt[,1]) lines(sgt[,1]) } gmm/man/getDat.Rd0000644000176200001440000000275612623157321013320 0ustar liggesusers\name{getDat} \alias{getDat} \title{Extracting data from a formula} \description{ It extract the data from a formula y~z with instrument h and put everything in a matrix. It helps redefine the function \eqn{g(\theta,x)} that is required by \code{\link{gmm}} and \code{\link{gel}}. } \usage{ getDat(formula, h, data, error=TRUE) } \arguments{ \item{formula}{A formula that defines the linear model to be estimated (see details).} \item{h}{A \eqn{n\times nh} matrix of intruments(see details).} \item{data}{A data.frame or a matrix with colnames (Optionnal).} \item{error}{If FALSE, the data is generated without giving any error message} } \details{The model to be estimated is based on the moment conditions \eqn{=0}. It adds a column of ones to z and h by default. They are removed if -1 is added to the formula. The error argument has been added for \code{\link{sysGmm}} with common coefficients because the check is only valid for equation by equation identification. } \value{ x: A \eqn{n \times l} matrix, where \eqn{l = ncol(y)+ncol(z)+ncol(h)+2} if "intercept" is TRUE and \eqn{ncol(y)+ncol(z)+xcol(h)} if "intercept" is FALSE. nh: dimension of h k: dimension of z ny: dimension of y } \examples{ n = 500 phi<-c(.2, .7) thet <- 0.2 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2, 0, 1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) x <- getDat(y ~ ym1 + ym2, H) } gmm/man/bread.Rd0000644000176200001440000000343614017527033013161 0ustar liggesusers\name{bread} \alias{bread.gmm} \alias{bread.gel} \alias{bread.tsls} \title{Bread for sandwiches} \description{ Computes the bread of the sandwich covariance matrix } \usage{ \method{bread}{gmm}(x, ...) \method{bread}{gel}(x, ...) \method{bread}{tsls}(x, ...) } \arguments{ \item{x}{A fitted model of class \code{gmm} or \code{gel}.} \item{...}{Other arguments when \code{bread} is applied to another class object} } \details{ When the weighting matrix is not the optimal one, the covariance matrix of the estimated coefficients is: \eqn{(G'WG)^{-1} G'W V W G(G'WG)^{-1}}, where \eqn{G=d\bar{g}/d\theta}, \eqn{W} is the matrix of weights, and \eqn{V} is the covariance matrix of the moment function. Therefore, the bread is \eqn{(G'WG)^{-1}}, which is the second derivative of the objective function. The method if not yet available for \code{gel} objects. } \value{ A \eqn{k \times k} matrix (see details). } \references{ Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. } \examples{ # See \code{\link{gmm}} for more details on this example. # With the identity matrix # bread is the inverse of (G'G) n <- 1000 x <- rnorm(n, mean = 4, sd = 2) g <- function(tet, x) { m1 <- (tet[1] - x) m2 <- (tet[2]^2 - (x - tet[1])^2) m3 <- x^3 - tet[1]*(tet[1]^2 + 3*tet[2]^2) f <- cbind(m1, m2, m3) return(f) } Dg <- function(tet, x) { jacobian <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2], -6*tet[1]*tet[2]), nrow=3,ncol=2) return(jacobian) } res <- gmm(g, x, c(0, 0), grad = Dg,weightsMatrix=diag(3)) G <- Dg(res$coef, x) bread(res) solve(crossprod(G)) } gmm/man/coef.Rd0000644000176200001440000000200311305632271013003 0ustar liggesusers\name{coef} \alias{coef.gel} \alias{coef.gmm} \title{Coefficients of GEL or GMM} \description{ It extracts the coefficients from \code{gel} or \code{gmm} objects. } \usage{ \method{coef}{gmm}(object, ...) \method{coef}{gel}(object, lambda = FALSE, ...) } \arguments{ \item{object}{An object of class \code{gel} or \code{gmm} returned by the function \code{\link{gel}} or \code{\link{gmm}}} \item{lambda}{If set to TRUE, the lagrange multipliers are extracted instead of the vector of coefficients} \item{...}{Other arguments when \code{coef} is applied to an other class object} } \value{ Vector of coefficients } \examples{ ################# n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H t0 <- c(0,.5,.5) res <- gel(g, x, t0) coef(res) coef(res, lambda = TRUE) ################### res <- gmm(g, x) coef(res) } gmm/man/confint.Rd0000644000176200001440000000724413107655275013556 0ustar liggesusers\name{confint} \alias{confint.gel} \alias{confint.ategel} \alias{confint.gmm} \alias{print.confint} \title{Confidence intervals for GMM or GEL} \description{ It produces confidence intervals for the coefficients from \code{gel} or \code{gmm} estimation. } \usage{ \method{confint}{gel}(object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, ...) \method{confint}{gmm}(object, parm, level = 0.95, ...) \method{confint}{ategel}(object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, robToMiss=TRUE, ...) \method{print}{confint}(x, digits = 5, ...) } \arguments{ \item{object}{An object of class \code{gel} or \code{gmm} returned by the function \code{\link{gel}} or \code{\link{gmm}}} \item{parm}{A specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.} \item{level}{The confidence level} \item{lambda}{If set to TRUE, the confidence intervals for the Lagrange multipliers are produced.} \item{type}{'Wald' is the usual symetric confidence interval. The thee others are based on the inversion of the LR, LM, and J tests.} \item{fact}{This parameter control the span of search for the inversion of the test. By default we search within plus or minus 3 times the standard error of the coefficient estimate.} \item{corr}{This numeric scalar is meant to apply a correction to the critical value, such as a Bartlett correction. This value depends on the model (See Owen; 2001)} \item{x}{An object of class \code{confint} produced by \code{confint.gel} and \code{confint.gmm}} \item{digits}{The number of digits to be printed} \item{robToMiss}{If \code{TRUE}, the confidence interval is based on the standard errors that are robust to misspecification} \item{...}{Other arguments when \code{confint} is applied to another classe object} } \value{ It returns a matrix with the first column being the lower bound and the second the upper bound.} \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. \emph{Journal of Business and Economic Statistics}, \bold{14} 262-280. Owen, A.B. (2001), Empirical Likelihood. \emph{Monographs on Statistics and Applied Probability 92, Chapman and Hall/CRC} } \examples{ ################# n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H t0 <- c(0,.5,.5) resGel <- gel(g, x, t0) confint(resGel) confint(resGel, level = 0.90) confint(resGel, lambda = TRUE) ######################## resGmm <- gmm(g, x) confint(resGmm) confint(resGmm, level = 0.90) ## Confidence interval with inversion of the LR, LM or J test. ############################################################## set.seed(112233) x <- rt(40, 3) y <- x+rt(40,3) # Simple interval on the mean res <- gel(x~1, ~1, method="Brent", lower=-4, upper=4) confint(res, type = "invLR") confint(res) # Using a Bartlett correction k <- mean((x-mean(x))^4)/sd(x)^4 s <- mean((x-mean(x))^3)/sd(x)^3 a <- k/2-s^2/3 corr <- 1+a/40 confint(res, type = "invLR", corr=corr) # Interval on the slope res <- gel(y~x, ~x) confint(res, "x", type="invLR") confint(res, "x") } gmm/man/FinRes.Rd0000644000176200001440000000161711313546054013271 0ustar liggesusers\name{FinRes} \alias{FinRes.baseGmm.res} \title{Method to finalize the result of the momentEstim method} \description{ It computes the final results that will be needed to create the object of class \code{gmm}.). } \usage{ \method{FinRes}{baseGmm.res}(z, object, ...) } \arguments{ \item{z}{An object of class determined by the method \code{momentEstim}.} \item{object}{An object produced my \code{getModel}} \item{...}{Other argument to be passed to other \code{FinRes} methods.} } \value{ It returns an object of class \code{gmm}. See \code{\link{gmm}} for more details. } \references{ Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. \emph{Econometrica}, \bold{50}, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators. \emph{Journal of Business and Economic Statistics}, \bold{14} 262-280. } gmm/man/ATEgel.Rd0000644000176200001440000001514414017527033013204 0ustar liggesusers\name{ATEgel} \alias{ATEgel} \alias{checkConv} \title{ATE with Generalized Empirical Likelihood estimation} \description{ Function to estimate the average treatment effect with the sample being balanced by GEL. } \usage{ ATEgel(g, balm, w=NULL, y=NULL, treat=NULL, tet0=NULL,momType=c("bal","balSample","ATT"), popMom = NULL, family=c("linear","logit", "probit"), type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD", "RCUE"), tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, optfct = c("optim", "nlminb"), optlam = c("nlminb", "optim", "iter", "Wu"), data=NULL, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, ...) checkConv(obj, tolConv=1e-4, verbose=TRUE, ...) } \arguments{ \item{g}{A formula as \code{y~z}, where code{y} is the response and \code{z} the treatment indicator. If there is more than one treatment, more indicators can be added or \code{z} can be set as a factor. It can also be of the form \code{g(theta, y, z)} for non-linear models. It is however, not implemented yet.} \item{obj}{Object of class \code{"ategel"} produced y \code{ATEgel}} \item{balm}{A formula for the moments to be balanced between the treated and control groups (see details)} \item{y}{The response variable when \code{g} is a function. Not implemented yet} \item{treat}{The treatment indicator when \code{g} is a function. Not implemented yet} \item{w}{A formula to add covariates to the main regression. When \code{NULL}, the default value, the main regression only include treatment indicators.} \item{tet0}{A \eqn{3 \times 1} vector of starting values. If not provided, they are obtained using an OLS regression} \item{momType}{How the moments of the covariates should be balanced. By default, it is simply balanced without restriction. Alternatively, moments can be set equal to the sample moments of the whole sample, or to the sample moments of the treated group. The later will produce the average treatment effect of the treated (ATT)} \item{popMom}{A vector of population moments to use for balancing. It can be used of those moments are available from a census, for example. When available, it greatly improves efficiency.} \item{family}{By default, the outcome is linearly related to the treatment indicators. If the outcome is binary, it is possible to use the estimating equations of either the logit or probit model.} \item{type}{"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continuous updated estimator, "ETEL" for exponentially tilted empirical likelihood of Schennach(2007), "HD" for Hellinger Distance of Kitamura-Otsu-Evdokimov (2013), and "ETHD" for the exponentially tilted Hellinger distance of Antoine-Dovonon (2015). "RCUE" is a restricted version of "CUE" in which the probabilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution.} \item{tol_lam}{Tolerance for \eqn{\lambda} between two iterations. The algorithm stops when \eqn{\|\lambda_i -\lambda_{i-1}\|} reaches \code{tol_lamb} (see \code{\link{getLamb}}) } \item{maxiterlam}{The algorithm to compute \eqn{\lambda} stops if there is no convergence after "maxiterlam" iterations (see \code{\link{getLamb}}).} \item{tol_obj}{Tolerance for the gradiant of the objective function to compute \eqn{\lambda} (see \code{\link{getLamb}}).} \item{optfct}{Algorithm used for the parameter estimates} \item{tol_mom}{It is the tolerance for the moment condition \eqn{\sum_{t=1}^n p_t g(\theta(x_t)=0}, where \eqn{p_t=\frac{1}{n}D\rho()} is the implied probability. It adds a penalty if the solution diverges from its goal.} \item{optlam}{Algorithm used to solve for the lagrange multiplier in \code{\link{getLamb}}. The algorithm Wu is only for \code{type="EL"}. The value of \code{optlam} is ignored for "CUE" because in that case, the analytical solution exists.} \item{data}{A data.frame or a matrix with column names (Optional). } \item{Lambdacontrol}{Controls for the optimization of the vector of Lagrange multipliers used by either \code{\link{optim}}, \code{\link{nlminb}} or \code{\link{constrOptim}}} \item{model, X, Y}{logicals. If \code{TRUE} the corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula.} \item{verbose}{If TRUE, a summary of the convergence is printed} \item{tolConv}{The tolerance for comparing moments between groups} \item{...}{More options to give to \code{\link{optim}} or \code{\link{nlminb}}. In \code{checkConv}, they are options passed to \code{\link{getImpProb}}.} } \details{ We want to estimate the model \eqn{Y_t = \theta_1 + \theta_2 treat + \epsilon_t}, where \eqn{\theta_2} is the treatment effect. GEL is used to balance the sample based on the argument \code{x} above. For example, if we want the sample mean of \code{x1} and \code{x2} to be balanced between the treated and control, we set \code{x} to \code{~x1+x2}. If we want the sample mean of \code{x1}, \code{x2}, \code{x1*x2}, \code{x1^2} and \code{x2^2}, we set \code{x} to \code{~x1*x2 + I(x1^2) + I(x2^2)}. } \value{ 'gel' returns an object of 'class' '"ategel"' The functions 'summary' is used to obtain and print a summary of the results. The object of class "ategel" is a list containing the same elements contained in objects of class \code{\link{gel}}. } \references{ Lee, Seojeong (2016), Asymptotic refinements of misspecified-robust bootstrap for GEL estimators, \emph{Journal of Econometrics}, \bold{192}, 86--104. Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood. \emph{Econometrica}, \bold{35}, 634-672. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling. \emph{Survey Methodology}, \bold{31}(2), page 239. Chausse, P. (2010), Computing Generalized Method of Moments and Generalized Empirical Likelihood with R. \emph{Journal of Statistical Software}, \bold{34}(11), 1--35. URL \url{https://www.jstatsoft.org/v34/i11/}. Chausse, P. and Giurcanu, M. and Luta, G. (2021) Estimating the Average Causal Effect using Generalized Empirical Likelihood Methods, Work in progress. } \examples{ data(nsw) # Scale income nsw$re78 <- nsw$re78/1000 nsw$re75 <- nsw$re75/1000 res <- ATEgel(re78~treat, ~age+ed+black+hisp+married+nodeg+re75, data=nsw,type="ET") summary(res) chk <- checkConv(res) res2 <- ATEgel(re78~treat, ~age+ed+black+hisp+married+nodeg+re75, data=nsw,type="ET", momType="balSample") summary(res2) chk2 <- checkConv(res2) } gmm/man/getImpProb.Rd0000644000176200001440000000247513460072542014157 0ustar liggesusers\name{getImpProb} \alias{getImpProb} \alias{getImpProb.gel} \title{Implied Probabilities} \description{ It computes the implied probabilities from objects of class \code{gel} with additional options. } \usage{ \method{getImpProb}{gel}(object, posProb=TRUE, normalize=TRUE, checkConv=FALSE,...) } \arguments{ \item{object}{Object of class \code{gel}.} \item{posProb}{Should the implied probabilities be transformed into positive probabilities?} \item{normalize}{Should we normalize the probabilities so that they sum to one?} \item{checkConv}{Should we add the attribute convergence to check the sum of the probabilities and the weighted sum of the moment conditions?} \item{...}{Additional arguments to pass to other methods} } \value{ A vector af implied probabilities. } \references{ Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators. \emph{Econometrica}, \bold{72}, 219-255. } \examples{ ################# n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H t0 <- c(0,.5,.5) res <- gel(g, x, t0) pt <- getImpProb(res) } gmm/man/plot.Rd0000644000176200001440000000477011305632271013062 0ustar liggesusers\name{plot} \alias{plot.gel} \alias{plot.gmm} \title{Plot Diagnostics for gel and gmm objects} \description{ It is a plot method for \code{gel} or \code{gmm} objects. } \usage{ \method{plot}{gel}(x, which = c(1L:4), main = list("Residuals vs Fitted values", "Normal Q-Q", "Response variable and fitted values","Implied probabilities"), panel = if(add.smooth) panel.smooth else points, ask = prod(par("mfcol")) < length(which) && dev.interactive(), ..., add.smooth = getOption("add.smooth")) \method{plot}{gmm}(x, which = c(1L:3), main = list("Residuals vs Fitted values", "Normal Q-Q", "Response variable and fitted values"), panel = if(add.smooth) panel.smooth else points, ask = prod(par("mfcol")) < length(which) && dev.interactive(), ..., add.smooth = getOption("add.smooth")) } \arguments{ \item{x}{\code{gel} or \code{gmm} object, typically result of \code{\link{gel}} or \code{\link{gmm}}.} \item{which}{if a subset of the plots is required, specify a subset of the numbers \code{1:4} for \code{gel} or \code{1:3} for \code{gmm}.} \item{main}{Vector of titles for each plot. } \item{panel}{panel function. The useful alternative to \code{\link{points}}, \code{\link{panel.smooth}} can be chosen by \code{add.smooth = TRUE}.} \item{ask}{logical; if \code{TRUE}, the user is \emph{ask}ed before each plot, see \code{\link{par}(ask=.)}.} \item{\dots}{other parameters to be passed through to plotting functions.} \item{add.smooth}{logical indicating if a smoother should be added to most plots; see also \code{panel} above.} } \details{ It is a beta version of a plot method for \code{gel} objects. It is a modified version of \code{plot.lm}. For now, it is available only for linear models expressed as a formula. Any suggestions are welcome regarding plots or options to include. The first two plots are the same as the ones provided by \code{plot.lm}, the third is the dependant variable \eqn{y} with its mean \eqn{\hat{y}} (the fitted values) and the last plots the implied probabilities with the empirical density \eqn{1/T}. } \examples{ # GEL # n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n,list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H t0 <- c(0,.5,.5) res <- gel(g, x, t0) plot(res, which = 3) plot(res, which = 4) # GMM # res <- gmm(g, x) plot(res, which = 3) } gmm/man/estfun.Rd0000644000176200001440000000521514017527033013405 0ustar liggesusers\name{estfun} \alias{estfun.gmmFct} \alias{estfun.gmm} \alias{estfun.gel} \alias{estfun.tsls} \alias{model.matrix.tsls} \title{Extracts the empirical moment function} \description{ It extracts the matrix of empirical moments so that it can be used by the \code{\link{kernHAC}} function. } \usage{ \method{estfun}{gmmFct}(x, y = NULL, theta = NULL, ...) \method{estfun}{gmm}(x, ...) \method{estfun}{gel}(x, ...) \method{estfun}{tsls}(x, ...) \method{model.matrix}{tsls}(object, ...) } \arguments{ \item{x}{A function of the form \eqn{g(\theta,y)} or a \eqn{n \times q} matrix with typical element \eqn{g_i(\theta,y_t)} for \eqn{i=1,...q} and \eqn{t=1,...,n} or an object of class \code{gmm}. See \code{\link{gmm}} for more details. For \code{\link{tsls}}, it is an object of class \code{tsls}.} \item{object}{An object of class \code{tsls}.} \item{y}{The matrix or vector of data from which the function \eqn{g(\theta,y)} is computed if \code{g} is a function.} \item{theta}{Vector of parameters if \code{g} is a function.} \item{...}{Other arguments when \code{estfun} is applied to another class object} } \details{ For \code{estfun.gmmFct}, it returns a \eqn{n \times q} matrix with typical element \eqn{g_i(\theta,y_t)} for \eqn{i=1,...q} and \eqn{t=1,...,n}. It is only used by \code{gmm} to obtain the estimates. For \code{estfun.gmm}, it returns the matrix of first order conditions of \eqn{\min_\theta \bar{g}'W\bar{g}/2}, which is a \eqn{n \times k} matrix with the \eqn{t^{th}} row being \eqn{g(\theta, y_t)W G}, where \eqn{G} is \eqn{d\bar{g}/d\theta}. It allows to compute the sandwich covariance matrix using \code{\link{kernHAC}} or \code{\link{vcovHAC}} when \eqn{W} is not the optimal matrix. The method if not yet available for \code{gel} objects. For tsls, model.matrix and estfun are used by \code{vcov()} to compute different covariance matrices using the \code{\link{sandwich}} package. See \code{\link{vcov.tsls}}. \code{model.matrix} returns the fitted values frin the first stage regression and \code{esfun} the residuals. } \value{ A \eqn{n \times q} matrix (see details). } \references{ Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. } \examples{ n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n=n,list(order=c(2,0,1),ar=phi,ma=thet,sd=sd)),ncol=1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gmm(g, x,weightsMatrix = diag(5)) gt <- res$gt G <- res$G foc <- gt%*%G foc2 <- estfun(res) foc[1:5,] foc2[1:5,] } gmm/man/wage.Rd0000644000176200001440000000144212620733247013026 0ustar liggesusers\name{wage} \docType{data} \alias{wage} \title{Labor Data} \description{ Data used to measure return to education by Griliches (1976) } \usage{data(wage)} \format{ A data frame containing 20 cross-sectional vectors. \describe{ \item{AGE, AGE80}{Age in 1969 and 1980 respetively} \item{EXPR, EXPR80}{Working experience in 1969 and 1980 respetively} \item{IQ}{IQ measure of the individual} \item{KWW}{A test score} \item{LW, LW80}{Log wage in 1969 and 1980 respectively} \item{MED}{Mother education} \item{MRT, MRT80}{} \item{RNS, RNS80}{} \item{S, S80}{Schooling in 1969 and 1980 respetively} \item{SMSA, SMSA80}{} \item{TENURE, TENURE80}{Tenure in 1969 and 1980 respetively} \item{YEAR}{} } } \source{\url{http://fhayashi.fc2web.com/datasets.htm}} \keyword{datasets} gmm/man/nsw.Rd0000644000176200001440000000303613552350472012713 0ustar liggesusers\name{nsw} \docType{data} \alias{nsw} \title{Lalonde subsample of the National Supported Work Demonstration Data (NSW)} \description{ This data was collected to evaluate the National Supported Work (NSW) Demonstration project in Lalonde (1986). } \usage{data(nsw)} \format{ A data frame containing 9 variables. \describe{ \item{treat}{Treatment assignment} \item{age}{Age} \item{ed}{Years of Education} \item{black}{1 if Black, 0 otherwise} \item{hisp}{1 if Hispanic 0 otherwise} \item{married}{1 if married 0 otherwise} \item{nodeg}{1 if no college degree 0 otherwise} \item{re75}{1975 earnings} \item{re78}{1978 earnings} } } \details{ The dataset was obtained from the ATE package (see reference). } \source{ "NSW Data Files" from Rajeev Dehejia's website. URL: \url{http://users.nber.org/~rdehejia/data/.nswdata2.html} "National Supported Work Evaluation Study, 1975-1979: Public Use Files." from the Interuniversity Consortium for Political and Social Research. URL: \url{http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/7865} } \references{ Lalonde, R. (1986). "Evaluating the Econometric Evaluations of Training Programs," American Economic Review, 76(4), 604-620. Dehejia R. and Wahba S. (1999). "Causal Effects in Non-Experimental Studies: Re-Evaluating the Evaluation of Training Programs," JASA 94 (448), 1053-1062. Asad Haris and Gary Chan (2015). ATE: Inference for Average Treatment Effects using Covariate Balancing. R package version 0.2.0. \url{https://CRAN.R-project.org/package=ATE} } \keyword{datasets} gmm/man/print.Rd0000644000176200001440000000206312620733247013237 0ustar liggesusers\name{print} \alias{print.gmm} \alias{print.sysGmm} \alias{print.gel} \title{Printing a gmm or gel object} \description{ It is a printing method for \code{gmm} or \code{gel} objects. } \usage{ \method{print}{gmm}(x, digits = 5, ...) \method{print}{gel}(x, digits = 5, ...) \method{print}{sysGmm}(x, digits = 5, ...) } \arguments{ \item{x}{An object of class \code{gmm} or \code{gel} returned by the function \code{\link{gmm}} or \code{\link{gel}}} \item{digits}{The number of digits to be printed} \item{...}{Other arguments when print is applied to an other class object} } \value{ It prints some results from the estimation like the coefficients and the value of the objective function. } \examples{ # GMM # n = 500 phi<-c(.2,.7) thet <- 0 sd <- .2 x <- matrix(arima.sim(n = n, list(order = c(2,0,1), ar = phi, ma = thet, sd = sd)), ncol = 1) y <- x[7:n] ym1 <- x[6:(n-1)] ym2 <- x[5:(n-2)] H <- cbind(x[4:(n-3)], x[3:(n-4)], x[2:(n-5)], x[1:(n-6)]) g <- y ~ ym1 + ym2 x <- H res <- gmm(g, x) print(res) # GEL # t0 <- c(0,.5,.5) res <- gel(g,x,t0) print(res) } gmm/man/bwWilhelm.Rd0000644000176200001440000000563214017527033014036 0ustar liggesusers\name{bwWilhelm} \alias{bwWilhelm} \title{Wilhelm (2015) bandwidth selection} \description{ It computes the optimal bandwidth for the HAC estimation of the covariance matrix of the moment conditions. The bandwidth was shown by Wilhelm (2005) to be the one that minimizes the MSE of the GMM estimator. } \usage{ bwWilhelm(x, order.by = NULL, kernel = c("Quadratic Spectral", "Bartlett", "Parzen", "Tukey-Hanning"), approx = c("AR(1)", "ARMA(1,1)"), weights = NULL, prewhite = 1, ar.method = "ols", data = list()) } \arguments{ \item{x}{An object of class \code{gmm}.} \item{order.by}{Either a vector 'z' or a formula with a single explanatory variable like '~ z'. The observations in the model are ordered by the size of 'z'. If set to 'NULL' (the default) the observations are assumed to be ordered (e.g., a time series).} \item{kernel}{type of kernel used to compute the covariance matrix of the vector of sample moment conditions (see \code{\link{kernHAC}} for more details)} \item{approx}{A character specifying the approximation method if the bandwidth has to be chosen by \code{bwAndrews}.} \item{weights}{numeric. A vector of weights used for weighting the estimated coefficients of the approximation model (as specified by 'approx'). By default all weights are 1 except that for the intercept term (if there is more than one variable)} \item{prewhite}{logical or integer. Should the estimating functions be prewhitened? If \code{TRUE} or greater than 0 a VAR model of order \code{as.integer(prewhite)} is fitted via \code{ar} with method \code{"ols"} and \code{demean = FALSE}.} \item{ar.method}{character. The \code{method} argument passed to \code{\link{ar}} for prewhitening.} \item{data}{an optional data frame containing the variables in the 'order.by' model.} } \value{ The function 'bwWilhelm' returns the optimal bandwidth. } \references{ Wilhelm, D. (2015), Optimal Bandwidth Selection for Robust Generalized Method of Moments Estimation. \emph{Econometric Theory}, \bold{31}, 1054--1077 Zeileis A (2006), Object-oriented Computation of Sandwich Estimators. \emph{Journal of Statistical Software}, \bold{16}(9), 1--16. URL \url{https://www.jstatsoft.org/v16/i09/}. } \note{ The function was written by Daniel Wilhelm and is based on \link{bwAndrews}. } \examples{ data(Finance) f1 <- Finance[1:300, "rm"] f2 <- Finance[1:300, "hml"] f3 <- Finance[1:300, "smb"] y <- Finance[1:300,"WMK"] ## Silly example just to make it over-identified ############################################### res <- gmm(y ~ f1, ~ f1 + f2 + f3) summary(res) ## Set the bandwidth using the second step estimate ################################################ bw <- bwWilhelm(res) res2 <- update(res, bw=bw) summary(res2) ## Set the bandwidth using the first-step estimate as for bwAndrews ################################################################### res3 <- gmm(y ~ f1, ~ f1 + f2 + f3, bw=bwWilhelm) summary(res3) } gmm/DESCRIPTION0000644000176200001440000000234214020351121012526 0ustar liggesusersPackage: gmm Version: 1.6-6 Date: 2021-02-07 Title: Generalized Method of Moments and Generalized Empirical Likelihood Author: Pierre Chausse Maintainer: Pierre Chausse Description: It is a complete suite to estimate models based on moment conditions. It includes the two step Generalized method of moments (Hansen 1982; ), the iterated GMM and continuous updated estimator (Hansen, Eaton and Yaron 1996; ) and several methods that belong to the Generalized Empirical Likelihood family of estimators (Smith 1997; , Kitamura 1997; , Newey and Smith 2004; , and Anatolyev 2005 ). Depends: R (>= 2.10.0), sandwich NeedsCompilation: yes Suggests: knitr, mvtnorm, car, stabledist, MASS, timeDate, timeSeries Imports: stats, methods, grDevices, graphics License: GPL (>= 2) VignetteBuilder: knitr Repository: CRAN Repository/R-Forge/Project: gmm Repository/R-Forge/Revision: 184 Repository/R-Forge/DateTimeStamp: 2021-03-02 21:28:59 Date/Publication: 2021-03-05 06:40:17 UTC Packaged: 2021-03-02 21:47:29 UTC; rforge gmm/build/0000755000176200001440000000000014017531161012130 5ustar liggesusersgmm/build/vignette.rds0000644000176200001440000000057014017531161014471 0ustar liggesusers}RMO0 ڍ}8+Vܦx[D>4nIVImgy(0FhFcNp]`t˂r11hinZP`_ȽL 䉿5Hs k f<,4U =؁w9(M܄gvqԂO, a 6O!OG!4P'*Γ #])phu #include #include "gmm.h" static const R_FortranMethodDef fortranMethods[] = { {"wu", (DL_FUNC) &F77_SUB(wu), 9}, {"lamcuep", (DL_FUNC) &F77_SUB(lamcuep), 9}, {NULL, NULL, 0} }; void R_init_gmm(DllInfo *dll) { R_registerRoutines(dll, NULL, NULL, fortranMethods, NULL); R_useDynamicSymbols(dll, FALSE); } gmm/src/Makevars0000644000176200001440000000006513464357722013333 0ustar liggesusersPKG_LIBS=$(LAPACK_LIBS) $(BLAS_LIBS) $(FLIBS) -lgomp gmm/src/gmm.h0000644000176200001440000000053013663503426012560 0ustar liggesusers#ifndef R_GMM_H #define R_GMM_H #include void F77_SUB(wu)(double *gt, double *tol, int *maxit, int *n, int *q, double *k, int *conv, double *obj, double *lam); void F77_SUB(lamcuep)(double *gt, int *n, int *q, double *k, int *maxit, int *conv, double *lam, double *pt, double *obj); #endif gmm/src/lambda_met.f0000644000176200001440000000745113467070535014076 0ustar liggesusers subroutine prep(gt, lam, n, q, d2) integer n, q, i, info, ip(q) double precision gt(n,q), lam(q), d2(q), dd(q,q) double precision tmp(n), tmp2(n), tmp3(n,q) call dgemv('n', n, q, 1.0d0, gt, n, lam, 1, 0.0d0, tmp, 1) tmp = 1/(1+tmp) call dgemv('t', n, q, 1.0d0, gt, n, tmp, 1, 0.0d0, d2, 1) tmp2 = tmp**2 do i=1,q tmp3(:,i) = -gt(:,i)*tmp2 end do call dgemm('t','n',q,q,n,1.0d0, gt, n, tmp3, n, 0.0d0, dd, q) call dgesv(q, 1, dd, q, ip, d2, q, info) end subroutine wu(gt, tol, maxit, n, q, k, conv, obj, lam) integer n, q, i, conv, maxit double precision obj, lam(q), tol, gt(n,q), k double precision dif, d2(q), r, tmp(n), tmp2(q) lam = 0.0d0 i = 1 dif = 1.0d0 do while (dif>tol .and. i <= maxit) call prep(gt, lam, n, q, d2) dif = maxval(abs(d2)) r = 1.0d0 do while (r>0) r = 0.0d0 tmp2 = lam-d2 call dgemv('n', n, q, 1.0d0, gt, n, tmp2, 1, 0.0d0, * tmp, 1) if (minval(tmp)<=-1) then r = r+1 end if if (r>0) then d2 = d2/2 end if end do lam = tmp2 i = i+1 end do if (i>=maxit) then lam = 0.0d0 conv = 1 else lam = -lam conv = 0 end if obj = sum(log(1+tmp*k))/n end subroutine ols(x, y, n, m, lwork, nrhs, info, coef) integer n, m, lwork, nrhs, info double precision x(n,m), y(n,nrhs), work(lwork) double precision coef(m,nrhs) double precision xtmp(n,m), ytmp(n,nrhs) xtmp = x ytmp = y call dgels('n', n, m, nrhs, xtmp, n, ytmp, n, work, -1, info) lwork = min(m*n, int(work(1))) if (info == 0) then call dgels('n',n,m,nrhs,xtmp,n,ytmp,n,work,lwork,info) coef = ytmp(1:m,:) end if end subroutine lamcue(gt, n, q, k, lam, pt, obj) integer n, q, lwork, info double precision gt(n,q), lam(q), one(n), pt(n), obj, k lwork = q*3 one = -1.0d0 call ols(gt, one, n, q, lwork, 1, info, lam) call dgemv('n', n, q, 1.0d0, gt, n, lam, 1, 0.0d0, * pt, 1) pt = pt*k obj = sum(-pt-(pt**2)/2)/n pt = 1+pt where (pt < 0) pt = 0.0d0 end where pt = pt/sum(pt) end subroutine getpt(gt,n,q,k,lam,pt) integer n, q double precision gt(n,q), lam(q), k, pt(n) call dgemv('n', n, q, 1.0d0, gt, n, lam, 1, 0.0d0, * pt, 1) pt = 1.0d0 + pt*k where (pt < 0) pt = 0.0d0 end where pt = pt/sum(pt) end subroutine lamcuep(gt, n, q, k, maxit, conv, lam, pt, obj) integer n, q, i, maxit, n0, n1, conv, ind(n), wi(n) double precision gt(n,q), lam(q), pt(n), obj, pt0(n), k double precision pt2(n) logical w(n) call lamcue(gt, n, q, k, lam, pt, obj) ind = (/ (i, i=1,n) /) i = 1 conv = 0 do w = pt > 0 n1 = count(w) n0 = n-n1 if (n1 < (q+1)) then pt = 1.0d0/n conv = 2 obj = 0.0d0 lam = 0.0d0 exit end if if (i > maxit) then pt = 1.0d0/n conv = 1 obj = 0.0d0 lam = 0.0d0 exit end if wi(1:n1) = pack(ind, w) call lamcue(gt(wi(1:n1),:), n1, q, k, lam, * pt0(1:n1), obj) call getpt(gt,n,q,k,lam,pt2) if (all(pt==pt2)) then exit end if pt = pt2 i = i+1 end do if (conv == 0) then obj = obj*n1/n + dble(n0)/(2*n) end if end gmm/vignettes/0000755000176200001440000000000014017531161013041 5ustar liggesusersgmm/vignettes/empir.bib0000644000176200001440000004743414010606726014652 0ustar liggesusers @article{jagannathan-skoulakis02, AUTHOR={Jagannathan, R. and Skoulakis, G.}, TITLE={Generalized Method of Moments: Applications in Finance}, JOURNAL={Journal of Business and Economic Statistics}, VOLUME={20}, PAGES={470-481}, YEAR={2002}, Number={4} } @article{garcia-renault-veredas06, AUTHOR={Garcia, R. and Renault, E. and Veredas, D.}, TITLE={Estimation of Stable Distribution by Indirect Inference}, JOURNAL={Working Paper: UCL and CORE}, VOLUME={}, PAGES={}, YEAR={2006}, Number={} } @book{nolan09, author = {J. 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E. and Berkowitz, J.}, TITLE={Dynamic Equilibrium Economies: A Framework for Comparing Models and Data}, JOURNAL={Review of Economic Studies}, VOLUME={65}, PAGES={433-451}, YEAR={1998} } @article{Berkowitz01, NUMBER={}, AUTHOR={Berkowitz, J.}, TITLE={Generalized Spectral Estimation of the Consumption based Pricing Model}, JOURNAL={Journal of Econometrics}, VOLUME={104}, PAGES={269-288}, YEAR={2001} } @article{christiano-vigfusson03, NUMBER={4}, AUTHOR={Christiano, J. L. and Vigfusson, R. J.}, TITLE={Maximum Likelihood in the Frequency Domain: The Importance of Time-to-Plan}, JOURNAL={Journal of Monetary Economics}, VOLUME={50}, PAGES={}, YEAR={2003} } @article{seidman-vogel89, NUMBER={}, AUTHOR={Seidman, T. I. and Vogel, C. R.}, TITLE={Well Posedness and Convergence of some Regularisation Methods for Non-linear Ill-posed Problems}, JOURNAL={Inverse Problems}, VOLUME={5}, PAGES={227-238}, YEAR={1989} } @Book{hamilton94, author = {Hamilton, J. D.}, publisher = {Princeton University Press}, title = {Time Series Analysis}, year = {1994}, } gmm/vignettes/gmm_with_R.rnw0000644000176200001440000017532314011255710015675 0ustar liggesusers\documentclass[11pt,letterpaper]{article} \usepackage{amsthm} \usepackage[hmargin=2cm,vmargin=2.5cm]{geometry} \newtheorem{theorem}{Theorem} \newtheorem{col}{Corollary} \newtheorem{lem}{Lemma} \usepackage[utf8x]{inputenc} \newtheorem{ass}{Assumption} \usepackage{amsmath} \usepackage{verbatim} \usepackage[round]{natbib} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \usepackage{hyperref} \hypersetup{ colorlinks, citecolor=black, filecolor=black, linkcolor=black, urlcolor=black } \bibliographystyle{plainnat} \newcommand{\E}{\mathrm{E}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\Prob}{\mathrm{Pr}} \newcommand{\Var}{\mathrm{Var}} \let\proglang=\textsf \newcommand{\pkg}[1]{{\fontseries{m}\fontseries{b}\selectfont #1}} \newcommand{\Vect}{\mathrm{Vec}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\conP}{\overset{p}{\to}} \newcommand{\conD}{\overset{d}{\to}} \newcommand\Real{ \mathbb{R} } \newcommand\Complex{ \mathbb{C} } \newcommand\Natural{ \mathbb{N} } \newcommand\rv{{\cal R}} \newcommand\Q{\mathbb{Q}} \newcommand\PR{{\cal R}} \newcommand\T{{\cal T}} \newcommand\Hi{{\cal H}} \newcommand\La{{\cal L}} \newcommand\plim{plim} \renewcommand{\epsilon}{\varepsilon} \begin{document} \author{Pierre Chauss\'e} \title{Computing Generalized Method of Moments and Generalized Empirical Likelihood with \proglang{R}} \maketitle \abstract{This paper shows how to estimate models by the generalized method of moments and the generalized empirical likelihood using the \proglang{R} package \textbf{gmm}. A brief discussion is offered on the theoretical aspects of both methods and the functionality of the package is presented through several examples in economics and finance. It is a modified version of \cite{chausse10} published in the Journal of Statistical Software. It has been adapted to the version 1.4-0. \textbf{Notice that the maintenance of the package is converging to zero. The new \pkg{momentfit} package, available on CRAN, will soon replace the \pkg{gmm} package.}} %\VignetteIndexEntry{Computing Generalized Empirical Likelihood and Generalized Method of Moments with R} %\VignetteDepends{gmm,mvtnorm,stabledist, car, MASS, timeDate, timeSeries} %\VignetteKeywords{generalized empirical likelihood, generalized method of moments, empirical likelihood, continuous updated estimator, exponential tilting, exponentially tilted empirical likelihood} %\VignettePackage{gmm} %\VignetteEngine{knitr::knitr} <>= library(knitr) opts_chunk$set(size='footnotesize', fig.height=5, out.width='70%') @ \section{Introduction} The generalized method of moments (GMM) has become an important estimation procedure in many areas of applied economics and finance since \cite{hansen82} introduced the two step GMM (2SGMM). It can be seen as a generalization of many other estimation methods like least squares (LS), instrumental variables (IV) or maximum likelihood (ML). As a result, it is less likely to be misspecified. The properties of the estimators of LS depend on the exogeneity of the regressors and the circularity of the residuals, while those of ML depend on the choice of the likelihood function. GMM is much more flexible since it only requires some assumptions about moment conditions. In macroeconomics, for example, it allows to estimate a structural model equation by equation. In finance, most data such as stock returns are characterized by heavy-tailed and skewed distributions. Because it does not impose any restriction on the distribution of the data, GMM represents a good alternative in this area as well. As a result of its popularity, most statistical packages like \proglang{Matlab}, \proglang{Gauss} or \proglang{Stata} offer tool boxes to use the GMM procedure. It is now possible to easily use this method in \proglang{R} with the new \pkg{gmm} package. Although GMM has good potential theoretically, several applied studies have shown that the properties of the 2SGMM may in some cases be poor in small samples. In particular, the estimators may be strongly biased for certain choices of moment conditions. In response to this result, \cite{hansen-heaton-yaron96} proposed two other ways to compute GMM: the iterative GMM (ITGMM) and the continuous updated GMM (CUE)\footnote{See also \cite{hall05} for a detailed presentation of most recent developments regarding GMM.}. Furthermore, another family of estimation procedures inspired by \cite{owen01}, which also depends only on moment conditions, was introduced by \cite{smith97}. It is the generalized empirical likelihood (GEL). So far, this method has not reached the popularity of GMM and it was not included in any statistical package until \pkg{gmm} was developed for \proglang{R} which also includes a GEL procedure. Asymptotic properties of GMM and generalized empirical likelihood (GEL) are now well established in the econometric literature. \cite{newey-smith04} and \cite{anatolyev05} have compared their second order asymptotic properties. In particular, they show that the second order bias of the empirical likelihood (EL) estimator, which is a special case of GEL, is smaller than the bias of the estimators from the three GMM methods. Furthermore, as opposed to GMM, the bias does not increase with the number of moment conditions. Since the efficiency improves when the number of conditions goes up, this is a valuable property. However, these are only asymptotic results which do not necessarily hold in small sample as shown by \cite{guggenberger08}. In order to analyze small sample properties, we have to rely on Monte Carlo simulations. However, Monte Carlo studies on methods such as GMM or GEL depend on complicated algorithms which are often home made. Because of that, results from such studies are not easy to reproduce. The solution should be to use a common tool which can be tested and improved upon by the users. Because it is open source, \proglang{R} offers a perfect platform for such tool. The \pkg{gmm} package allows to estimate models using the three GMM methods, the empirical likelihood and the exponential tilting, which belong to the family of GEL methods, and the exponentially tilted empirical likelihood which was proposed by \cite{schennach07}, Also it offers several options to estimate the covariance matrix of the moment conditions. Users can also choose between \textit{optim}, if no restrictions are required on the coefficients of the model to be estimated, and either \textit{nlminb} or \textit{constrOptim} for constrained optimizations. The results are presented in such a way that \proglang{R} users who are familiar with \textit{lm} objects, find it natural. In fact, the same methods are available for \textit{gmm} and \textit{gel} objects produced by the estimation procedures. The paper is organized as follows. Section 2 presents the theoretical aspects of the GMM method along with several examples in economics and finance. Through these examples, the functionality of the \pkg{gmm} packages is presented in details. Section 3 presents the GEL method with some of the examples used in section 2. Section 4 concludes and Section 5 gives the computational details of the package. \section{Generalized method of moments} This section presents an overview of the GMM method. It is intended to help the users understand the options that the \pkg{gmm} package offers. For those who are not familiar with the method and require more details, see \cite{hansen82} and \cite{hansen-heaton-yaron96} for the method itself, \cite{newey-west94} and \cite{andrews91} for the choice of the covariance matrix or \cite{hamilton94}. We want to estimate a vector of parameters $\theta_0 \in \Real^p$ from a model based on the following $q\times 1$ vector of unconditional moment conditions: \begin{equation}\label{mcond} E[g(\theta_0,x_i)] = 0 , \end{equation} where $x_i$ is a vector of cross-sectional data, time series or both. In order for GMM to produce consistent estimates from the above conditions, $\theta_0$ has to be the unique solution to $E[g(\theta,x_i)]=0$ and be an element of a compact space. Some boundary assumptions on higher moments of $g(\theta,x_i)$ are also required. However, it does not impose any condition on the distribution of $x_i$, except for the degree of dependence of the observations when it is a vector of time series. Several estimation methods such as least squares (LS), maximum likelihood (ML) or instrumental variables (IV) can also be seen as being based on such moment conditions, which make them special cases of GMM. For example, the following linear model: \[ Y = X\beta + u , \] where $Y$ and $X$ are respectively $n\times 1$ and $n\times k$ matrices, can be estimated by LS. The estimate $\hat{\beta}$ is obtained by solving $\min_\beta \|u\|^2$ and is therefore the solution to the following first order condition: \[ \frac{1}{n}X'u(\beta) = 0 , \] which is the estimate of the moment condition $E(X_iu_i(\beta))=0$. The same model can be estimated by ML in which case the moment condition becomes: \[ E\left[\frac{d l_i(\beta)}{d\beta}\right]=0 , \] where $l_i(\beta)$ is the density of $u_i$. In presence of endogeneity of the explanatory variable $X$, which implies that $E(X_iu_i)\neq 0$, the IV method is often used. It solves the endogeneity problem by substituting $X$ by a matrix of instruments $H$, which is required to be correlated with $X$ and uncorrelated with $u$. These properties allow the model to be estimated by the conditional moment condition $E(u_i|H_i)=0$ or its implied unconditional moment condition $E(u_iH_i)=0$. In general we say that $u_i$ is orthogonal to an information set $I_i$ or that $E(u_i|I_i)=0$ in which case $H_i$ is a vector containing functions of any element of $I_i$. The model can therefore be estimated by solving \[ \frac{1}{T}H'u(\beta)=0 . \] When there is no assumption on the covariance matrix of $u$, the IV corresponds to GMM. If $E(X_iu_i)=0$ holds, generalized LS with no assumption on the covariance matrix of $u$ other than boundary ones is also a GMM method. For the ML procedure to be viewed as GMM, the assumption on the distribution of $u$ must be satisfied. If it is not, but $E(dl_i(\theta_0)/d\theta)=0$ holds, as it is the case for linear models with non normal error terms, the pseudo-ML which uses a robust covariance matrix can be seen as being a GMM method. Because GMM depends only on moment conditions, it is a reliable estimation procedure for many models in economics and finance. For example, general equilibrium models suffer from endogeneity problems because these are misspecified and they represent only a fragment of the economy. GMM with the right moment conditions is therefore more appropriate than ML. In finance, there is no satisfying parametric distribution which reproduces the properties of stock returns. The family of stable distributions is a good candidate but only the densities of the normal, Cauchy and Lévy distributions, which belong to this family, have a closed form expression. The distribution-free feature of GMM is therefore appealing in that case. Although GMM estimators are easily consistent, efficiency and bias depend on the choice of moment conditions. Bad instruments implies bad information and therefore low efficiency. The effects on finite sample properties are even more severe and are well documented in the literature on weak instruments. \cite{newey-smith04} show that the bias increases with the number of instruments but efficiency decreases. Therefore, users need to be careful when selecting the instruments. \cite{carrasco07} gives a good review of recent developments on how to choose instruments in her introduction. In general, the moment conditions $E(g(\theta_0,x_i))=0$ is a vector of nonlinear functions of $\theta_0$ and the number of conditions is not limited by the dimension of $\theta_0$. Since efficiency increases with the number of instruments $q$ is often greater than $p$, which implies that there is no solution to \[ \bar{g}(\theta) \equiv \frac{1}{n}\sum_{i=1}^n g(\theta,x_i)=0. \] The best we can do is to make it as close as possible to zero by minimizing the quadratic function $\bar{g}(\theta)'W\bar{g}(\theta)$, where $W$ is a positive definite and symmetric $q\times q$ matrix of weights. The optimal matrix $W$ which produces efficient estimators is defined as: \begin{equation}\label{optw} W^* = \left\{\lim_{n\rightarrow \infty} Var(\sqrt{n} \bar{g}(\theta_0)) \equiv \Omega(\theta_0) \right\}^{-1}. \end{equation} This optimal matrix can be estimated by an heteroskedasticity and auto-correlation consistent (HAC) matrix like the one proposed by \cite{newey-west87a}. The general form is: \begin{equation}\label{optw_hat} \hat{\Omega} = \sum_{s=-(n-1)}^{n-1} k_h(s) \hat{\Gamma}_s(\theta^*), \end{equation} where $k_h(s)$ is a kernel, $h$ is the bandwidth which can be chosen using the procedures proposed by \cite{newey-west87a} and \cite{andrews91}, \[ \hat{\Gamma}_s(\theta^*) = \frac{1}{n}\sum_i g(\theta^*,x_i)g(\theta^*,x_{i+s})' \] and $\theta^*$ is a convergent estimate of $\theta_0$. There are many choices for the HAC matrix. They depend on the kernel and bandwidth selection. Although the choice does not affect the asymptotic properties of GMM, very little is known about the impacts in finite samples. The GMM estimator $\hat{\theta}$ is therefore defined as: \begin{equation}\label{gmm} \hat{\theta} = \arg\min_{\theta} \bar{g}(\theta)'\hat{\Omega}(\theta^*)^{-1}\bar{g}(\theta) \end{equation} The original version of GMM proposed by \cite{hansen82} is called two-step GMM (2SGMM). It computes $\theta^*$ by minimizing $\bar{g}(\theta)'\bar{g}(\theta)$. The algorithm is therefore: \begin{itemize} \item[1-] Compute $\theta^* = \arg\min_\theta \bar{g}(\theta)'\bar{g}(\theta)$ \item[2-] Compute the HAC matrix $\hat{\Omega}(\theta^*)$ \item[3-] Compute the 2SGMM $\hat{\theta} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta^*)\big]^{-1}\bar{g}(\theta)$ \end{itemize} In order to improve the properties of 2SGMM, \cite{hansen-heaton-yaron96} suggest two other methods. The first one is the iterative version of 2SGMM (ITGMM) and can be computed as follows: \begin{itemize} \item[1-] Compute $\theta^{(0)} = \arg\min_\theta \bar{g}(\theta)'\bar{g}(\theta)$ \item[2-] Compute the HAC matrix $\hat{\Omega}(\theta^{(0)})$ \item[3-] Compute the $\theta^{(1)} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta^{(0)})\big]^{-1}\bar{g}(\theta)$ \item[4-] If $\| \theta^{(0)}-\theta^{(1)}\|< tol$ stops, else $\theta^{(0)}=\theta^{(1)}$ and go to 2- \item[5-] Define the ITGMM estimator $\hat{\theta}$ as $\theta^{(1)}$ \end{itemize} where $tol$ can be set as small as we want to increase the precision. In the other method, no preliminary estimate is used to obtain the HAC matrix. The latter is treated as a function of $\theta$ and is allowed to change when the optimization algorithm computes the numerical derivatives. It is therefore continuously updated as we move toward the minimum. For that, it is called the continuous updated estimator (CUE). This method is highly nonlinear. It is therefore crucial to choose a starting value that is not too far from the minimum. A good choice is the estimate from 2SGMM which is known to be root-n convergent. The algorithm is: \begin{itemize} \item[1-] Compute $\theta^*$ using 2SGMM \item[2-] Compute the CUE estimator defined as \[ \hat{\theta} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta)\big]^{-1}\bar{g}(\theta) \] using $\theta^*$ as starting value. \end{itemize} According to \cite{newey-smith04} and \cite{anatolyev05}, 2SGMM and ITGMM are second order asymptotically equivalent. On the other hand, they show that the second order asymptotic bias of CUE is smaller. The difference in the bias comes from the randomness of $\theta^*$ in $\Omega(\theta^*)$. Iterating only makes $\theta^*$ more efficient. These are second order asymptotic properties. They are informative but may not apply in finite samples. In most cases, we have to rely on numerical simulations to analyze the properties in small samples. Given some regularity conditions, the GMM estimator converges as $n$ goes to infinity to the following distribution: \[ \sqrt{n}(\hat{\theta}-\theta_0) \stackrel{L}{\rightarrow} N(0,V), \] where \[ V = E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right)'\Omega(\theta_0)^{-1}E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right) \] Inference can therefore be performed on $\hat{\theta}$ using the assumption that it is approximately distributed as $N(\theta_0,\hat{V}/n)$. If $q>p$, we can perform a J-test to verify if the moment conditions hold. The null hypothesis and the statistics are respectively $H0:E[g(\theta,x_i)]=0$ and: \[ n\bar{g}(\hat{\theta})'[\hat{\Omega}(\theta^*)]^{-1}\bar{g}(\hat{\theta}) \stackrel{L}{\rightarrow} \chi^2_{q-p}. \] \section{GMM with R} The \pkg{gmm} package can be loaded the usual way. <<>>= library(gmm) @ The main function is \textit{gmm()} which creates an object of class \textit{gmm}. Many options are available but in many cases they can be set to their default values. They are explained in details below through examples. The main arguments are \textit{g} and \textit{x}. For a linear model, \textit{g} is a formula like \textit{y~z1+z2} and \textit{x} the matrix of instruments. In the nonlinear case, they are respectively the function $g(\theta,x_i)$ and its argument. The available methods are \textit{coef}, \textit{vcov}, \textit{summary}, \textit{residuals}, \textit{fitted.values}, \textit{plot}, \textit{confint}. The model and data in a \textit{data.frame} format can be extracted by the generic function \textit{model.frame}. \subsection{Estimating the parameters of a normal distribution} This example\footnote{Thanks to Dieter Rozenich for his suggestion.}, is not something we want to do in practice, but its simplicity allows us to understand how to implement the \textit{gmm()} procedure by providing the gradient of $g(\theta,x_i)$. It is also a good example of the weakness of GMM when the moment conditions are not sufficiently informative. In fact, the ML estimators of the mean and the variance of a normal distribution are more efficient because the likelihood carries more information than few moment conditions. For the two parameters of a normal distribution $(\mu,\sigma)$ we have the following vector of moment conditions: \[ E[g(\theta,x_i)] \equiv E\left[ \begin{array}{c} \mu - x_{i} \\ \sigma^2 - (x_{i}-\mu)^2 \\ x_{i}^{3} - \mu (\mu^2+3\sigma^{2}) \end{array} \right] = 0 , \] where the first two can be directly obtained by the definition of $(\mu,\sigma)$ and the last comes from the third derivative of the moment generating function evaluated at 0. We first need to create a function $g(\theta,x)$ which returns an $n\times 3$ matrix: <<>>= g1 <- function(tet,x) { m1 <- (tet[1]-x) m2 <- (tet[2]^2 - (x - tet[1])^2) m3 <- x^3-tet[1]*(tet[1]^2+3*tet[2]^2) f <- cbind(m1,m2,m3) return(f) } @ The following is the gradient of $\bar{g}(\theta)$: \[ G\equiv \frac{\partial \bar{g}(\theta)}{\partial \theta} = \left( \begin{array}{cc} 1 & 0\\ 2(\bar{x}-\mu) & 2\sigma\\ -3(\mu^{2}+\sigma^{2}) & -6\mu\sigma \end{array} \right). \] If provided, it will be used to compute the covariance matrix of $\hat{\theta}$. It can be created as follows: <<>>= Dg <- function(tet,x) { G <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2],-6*tet[1]*tet[2]), nrow=3,ncol=2) return(G) } @ First we generate normally distributed random numbers: <<>>= set.seed(123) n <- 200 x1 <- rnorm(n, mean = 4, sd = 2) @ We then run \textit{gmm} using the starting values $(\mu_0,\sigma^2_0)=(0,0)$ <<>>= print(res <- gmm(g1,x1,c(mu = 0, sig = 0), grad = Dg)) @ The \textit{summary} method prints more results from the estimation: <<>>= summary(res) @ The section "Initial values of the coefficients" shows the first step estimates used to either compute the weighting matrix in the 2-step GMM or the fixed bandwidth in CUE or iterative GMM. The J-test of over-identifying restrictions can also be extracted by using the method \textit{specTest}: <<>>= specTest(res) @ A small simulation using the following function shows that ML produces estimators with smaller mean squared errors than GMM based on the above moment conditions. However, it is not GMM but the moment conditions that are not efficient, because ML is GMM with the likelihood derivatives as moment conditions. <<>>= sim_ex <- function(n,iter) { tet1 <- matrix(0,iter,2) tet2 <- tet1 for(i in 1:iter) { x1 <- rnorm(n, mean = 4, sd = 2) tet1[i,1] <- mean(x1) tet1[i,2] <- sqrt(var(x1)*(n-1)/n) tet2[i,] <- gmm(g1,x1,c(0,0),grad=Dg)$coefficients } bias <- cbind(rowMeans(t(tet1)-c(4,2)),rowMeans(t(tet2)-c(4,2))) dimnames(bias)<-list(c("mu","sigma"),c("ML","GMM")) Var <- cbind(diag(var(tet1)),diag(var(tet2))) dimnames(Var)<-list(c("mu","sigma"),c("ML","GMM")) MSE <- cbind(rowMeans((t(tet1)-c(4,2))^2),rowMeans((t(tet2)-c(4,2))^2)) dimnames(MSE)<-list(c("mu","sigma"),c("ML","GMM")) return(list(bias=bias,Variance=Var,MSE=MSE)) } @ The following results can be reproduced with $n=50$, $iter=2000$ and by setting \textit{set.seed(345)}: \begin{center} \begin{tabular}{|c|c|c|c||c|c|c|} \hline &\multicolumn{3}{c||}{$\mu$}&\multicolumn{3}{c}{$\sigma$} \\ \hline & Bias & Variance & MSE & Bias & Variance &MSE \\ \hline GMM& 0.0020 &0.0929 & 0.0928& -0,0838 & 0.0481& 0.0551\\ ML& 0.0021 &0.0823 & 0.0822 & -0.0349 & 0.0411& 0.0423 \\ \hline \end{tabular} \end{center} \subsection{Estimating the parameters of a stable distribution} The previous example showed that ML should be used when the true distribution is known. However, when the density does not have a closed form expression, we have to consider other alternatives. \cite{garcia-renault-veredas06} propose to use indirect inference and perform a numerical study to compare it with several other methods. One of them is GMM for a continuum of moment conditions and was suggested by \cite{carrasco-florens02}. It uses the fact that the characteristic function $E(e^{ix_i\tau})$, where $i$ is the imaginary number and $\tau\in\Real$, has a closed form expression (for more details on stable distribution, see \cite{nolan09}). The \pkg{gmm} package does not yet deal with continuum of moment conditions but we can choose a certain grid $\{\tau_1,..., \tau_q\}$ over a given interval and estimate the parameters using the following moment conditions: \[ E\left[e^{ix_i\tau_l}-\Psi(\theta;\tau_l) \right]=0~~\mbox{for}~~l=1,...,q~, \] where $\Psi(\theta;\tau_l)$ is the characteristic function. There is more than one way to define a stable distribution and it depends on the choice of parametrization. We will follow the notation of \cite{nolan09} and consider stable distributions $S(\alpha,\beta,\gamma,\delta;1)$, where $\alpha\in (0,2]$ is the characteristic exponent and $\beta\in[-1,1]$, $\gamma>0$ and $\delta\in\Real$ are respectively the skewness, the scale and the location parameters. The last argument defines which parametrization we use. The \pkg{stabledist} package of \cite{stabledist} offers a function to generate random variables from stable distributions and uses the same notation. This parametrization implies that: \[ \Psi(\theta;\tau_l) = \left\{ \begin{array}{lcr} \exp{(-\gamma^\alpha|\tau_l|^\alpha[1-i\beta(\tan{\frac{\pi\alpha}{2}})(\mathrm{sign}(\tau_l))] + i\delta \tau_l)} &\mbox{for}&\alpha\neq 1\\ \exp{(-\gamma|\tau_l|[1+i\beta\frac{2}{\pi}(\mathrm{sign}(\tau_l))\log{|\tau_l|}] + i\delta \tau_l)} &\mbox{for}&\alpha= 1\\ \end{array} \right. , \] The function \textit{charStable} included in the package computes the characteristic function and can be used to construct $g(\theta,x_i)$. To avoid dealing with complex numbers, it returns the imaginary and real parts in separate columns because both should have zero expectation. The function is: <<>>= g2 <- function(theta,x) { tau <- seq(1,5,length.out=10) pm <- 1 x <- matrix(c(x),ncol=1) x_comp <- x%*%matrix(tau,nrow=1) x_comp <- matrix(complex(ima=x_comp),ncol=length(tau)) emp_car <- exp(x_comp) the_car <- charStable(theta,tau,pm) gt <- t(t(emp_car) - the_car) gt <- cbind(Im(gt),Re(gt)) return(gt) } @ The parameters of a simulated random vector can be estimated as follows (by default, $\gamma$ and $\delta$ are set to $1$ and $0$ respectively in \textit{rstable}). For the example, the starting values are the ones of a normal distribution with mean 0 and variance equals to \textit{var(x)}: <<>>= library(stabledist) set.seed(345) x2 <- rstable(500,1.5,.5,pm=1) t0 <- c(alpha = 2, beta = 0, gamma = sd(x2)/sqrt(2), delta = 0) print(res <- gmm(g2,x2,t0)) @ The result is not very close to the true parameters. But we can see why by looking at the J-test that is provided by the \textit{summary} method: <<>>= summary(res) @ The null hypothesis that the moment conditions are satisfied is rejected. For nonlinear models, a significant J-test may indicate that we have not reached the global minimum. Furthermore, the standard deviation of the coefficient of $\delta$ indicates that the covariance matrix is nearly singular. Notice also that the convergence code is equal to 1, which indicates that the algorithm did not converge. We could try different starting values, increase the number of iterations in the control option of \textit{optim} or use \textit{nlminb} which allows to put restrictions on the parameter space. The former would work but the latter will allow us to see how to select another optimizer. The option \textit{optfct} can be modified to use this algorithm instead of \textit{optim}. In that case, we can specify the upper and lower bounds of $\theta$. <<>>= res2 <- gmm(g2,x2,t0,optfct="nlminb",lower=c(0,-1,0,-Inf),upper=c(2,1,Inf,Inf)) summary(res2) @ Although the above modification solved the convergence problem, there is another issue that we need to address. The first step estimate used to compute the weighting matrix is almost identical to the starting values. There is therefore a convergence problem in the first step. In fact, choosing the initial $\alpha$ to be on the boundary was not a wise choice. Also, it seems that an initial value of $\beta$ equals to zero makes the objective function harder to minimize. Having a gobal minimum for the first step estimate is important if we care about efficiency. A wrong estimate will cause the weighting matrix not being a consistent estimate of the optimal matrix. The information about convergence is included in the argument 'initialAlgoInfo' of the gmm object. We conclude this example by estimating the parameters for a vector of stock returns from the data set \textit{Finance} that comes with the \pkg{gmm} package. <>= data(Finance) x3 <- Finance[1:1500,"WMK"] t0<-c(alpha = 1.8, beta = 0.1, gamma = sd(x3)/sqrt(2),delta = 0) res3 <- gmm(g2,x3,t0,optfct="nlminb") summary(res3) @ For this sub-sample, the hypothesis that the return follows a stable distribution is rejected. The normality assumption can be analyzed by testing $H_0:\alpha=2,\beta=0$ using \textit{linearHypothesis} from the \pkg{car} package: <<>>= library(car) linearHypothesis(res3,cbind(diag(2),c(0,0),c(0,0)),c(2,0)) @ It is clearly rejected. The result is even stronger if the whole sample is used. \subsection{A linear model with iid moment conditions} We want to estimate a linear model with an endogeneity problem. It is the model used by \cite{carrasco07} to compare several methods which deal with the many instruments problem. We want to estimate $\delta$ from: \[ y_i = \delta W_i + \epsilon_i \] with $\delta=0.1$ and \[ W_i = e^{-x_i^2} + u_i , \] where $(\epsilon_i,u_i) \sim ~iidN(0,\Sigma)$ with \[ \Sigma = \left( \begin{array}{cc} 1&0.5\\ 0.5&1 \end{array} \right) \] Any function of $x_i$ can be used as an instrument because it is orthogonal to $\epsilon_i$ and correlated with $W_i$. There is therefore an infinite number of possible instruments. For this example, $(x_i,x_i^2,x_i^3)$ will be the selected instruments and the sample size is set to $n=400$: <<>>= library(mvtnorm) set.seed(112233) sig <- matrix(c(1,.5,.5,1),2,2) n <- 400 e <- rmvnorm(n,sigma=sig) x4 <- rnorm(n) w <- exp(-x4^2) + e[,1] y <- 0.1*w + e[,2] @ where \textit{rmvnorm} is a multivariate normal distribution random generator which is included in the package \pkg{mvtnorm} (\cite{mvtnorm}). For a linear model, the $g$ argument is a formula that specifies the right- and left-hand sides as for \textit{lm} and $x$ is the matrix of instruments: <<>>= h <- cbind(x4, x4^2, x4^3) g3 <- y~w @ By default, an intercept is added to the formula and a vector of ones to the matrix of instruments. It implies the following moment conditions: \[ E\left(\begin{array}{c} (y_i-\alpha-\delta W_i)\\ (y_i-\alpha-\delta W_i)x_i\\ (y_i-\alpha-\delta W_i)x_i^2\\ (y_i-\alpha-\delta W_i)x_i^3 \end{array} \right)=0 \] In order the remove the intercept, -1 has to be added to the formula. In that case there is no column of ones added to the matrix of instruments. To keep the condition that the expected value of the error terms is zero, the column of ones needs to be included manually. We know that the moment conditions of this example are iid. Therefore, we can add the option \textit{vcov="iid"}. This option tells \textit{gmm} to estimate the covariance matrix of $\sqrt{n}\bar{g}(\theta^*)$ as follows: \[ \hat{\Omega}(\theta^*) = \frac{1}{n}\sum_{i=1}^n g(\theta^*,x_i)g(\theta^*,x_i)' \] However, it is recommended not to set this option to ``iid" in practice with real data because one of the reasons we want to use GMM is to avoid such restrictions. Finally, it is not necessary to provide the gradient when the model is linear since it is already included in \textit{gmm}. The first results are: <<>>= summary(res <- gmm(g3,x=h)) @ By default, the 2SGMM is computed. Other methods can be chosen by modifying the option ``type". The second possibility is ITGMM: <<>>= res2 <- gmm(g3,x=h,type='iterative',crit=1e-8,itermax=200) coef(res2) @ The procedure iterates until the difference between the estimates of two successive iterations reaches a certain tolerance level, defined by the option \textit{crit} (default is $10^{-7}$), or if the number of iterations reaches \textit{itermax} (default is 100). In the latter case, a message is printed to indicate that the procedure did not converge. The third method is CUE. As you can see, the estimates from ITGMM is used as starting values. However, the starting values are required only when \textit{g} is a function. When \textit{g} is a formula, the default starting values are the ones obtained by setting the matrix of weights equal to the identity matrix. <<>>= res3 <- gmm(g3,x=h,res2$coef,type='cue') coef(res3) @ It is possible to produce confidence intervals by using the method \textit{confint}: <<>>= confint(res3,level=.90) @ Whether \textit{optim} or \textit{nlminb} is used to compute the solution, it is possible to modify their default options by adding \textit{control=list()}. For example, you can keep track of the convergence with \textit{control=list(trace=TRUE)} or increase the number of iterations with \textit{control=list(maxit=1000)}. You can also choose the \textit{BFGS} algorithm with \textit{method="BFGS"} (see \textit{help(optim)} for more details). The methods \textit{fitted} and \textit{residuals} are also available for linear models. We can compare the fitted values of \textit{lm} with the ones from \textit{gmm} to see why this model cannot be estimated by LS. \begin{center} <<>>= plot(w,y,main="LS vs GMM estimation") lines(w,fitted(res),col=2) lines(w,fitted(lm(y~w)),col=3,lty=2) lines(w,.1*w,col=4,lty=3) legend("topleft",c("Data","Fitted GMM","Fitted LS","True line"),pch=c(1,NA,NA,NA),col=1:3,lty=c(NA,1,2,3)) @ \end{center} The LS seems to fit the model better. But the graphics hides the endogeneity problem. LS overestimates the relationship between $y$ and $w$ because it does not take into account the fact that some of the correlation is caused by the fact that $y_i$ and $w_i$ are positively correlated with the error term $\epsilon_i$. Finally, the \textit{plot} method produces some graphics to analyze the properties of the residuals. It can only be applied to \textit{gmm} objects when \textit{g} is a formula because when \textit{g} is a function, residuals are not defined. \subsection{Estimating the AR coefficients of an ARMA process} \label{ar} The estimation of auto-regressive coefficients of ARMA(p,q) processes is better performed by ML or nonlinear LS. But in Monte Carlo experiments, it is often estimated by GMM to study its properties. It gives a good example of linear models with endogeneity problems in which the moment conditions are serially correlated and possibly conditionally heteroskedastic. As opposed to the previous example, the choice of the HAC matrix becomes an important issue. We want to estimate the AR coefficients of the following process: \[ X_t = 1.4 X_{t-1} - 0.6X_{t-2} + u_t \] where $u_t = 0.6\epsilon_{t-1} -0.3 \epsilon_{t-2} + \epsilon_t$ and $\epsilon_t\sim iidN(0,1)$. This model can be estimated by GMM using any $X_{t-s}$ for $s>2$, because they are uncorrelated with $u_t$ and correlated with $X_{t-1}$ and $X_{t-2}$. However, as $s$ increases the quality of the instruments decreases since the stationarity of the process implies that the auto-correlation goes to zero. For this example, the selected instruments are $(X_{t-3},X_{t-4},X_{t-5},X_{t-6})$ and the sample size equals 400. The ARMA(2,2) process is generated by the function \textit{arima.sim}: <<>>= t <- 400 set.seed(345) x5 <- arima.sim(n=t,list(ar=c(1.4,-0.6),ma=c(0.6,-0.3))) x5t<-cbind(x5) for (i in 1:6) x5t<-cbind(x5t,lag(x5,-i)) x5t<-na.omit(x5t) g4<-x5t[,1]~x5t[,2]+x5t[,3] res<-gmm(g4,x5t[,4:7]) summary(res) @ The optimal matrix, when moment conditions are based on time series, is an HAC matrix which is defined by equation (\ref{optw_hat}). Several estimators of this matrix have been proposed in the literature. Given some regularity conditions, they are asymptotically equivalent. However, their impacts on the finite sample properties of GMM estimators may differ. The \pkg{gmm} package uses the \pkg{sandwich} package to compute these estimators which are well explained by \cite{zeileis06} and \cite{zeileis04}. We will therefore briefly summarize the available options. The option \textit{kernel} allows to choose between five kernels: Truncated, Bartlett, Parzen, Tukey-Hanning and Quadratic spectral\footnote{The first three have been proposed by \cite{white84}, \cite{newey-west87a} and \cite{gallant87} respectively and the last two, applied to HAC estimation, by \cite{andrews91}. But the latter gives a good review of all five.}. By default, the Quadratic Spectral kernel is used as it was shown to be optimal by \cite{andrews91} with respect to some mean squared error criterion. In most statistical packages, the Bartlett kernel is used for its simplicity. It makes the estimation of large models less computationally intensive. It may also make the \textit{gmm} algorithm more stable numerically when dealing with highly nonlinear models, especially with CUE. We can compare the results with different choices of kernel: <<>>= res2 <- gmm(g4,x=x5t[,4:7],kernel="Truncated") coef(res2) res3 <- gmm(g4,x=x5t[,4:7],kernel="Bartlett") coef(res3) res4 <- gmm(g4,x=x5t[,4:7],kernel="Parzen") coef(res4) res5<- gmm(g4,x=x5t[,4:7],kernel="Tukey-Hanning") coef(res5) @ The similarity of the results is not surprising since the matrix of weights should only affect the efficiency of the estimator. We can compare the estimated standard deviations using the method \textit{vcov}: <<>>= diag(vcov(res2))^.5 diag(vcov(res3))^.5 diag(vcov(res4))^.5 diag(vcov(res5))^.5 @ which shows, for this example, that the Bartlett kernel generates the estimates with the smallest variances. However, it does not mean it is better. We have to run simulations and compute the true variance if we want to compare them. In fact, we do not know which one produces the most accurate estimate of the variance. The second options is for the bandwidth selection. By default it is the automatic selection proposed by \cite{andrews91}. It is also possible to choose the automatic selection of \cite{newey-west94} by adding \textit{bw=bwNeweyWest} (without quotes because \textit{bwNeweyWest} is a function). A prewhitened kernel estimator can also be computed using the option \textit{prewhite=p}, where $p$ is the order of the vector auto-regressive (VAR) used to compute it. By default, it is set to \textit{FALSE}. \cite{andrews-monahan92} show that a prewhitened kernel estimator improves the properties of hypothesis tests on parameters. Finally, the \textit{plot} method can be applied to \textit{gmm} objects to do a Q-Q plot of the residuals: \begin{center} <<>>= plot(res,which=2) @ \end{center} or to plot the observations with the fitted values: \begin{center} <<>>= plot(res,which=3) @ \end{center} \subsection{Estimating a system of equations: CAPM} We want to test one of the implications of the capital asset pricing model (CAPM). This example comes from \cite{campbell-lo-mackinlay96}. It shows how to apply the \pkg{gmm} package to estimate a system of equations. The theory of CAPM implies that $\mu_i-R_f = \beta_i (\mu_m-R_f)$ $\forall i$, where $\mu_i$ is the expected value of stock i's return, $R_f$ is the risk free rate and $\mu_m$ is the expected value of the market porfolio's return. The theory can be tested by running the following regression: \[ (R_t-R_f) = \alpha + \beta (R_{mt}-R_f) + \epsilon_t, \] where $R_t$ is a $N\times 1$ vector of observed returns on stocks, $R_{mt}$ if the observed return of a proxy for the market portfolio, $R_f$ is the interest rate on short term government bonds and $\epsilon_t$ is a vector of error terms with covariance matrix $\Sigma_t$. When estimated by ML or LS, $\Sigma$ is assumed to be fixed. However, GMM allows $\epsilon_t$ to be heteroskedastic and serially correlated. One implication of the CAPM is that the vector $\alpha$ should be zero. It can be tested by estimating the model with $(R_{mt}-R_f)$ as instruments, and by testing the null hypothesis $H_0:~\alpha=0$. The data, which are included in the package, are the daily returns of twenty selected stocks from January 1993 to February 2009, the risk-free rate and the three factors of Fama and French\footnote{The symbols of the stocks taken from \url{http://ca.finance.yahoo.com/} are ("WMK", "UIS", "ORB", "MAT", "ABAX","T", "EMR", "JCS", "VOXX", "ZOOM", "ROG", "GGG", "PC", "GCO", "EBF", "F", "FNM", "NHP", "AA", "TDW"). The four other series can be found on K. R. French's web site: \url{http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html}}. The following test is performed using the returns of 5 stocks and a sample size of 500\footnote{The choice of sample size is arbitrary. The purpose is to show how to estimate a system of equations not to test the CAPM. Besides, the $\beta$'s seem to vary over time. It is therefore a good practice to estimate the model using short periods.}. <<>>= data(Finance) r <- Finance[1:500,1:5] rm <- Finance[1:500,"rm"] rf <- Finance[1:500,"rf"] z <- as.matrix(r-rf) zm <- as.matrix(rm-rf) res <- gmm(z~zm,x=zm) coef(res) R <- cbind(diag(5),matrix(0,5,5)) c <- rep(0,5) linearHypothesis(res,R,c,test = "Chisq") @ where the asymptotic chi-square is used since the default distribution requires a normality assumption. The same test could have been performed using the names of the coefficients: <>= test <- paste(names(coef(res)[1:5])," = 0",sep="") linearHypothesis(res,test) @ Another way to test the CAPM is to estimate the restricted model ($\alpha=0$), which is over-identified, and to perform a J-test. Adding $-1$ to the formula removes the intercept. In that case, a column of ones has to be added to the matrix of instruments: <<>>= res2<-gmm(z~zm-1,cbind(1,zm)) specTest(res2) @ which confirms the non-rejection of the theory. \subsection{Testing the CAPM using the stochastic discount factor representation} In some cases the theory is directly based on moment conditions. When it is the case, testing the validity of these conditions becomes a way of testing the theory. \cite{jagannathan-skoulakis02} present several GMM applications in finance and one of them is the stochastic discount factor (SDF) representation of the CAPM. The general theory implies that $E(m_tR_{it})=1$ for all $i$, where $m_t$ is the SDF and $R_{it}$ the gross return ($1 + r_{it}$). It can be shown that if the CAPM holds, $m_t=\theta_0 + \theta_0 R_{mt}$ which implies the following moment conditions: \[ E\Big[R_{it}(\theta_0-\theta_1R_{mt})-1 \Big] = 0~~\mbox{for}~~ i=1,...,N \] which can be tested as follows: <<>>= g5 <- function(tet, x) { gmat <- (tet[1] + tet[2] * (1 + c(x[, 1]))) * (1 + x[, 2:6]) - 1 return(gmat) } res_sdf <- gmm(g5, x = as.matrix(cbind(rm, r)), c(0, 0)) specTest(res_sdf) @ which is consistent with the two previous tests. \subsection{Estimating continuous time processes by discrete time approximation} This last example also comes from \cite{jagannathan-skoulakis02}. We want to estimate the coefficients of the following continuous time process which is often used in finance for interest rates: \[ dr_t = (\alpha + \beta r_t)dt + \sigma r_t^\gamma dW_t, \] where $W_t$ is a standard Brownian motion. Special cases of this process are the Brownian motion with drift ($\beta=0$ and $\gamma = 0$), the Ornstein-Uhlenbeck process ($\gamma=0$) and the Cox-Ingersoll-Ross or square root process ( $\gamma = 1/2$). It can be estimated using the following discrete time approximation: \[ r_{t+1}-r_t = \alpha + \beta r_t + \epsilon_{t+1} \] with \[ E_t \epsilon_{t+1}=0,~~\mbox{and}~~E_t(\epsilon_{t+1}^2) = \sigma^2r_t^{2\gamma} \] Notice that ML cannot be used to estimate this model because the distribution depends on $\gamma$. In particular, it is normal for $\gamma=0$ and gamma for $\gamma=1/2$. It can be estimated by GMM using the following moment conditions: \[ E[g(\theta,x_t)] \equiv E\left(\begin{array}{c} \epsilon_{t+1}\\ \epsilon_{t+1}r_t\\ \epsilon_{t+1}^2-\sigma^2r_t^{2\gamma}\\ (\epsilon_{t+1}^2-\sigma^2r_t^{2\gamma})r_t \end{array} \right) = 0 \] The related \textit{g} function, with $\theta=\{\alpha, \beta, \sigma^2, \gamma\}$ is: <<>>= g6 <- function(theta, x) { t <- length(x) et1 <- diff(x) - theta[1] - theta[2] * x[-t] ht <- et1^2 - theta[3] * x[-t]^(2 * theta[4]) g <- cbind(et1, et1 * x[-t], ht, ht * x[-t]) return(g) } @ In order to estimate the model, the vector of interest rates needs to be properly scaled to avoid numerical problems. The transformed series is the annualized interest rates expressed in percentage. Also, the starting values are obtained using LS and some options for \textit{optim} need to be modified. <<>>= rf <- Finance[,"rf"] rf <- ((1 + rf/100)^(365) - 1) * 100 dr <- diff(rf) res_0 <- lm(dr ~ rf[-length(rf)]) tet0 <- c(res_0$coef, var(residuals(res_0)), 0) names(tet0) <- c("alpha", "beta", "sigma^2", "gamma") res_rf <- gmm(g6, rf, tet0, control = list(maxit = 1000, reltol = 1e-10)) coef(res_rf) @ \subsection{Comments on models with panel data} The \pkg{gmm} package is not directly built to easily deal with panel data. However, it is flexible enough to make it possible in most cases. To see that, let us consider the following model (see \cite{wooldridge02} for more details): \[ y_{it} = x_{it} \beta + a_i + \epsilon_{it}\mbox{ for } i=1,...,N\mbox{ and }t=1,...,T, \] where $x_{it}$ is $1\times k$, $\beta$ is $k\times 1$, $\epsilon_{it}$ is an error term and $a_i$ is an unobserved component which is specific to individual $i$. If $a_i$ is correlated with $x_{it}$, it can be removed by subtracting the average of the equation over time, which gives: \[ (y_{it}-\bar{y}_i) = (x_{it}-\bar{x}_i) \beta + (\epsilon_{it}-\bar{\epsilon}_i) \mbox{ for } i=1,...,N\mbox{ and } t=1,...,T, \] which can be estimated by \textit{gmm}. For example, if there are 3 individuals the following corresponds to the GMM fixed effects estimation: <>= y <- rbind(y1-mean(y1),y2-mean(y2),y3-mean(y3)) x <- rbind(x1-mean(x1),x2-mean(x2),x3-mean(x3)) res <- gmm(y~x,h) @ However, if $a_i$ is not correlated with $x_{it}$, the equation represents a random effects model. In that case, it is more efficient not to remove $a_i$ from the equation because of the information it carries about the individuals. The error terms are then combined in a single one, $\eta_{it}=(a_i + \epsilon_{it})$ to produce the linear model: \[ y_{it} = x_{it} \beta + \eta_{it} \] This model cannot be efficiently estimated by OLS because the presence of the common factor $a_i$ at each period implies that $\eta_{it}$ is serially correlated. However, GMM is well suited to deal with such specifications. The following will therefore produce a GMM random effects estimation: <>= y <- rbind(y1,y2,y3) x <- rbind(x1,x2,x3) res <- gmm(y~x,h) @ The package \pkg{plm} of \cite{plm} offers several functions to manipulate panel data. It could therefore be combined with \pkg{gmm} when estimating such models. It also offers a way to estimate them with its own GMM algorithm for panel data. \subsection{GMM and the sandwich package} In the \pkg{gmm} package, the estimation of the optimal weighting matrices are obtained using the \pkg{sandwich} package of \cite{zeileis06}. For example, the weighting matrix of the two-step GMM defined as: \[ W = \left[\lim_{n\rightarrow \infty} Var{(\sqrt{n}\bar{g})}\right]^{-1} \] is estimated as follows: <>= gt <- g(t0, x) V <- kernHAC(lm(gt~1),sandwich = FALSE) W <- solve(V) @ where $t0$ is any consistent estimate. As long as the optimal matrix is used, the covariance matrix of the coefficients can be estimated as follows: \[ (\hat{G}'W\hat{G})^{-1}/n \equiv (\hat{G}'\hat{V}^{-1}\hat{G})^{-1}/n, \] where $\hat{G}=d\bar{g}(\hat{\theta})/d\theta$ and $\hat{V}$ is obtained using \textit{kernHAC()}. It is not a sandwich covariance matrix and is computed using the \textit{vcov()} method included in \pkg{gmm}. However, if any other weighting matrix is used, say $W$, the estimated covariance matrix of the coefficients must then be estimated as follows: \[ (\hat{G}'W\hat{G})^{-1}\hat{G}'W\hat{V}W\hat{G}(\hat{G}'W\hat{G})^{-1}/n. \] A \textit{bread()} and \textit{estfun()} methods are available for \textit{gmm} objects which allows to compute the above matrix using the \pkg{sandwich} package. The \textit{bread()} method computes $(\hat{G}'W\hat{G})^{-1}$ while the \textit{estfun()} method returns a $T\times q$ matrix with the $t^{th}$ row equals to $g(\hat{\theta}, x_t) W \hat{G}$. The \textit{meatHAC()} method applied to the latter produces the right meat. Let us consider the example of section (\ref{ar}). Suppose we want to use the identity matrix to eliminate one source of bias, at the cost of lower efficiency. In that case, a consistent estimate of the covariance matrix is \[ (\hat{G}'\hat{G})^{-1}\hat{G}'\hat{V}\hat{G}(\hat{G}'\hat{G})^{-1}/n, \] which can be computed as: <<>>= print(res<-gmm(g4,x5t[,4:7],wmatrix="ident")) diag(vcovHAC(res))^.5 @ which is more robust than using \textit{vcov()}: <<>>= diag(vcov(res))^.5 @ Notice that it is possible to fixe $W$. Therefore, the above results can also be obtained as: <<>>= print(res<-gmm(g4,x5t[,4:7], weightsMatrix = diag(5))) @ In this case, the choice of the type of GMM is irrelevant since the weighting matrix is fixed. \section{Generalized empirical likelihood} The GEL is a new family of estimation methods which, as GMM, is based on moment conditions. It follows \cite{owen01} who developed the idea of empirical likelihood estimation which was meant to improve the confidence regions of estimators. We present here a brief discussion on the method without going into too much details. For a complete review, see \cite{smith97}, \cite{newey-smith04} or \cite{anatolyev05}. The estimation is based on \[ E(g(\theta_0,x_i))=0, \] which can be estimated in general by \[ \tilde{g}(\theta) = \sum_{i=1}^n p_i g(\theta,x_i) =0, \] where $p_i$ is called the implied probability associated with the observation $x_i$. For the GEL method, it is assumed that $q>p$ because otherwise it would correspond to GMM. Therefore, as it is the case for GMM, there is no solution to $\bar{g}(\theta)=0$. However, there is a solution to $\tilde{g}(\theta)=0$ for some choice of the probabilities $p_i$ such that $\sum_i p_i=1$. In fact, there is an infinite number of solutions since there are $(n+q)$ unknowns and only $q+1$ equations. GEL selects among them the one for which the distance between the vector of probabilities $p$ and the empirical density $1/n$ is minimized. The empirical likelihood of \cite{owen01} is a special case in which the distance is the likelihood ratio. The other methods that belong to the GEL family of estimators use different metrics. If the moment conditions hold, the implied probabilities carry a lot of information about the stochastic properties of $x_i$. For GEL, the estimations of the expected value of the Jacobian and the covariance matrix of the moment conditions, which are required to estimate $\theta$, are based on $p_i$ while in GMM they are estimated using $1/n$. \cite{newey-smith04} show that this difference explains partially why the second order properties of GEL are better. Another difference between GEL and GMM is how they deal with the fact that $g(\theta,x_i)$ can be a conditionally heteroskedastic and weakly dependent process. GEL does not require to compute explicitly the HAC matrix of the moment conditions. However, if it does not take it into account, its estimators may not only be inefficient but may also fail to be consistent. \cite{smith01} proposes to replace $g(\theta,x_i)$ by: \[ g^w(\theta,x_i) = \sum_{s=-m}^m w(s)g(\theta,x_{i-s}) \] where $w(s)$ are kernel based weights that sum to one (see also \cite{kitamura-stutzer97} and \cite{smith97}). The sample moment conditions become: \begin{equation}\label{gel_mcond} \tilde{g}(\theta) = \sum_{i=1}^n p_i g^w(\theta,x_i) =0 \end{equation} The estimator is defined as the solution to the following constrained minimization problem: \begin{eqnarray} \hat{\theta}_n &=& \arg\min_{\theta,p_i} \sum_{i=1}^n h_n(p_i) ,\\ && \mbox{subject to } \\ && \sum_{i=1}^n p_ig^w(\theta,x_i) = 0 ~~\mbox{and} \label{const}\\ && \sum_{i=1}^n p_i=1, \end{eqnarray} where $h_n(p_i)$ has to belong to the following Cressie-Read family of discrepancies: \[ h_n(p_i) = \frac{[\gamma(\gamma+1)]^{-1}[(np_i)^{\gamma+1}-1]}{n} . \] \cite{smith97} showed that the empirical likelihood method (EL) of \cite{owen01} ($\gamma=0$) and the exponential tilting of \cite{kitamura-stutzer97} ($\gamma=-1$) belong to the GEL family of estimators while \cite{newey-smith04} show that it is also the case for the continuous updated estimator of \cite{hansen-heaton-yaron96} ($\gamma=1$). What makes them part of the same GEL family of estimation methods is the existence of a dual problem which is defined as: \begin{equation}\label{gel_obj} \hat{\theta} = \arg\min_{\theta}\left[\max_{\lambda} P_n(\theta,\lambda) = \frac{1}{n}\sum_{i=1}^n\rho\left(\lambda'g^w(\theta,x_i)\right)\right] \end{equation} where $\lambda$ is the Lagrange multiplier associated with the constraint (\ref{const}) and $\rho(v)$ is a strictly concave function normalized so that $\rho'(0)=\rho''(0)=-1$. It can be shown that $\rho(v)=\ln{(1-v)}$ corresponds to EL , $\rho(v)=-\exp{(v)}$ to ET and to CUE if it is quadratic. The equivalence of the primal and dual problems can easily be verified by showing that they both share the same following first order conditions: \begin{equation}\label{lam} \sum_{i=1}^n p_i g^w(\theta,x_i) = 0, \end{equation} \begin{equation} \sum_{i=1}^n p_i \lambda'\left(\frac{\partial g^w(\theta,x_i)}{\partial \theta}\right) = 0 , \end{equation} with \begin{equation} p_i = \frac{1}{n}\rho'\left(\lambda'g^w(\theta,x_i)\right) . \end{equation} Equation (\ref{gel_obj}) represents a saddle point problem. The solution is obtained by solving simultaneously two optimization problems. We can solve for $\theta$ by minimizing $P_n(\theta,\lambda(\theta))$, where $\lambda(\theta)$ is the solution to $\arg\max_\lambda P_n(\theta,\lambda)$ for a given $\theta$. Therefore an optimization algorithm needs to be called inside the $P_n(\theta,\lambda)$ function. It makes the GEL very hard to implement numerically. For example, \cite{guggenberger08}, who analyzes the small sample properties of GEL, uses an iterative procedure based on the Newton method for $\lambda$ and a grid search for $\theta$ in order to confidently reach the absolute minimum. Using such iterative procedures for $\lambda$ makes the problem less computationally demanding and does not seem to affect the properties of the estimator of $\theta_0$. Indeed, \cite{guggenberger-hahn05} show that going beyond two iterations for $\lambda$ does not improve the second order asymptotic properties of the estimator of $\theta_0$. The function \textit{gel} offers two options. By default, $\lambda(\theta)$ is obtained by the following iterative method: \[ \lambda_l = \lambda_{l-1} - \left[\frac{1}{n}\sum_{i=1}^n \rho''(\lambda_{l-1}'g_t)g_tg_t'\right]^{-1}\left[\frac{1}{n} \sum_{i=1}^n \rho'(\lambda_{l-1}'g_i) g_i \right] \] starting with $\lambda=0$, which corresponds to its asymptotic value. The algorithm stops when $\|\lambda_l-\lambda_{l-1}\|$ reaches a certain tolerance level. The second option is to let \textit{optim} solve the problem. Then, as for \textit{gmm}, the minimization problem is solved either by \textit{optim}, \textit{nlminb} or \textit{constrOptim}. In order to test the over-identifying restrictions, \cite{smith04} proposes three tests which are all asymptotically distributed as a $\chi^2_{q-p}$. The first one is the J-test: \[ n \bar{g}^w(\hat{\theta})'[\hat{\Omega}(\hat{\theta})]^{-1}\bar{g}^w(\hat{\theta}), \] the second is a Lagrange multiplier test (LM): \[ LM = n\hat{\lambda}'\hat{\Omega}(\hat{\theta})\hat{\lambda} \] and the last one is a likelihood ratio test (LR): \[ LR = 2\sum_{i=1}^n\left[ \rho\left(\hat{\lambda}'g^w(\hat{\theta},x_i)\right) - \rho(0)\right] \] \section{GEL with R} \subsection{Estimating the parameters of a normal distribution} For this example, we can leave the option \textit{smooth} at its default value, which is \textit{FALSE}, because of the iid properties of $x$. A good starting value is very important for GEL. The best choice is the sample mean and the standard deviation. By default the option \textit{type} is set to \textit{EL}. The same methods that apply to \textit{gmm} objects, can also be applied to \textit{gel} objects. <<>>= tet0 <- c(mu = mean(x1), sig = sd(x1)) res_el <- gel(g1,x1,tet0) summary(res_el) @ Each Lagrange multiplier represents a shadow price of the constraint implied by moment condition. A binding constraint will produce a multiplier different from zero. Therefore, its value informs us on the validity of the moment condition. In the above results, the $\lambda$'s are significantly different from zero which would normally suggest that the moment conditions associated with them are violated. As a result, the LM test also rejects the null hypothesis since it is based on the $\lambda$'s. Notice that \textit{summary} reports two convergence codes, one for $\lambda$ et another for $\theta$. The ET and CUE estimates can be obtained as follows: <<>>= res_et <- gel(g1,x1,tet0,type="ET") coef(res_et) @ <<>>= res_cue <- gel(g1,x1,tet0,type="CUE") coef(res_cue) @ A fourth method is available which is called the exponentially tilted empirical likelihood (ETEL) and was proposed by \cite{schennach07}. However, it does not belong to the family of GEL estimators. It solves the problem of misspecified models. In such models there may not exist any pseudo value to which $\hat{\theta}$ converges as the sample size increases. ETEL uses the $\rho()$ of ET to solve for $\lambda$ and the $\rho()$ of EL to solve for $\theta$. \cite{schennach07} shows that ETEL shares the same asymptotic properties of EL without having to impose restrictions on the domain of $\rho(v)$ when solving for $\lambda$. <<>>= res_etel <- gel(g1,x1,c(mu=1,sig=1),type="ETEL") coef(res_etel) @ The type ETEL is experimental for now. Although it is supposed to be more stable because no restrictions are required to solve for $\lambda$, once we substitute $\lambda(\theta)$ in the EL objective function to estimate $\theta$, we still need to restrict $\lambda'g_t$ to avoid having NA's. The solution used in gel() is to obtain $\lambda(\theta)$ with \textit{constrOptim} with the restriction $\lambda'gt > 1$ even if it is not required by ET ($\rho(v)=-\exp{(v)}$). It is however sensitive to starting values. That's the reason why we used different ones above. \subsection{Estimating the AR coefficients of an ARMA process} Because the moment conditions are weakly dependent, we need to set the option \textit{smooth=TRUE}. Before going to the estimation procedure, we need to understand the relationship between the smoothing kernel and the HAC estimator. The reason why we need to smooth the moment function is that GEL estimates the covariance matrix of $\bar{g}(\theta,x_t)$, as if we had iid observations, using the expression $(1/T)\sum_{t=1}^T(g_tg_t')$. We can show that substituting $g_t$ by $g^w_t$ in this expression results in an HAC estimator. However, the relationship between the smoothing kernel and the kernel that appears in the HAC estimator is not obvious. For example, we can show that if the smoothing kernel is Truncated, then the kernel in the HAC estimator is the Bartlett. Let us consider the truncated kernel with a bandwidth of 2. This implies that $w(s)=1/5$ for $|s|\leq 2$ and 0 otherwise. Then, the expression for the covariance matrix becomes: \begin{eqnarray*} \frac{1}{T}\sum_{t=1}^T g^w_t (g^w_t)' &=& \frac{1}{T}\sum_{t=1}^T \left( \sum_{s=-2}^2 \frac{1}{5} g_{t+s} \right) \left( \sum_{l=-2}^2 \frac{1}{5} g_{t+l}' \right),\\ &=& \frac{1}{25}\sum_{s=-2}^2\sum_{l=-2}^2\left( \frac{1}{T}\sum_{t=1}^T g_{t+s}g_{t+l}' \right),\\ &=& \frac{1}{25}\sum_{s=-2}^2\sum_{l=-2}^2\hat{\Gamma}_{s-l} ,\\ &=& \frac{1}{25}\sum_{s=-4}^4 (5-|s|)\hat{\Gamma}_{s},\\ &=& \sum_{s=-4}^4 \left(\frac{1}{5}-\frac{|s|}{25}\right)\hat{\Gamma}_{s},\\ &=& \sum_{s=-T+1}^{T-1} k_5(s) \hat{\Gamma}_{s}, \end{eqnarray*} where $k_5(s)$ is the Bartlett kernel with a bandwidth of 5 defined as \[ K_5(s) = \left\{ \begin{array}{ccc} 1/5 - |s|/25 & \mbox{if}&|s|\leq 5\\ 0 &\mbox{otherwise}& \end{array}\right. . \] See \cite{smith01} for more details. The model will therefore be estimated using the kernel Truncated. The GMM estimate with the identity matrix is selected as starting value. <>= tet0 <- gmm(g4,x=x5t[,4:7],wmatrix="ident")$coef res <- gel(g4,x=x5t[,4:7],tet0,smooth=TRUE,kernel="Truncated") summary(res) @ The \textit{specTest} method applied to a \textit{gel} object computes the three tests proposed by \cite{smith04}: <<>>= specTest(res) @ The \textit{plot} method produces one more graphics when applied to a \textit{gel} object. It shows the implied probabilities along with the empirical density $(1/T)$. It allows to see which observations have more influence: \begin{center} <<>>= plot(res,which=4) @ \end{center} We can also select \textit{optfct="nlminb"} or \textit{constraint=TRUE} in order to impose restrictions on the coefficients. The former sets lower and upper bounds for the coefficients, while the latter imposes linear constraints using the algorithm \textit{constrOptim}. In this example we want the sum of the AR coefficients to be less than one. \textit{constrOptim} imposes the constraint $ui\theta-ci \geq 0$. Therefore, we need to set: <>= ui=cbind(0,-1,-1) ci <- -1 @ and rerun the estimation as <>= res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE,kernel="Truncated", constraint=TRUE, ui=ui,ci=ci) @ The result, which is not shown, is identical. They are also many option to compute the $\lambda$'s. From version 1.4-0, the default algorithm is \textit{nlminb} because the gradient and Hessian matrix are well defined analytically which speed up convergence. The other choices are \textit{optim} or "iter" which uses a Newton method to solve the first order condition. If the option optlam is set to "optim" and the type is EL, \textit{contrOptim} is selected automatically to restrict $\lambda'g_t$ to be less than 1. It is also possible to change the default values in the control list of the optimizer with the option LambdaControl (see \textit{?nlminb} or \textit{?optim}). Here are some examples: <>= res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim") res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim", LambdaControl=list(trace=TRUE, parscale=rep(.1,5))) @ \subsection{Comments} The GEL method is very unstable numerically. This fact has been reported many times in the recent literature. The method has been included in the \pkg{gmm} package because recent theoretical evidence suggests that it may produce better estimators than GMM. Because \proglang{R} is an open source statistical package, it offers a good platform to experiment with numerical properties of estimators. \section{Conclusion} The \pkg{gmm} package offers complete and flexible algorithms to estimate models by GMM and GEL. Several options are available which allow to choose among several GMM and GEL methods and many different HAC matrix estimators. In order to estimate the vector of parameters, users can select their preferred optimization algorithm depending on whether inequality constraints are required. For the vector of Lagrange multiplier of GEL, it can be computed by an iterative procedure based on the Newton method which increases the speed of convergence and reduce the instability of the estimation procedure. It could then easily be used by those who are interested in studying the numerical properties of both methods. The package also offers an interface which is comparable to the least squares method \textit{lm}. Linear model are estimated using formula and methods such as \textit{summary}, \textit{vcov}, \textit{coef}, \textit{confint}, \textit{plot}. \textit{residuals} or \textit{fitted} are available for the objects of class \textit{gmm} and \textit{gel}. \proglang{R} users will therefore have little difficulty in using the package. \section{Computational Details} The package \pkg{gmm} is written entirely in \proglang{R} and S3-classes with methods are used. It can be found on the comprehensive \proglang{R} archive network (CRAN,\url{http://CRAN.R-project.org/}). It is also hosted on R-Forge (\url{http://r-forge.r-project.org/projects/gmm}). It is shipped with a NAMESPACE. The version used to produce this paper is 1.4-0. It depends on the \pkg{sandwich} package of \cite{zeileis06}, which is used to compute de HAC matrices. The packages \pkg{car} (\cite{car}), \pkg{mvtnorm} (\cite{mvtnorm}), \pkg{stabledist} (\cite{stabledist}), \pkg{MASS} (\cite{MASS}), \pkg{timeDate} (\cite{timeDate}) and \pkg{timeSeries} (\cite{timeSeries}) are suggested in order to reproduce the examples. \section*{Acknowledgments} I am grateful to the three anonymous referees of the Journal of Statistical Software for great comments on the paper and the package. I also want to thank Achim Zeileis for his suggestions regarding the way the \pkg{sandwich} package can be used within \textit{gmm}. \bibliography{empir} \end{document} gmm/NEWS0000644000176200001440000004104113460072542011534 0ustar liggesusersChanges in version 1.6-2 o Fixed a bug with iterative GMM from the previous version o Added the option onlyCoefficients=FALSE to gmm() and gel(). If set to TRUE, no covariance matrix is computed. Only the coefficients objective and the lambdas (for gel only) are returned. o In previous versions, the HAC weighting matrix for linear models was computed without specifying that the first column of the instrument matrix was a column of ones (with its name being "(Intercept)"). It makes a difference when computing the bandwidth with bwNeweyWest. The new version, takes it into account. It is not possible to obtain the same sandwich covariance matrix with gmm and OLS. o In previous versions, the weighting matrix and, as a result, the standard errors were based on the small sample adjusted vcovHAC for weakly dependent processes. The adjustment is not justified for GMM. The option adjust=FALSE is therefore added to vcovHAC in .myKernHAC() o Added a method getImpProb, which extract the implied probabilities. It allows for negative probabilities as it often happens with CUE o lambda for CUE is solved analytically (it is minus the projection of gt on a vector of ones). The option RCue for optlam can be used to restrict the implied probabilities to be non-negative. Changes in version 1.6-1 o Added the function bwWilhelm() to compute the optimal bandwidth of Wilhelm(2015). o The argument $vcov of tsls object is the Classical covariance matrix. vcov() must be used to obtain a different covariance o Changed the default value of prewhite to 1 in gmm() because it is recommended for HAC estimation o gmm with optfct="optimize" was generating an error message. It has been fixed o A new set of functions to estimate the average treatment effect by GEL have been added. See ?ATEgel o Some other minor bugs that were reported by users were fixed. Changes in version 1.6-0 o tsls is now a real 2-stage least square. Before it was a 2-step optimal GMM with HCCM weighting matrix. o Fixed a typo in FinRes.R file. It was preventing to compute the proper vcov matrix for a very special case (fixed weights) o Added Helinger distance and Exponentially tilted Helinger estimator to gel() o Fixed the LR test of ETEL in gel() o Cleaned a lot of the codes. In particular, a single moment function and its gradiant function is used now. There is also a common weighting matrix generator function. The main goal is to make it more flexible and give the possibility of creating weighting matrix based on other assumption such as clustering. o For GEL with smooth=TRUE, the weights were not fixed which was creating problems of convergence. o Add the option of fixing the value of some coefficient in gel as it as already the case for gmmm. o It is no possible to provide gmm with the gradiant function when constrOptim is used. o There is now an evalGmm() function to create a gmm object with a given fixed vector. Not estimation is done, but the bandwidth and weights for the HAC estimator are generated. It is possible to give a different vector os parameters for the moment function and the weighting matrix. o There is also an evalGel(). You fix the parameters and only the lambda are computed. o Fixed NAMESPACE to avoid the notes given by CRAN. o Modified getDat to allow for ~1 as instruments. Useful for inference on the mean using empirical likelihood with gel(x~1,~1). o The Lambdas and specTest from gel are printed even if the model is just identified. It is consitent with gmm which prints the j-test for just identified models. The reason if to allow evalGel to generate the Lamdbas for testing purpose. o confint.gxx() has been modified. It now create an object and a print method has been created as well. o confint.gel includes the possibility of building a confidence interval using the inversion of one of the three specification tests. It is therefore possible de construct an Empirical Likelihood confidence interval as described in Owen 2001. See manual. o On the process to create a sysGmm for system og linear equations. The function sysGmm() will do the job. Not yet working. o getDat is modified so that the name of the dependent variable is the one included in the formula. o A set of functions have been added to estimate system of linear equations (this is a beta version not tested yet). o A set of labor data as been added to test the sysGmm function o A panel of Macro data has been added. Changes in version 1.5-2 o Added constrOptim to nonlinear gmm() with fixed W Changes in version 1.5-1 o Added importFrom element o Added element df.residual to gmm objects to allow F-type linearHypothesis o Removed the unnecessary ls estimations in the first stage 2SLS Changes in version 1.5-0 o Thanks to Eric Zivot. Many improvements are based on his testings of the package. o Fixed many bugs for the case in which vcov is set to iid o Added options to tsls(). There are options for different sandwich matrix when summary is called Changes in version 1.4-6 o Added the possibility of providing data.frame() with formula to gel() Changes in version 1.4-5 o Replaced the deprecated function real(), is.real() and as.real() by double() Changes in version 1.4-4 o Fixed an instability problem when the model is just identified and wmatrix is set to "ident" Changes in version 1.4-3 o Added the option "constrOptim" to optfct for nonlinear models o Added the option "eqConstFullVcov" to gmm() to allow the computation the covariance matrix of all coefficients including the constrained ones (see ?gmm for details and examples). o Removed the dependency on fBasics and replaced it with stabledist Changes in version 1.4-2 o Bugs fixed when running gel() with smooth=T and g() begina nonlinear function o Now, print.gel() and print.summary.gel() repports the bandwidth when smooth=T Changes in version 1.4-0 o The method for GMM-CUE has been modified. Before, the weights for the kernel estimation of the weighting matrix were flexible inside the optimizer which was making the algorithm long and unstable. It is now fixed using either the starting values provided by the user (for linear cases) or the first step GMM. o You can now trace the convergence of the iterative GMM with the option traceIter=T o The weights for the kernel are also fixed for iter-GMM. It is faster and I don't think it should change at each iteration. I am open to comments on that. o A function tsls() has been added to facilitate the estimation by 2SLS. The user does not have to worry about selecting the right gmm options o summary and print Methods have been added for tsls objects. The summary prints the f-statistics of the first stage LS o SmoothG for GEL uses now the kernapply() function which is more efficient. o Something the GMM converge to weird values which is sometimes caused by bad first step estimates used to compute the weighting matrix. Summary() prints the initial values to have more infortmation when convergence fails. o There was a bug in specTest() for GEL. The degrees of freedom for the J-test were wrong. It is fixed. o For GEL type CUE the implied probabilities are computed according to Antoine, Bonnal, and Renault (2007) which solves the problem of negative probabilities. o The data file Finance has been resaved which implies that the package depends now on R version ­2.10.0 and higher o The function rho() had been changed to .rho() because it is not useful outside the gel() function o The function getLamb() has been modified. It is now more efficient. The default for the gel() option optlam is now nlminb. The gradient and the hessian is provided which makes it much faster to solve for lambda. The argument of the function also changed. Instead of providing the function g() and the vector of coefficients, we provide the matrix gt. I may be useful to call the function sometimes to solve for lambda and it is easier that way. o The choice "iter" from the option optlam is kept for cases in which the optimizer fails to solve for lambda. It is often the case when type="ETEL" is chosen which tends to produce NA's. o For GEL of type "EL" and optfct="optim", the algorith constrOptim() is launched to make sure lambda'gt is always greater than 1 and avoid NA's when computing log(1-lambda'gt). The algorithm checks first with optim(), and swith to constrOptim() is optim() fails o Sometimes, problems happen in GMM estimation because of the bad first step estimates used to compute the weighting matrix. The first step estimates are usually computed using the identity matrix. The vector is now printed for better control. o Cleaned the codes. The data are in object$dat and we can get the moment matrix by calling gt <- object$g(object$coef,object$dat) for linear and non-linear models, where object is of class gmm. o We can now impose equality constraints of the type theta_i=c_i with the option eqConst=. See help(gmm) which includes examples. o The K statistics for weakly identified models of Kleiberben (2005) has been added. See ?KTest. o (gmm only) The bandwidth for the HAC estimation is set by the first step estimate. Its value, along with the weights, is saved in attr(obj$w0,"Spec"). If the user is not satisfied with this automatic bw selection, bw can be set to any fixed number when calling gmm(). See ?gmm for more details and examples. o The function gmmWithConst() reestimate an unrestricted model by imposing linear constraints, using the same specification. It also use the same bandwidth for the estimation of the HAC matrix o A regularized version of GEL based on Chausse(2011) is now implemented. The Lagrange multiplier can be regularized by setting the option "alpha=" to a number. The regularization can stabilize the GEL estimator. A preliminary paper will be available on my web site that analyze the properties of such modification. A method for selecting alpha will be implemented in future version. Changes in version 1.3-8 o A bug was found in the computation of linear GMM. The weighting matrix was not use properly. It is now fixed. Changes in version 1.3-7 o Until now there was no way to know whether optim or nlminb has converged in case of nonlinear GMM. The convergence code is now printed by print.gmm and the message and function evaluation by print.summary.gmm o For cases in which the weighting matrix is provided, you have the choice of assuming that it is just a fixed matrix not necessary equal to the inverse of the covariance matrix of the moment conditions (the default) or setting vcov="TrueFixed" if it is the inverse of the estimated covariance matrix. o If gradv is provided, it is not only used to compute the covariance matrix. It is also used to compute the gradiant of the objective function when the method is set to "CG" or "BFGS" in optim. For CUE, it is only used for the first step estimate because the gradiant can hardly be obtained analytically in that case. Changes in version 1.3-6 o Bug fixed when estimating model with timeSeries objects o When NA were present in the data and the instruments, gmm() was not working when omitting them created different dimension. If by chance it created the same dimensions, the data were not correctly aligned. Now it is aligned and the observations associated with the NA's are removed. When it happens, a warning message is displayed. Changes in version 1.3-5 o Bug fixed when estimating linear model with Z=X (OLS with robust matrix) Changes in version 1.3-4 o It is now possible to enter the instruments in x either as a matrix (like before) or as a formula. See details and examples in gmm help o A data argument is added to the gmm function. Therefore, it is no longer required that the variables in data.frames be attached before using gmm(). You just need to add the option data=your_data.frame. o 2SLS is now implemented in a more efficient way for linear models. Just add the option vcov="iid". An example as been included for that case Changes in version 1.3-3 o It is a very small modification to avoid errors in the installation. The function linear.hypothesis of the car package is now deprecated. They have be changed to linearHypothesis. Changes in version 1.3-2 o The functions HAC, weightsAndrews2, bwAndrews2 and all the others to compute the HAC matrix have been removed. The sandwich package is now used to compute these matrices. o The option centeredVcov has beed added to the gmm function. It allows to compute the covariance matrix using a centered moment function. o The option weightsMatrix has beed added to the gmm function. It allows to fixe the matrix of weights. o The methods bread() and estfun() are now available for gmm objects. It allows to compute a sandwich covariance matrix which is required if the weighting matrix is not the optimal one. o The results are obtained even if the covariance matrix cannot be computed o The vignette has been updated. Changes in version 1.3-0 o The method "getModel", "momentEstim" is now used also for the gel procedure. o The GEL procedure as been modified. For the case in which the option smooth=TRUE is used, the appropriate scaling parameters have been added following Smith(2004) in the estimation, the covariance matrix of the Lagrange multiplier and the tests of over-identifying restrictions. In this case, only the kernels Truncated and Bartlett are available. For the former, the optimal bandwidth of Andrews(91) for Bartlett is used and for the latter, it is the bandwidth proposed for the Parzen. See Smith(2004) for more details. Also, the option vcov as been removed since it is irrelevant for GEL. By setting smooth to FALSE, it is implied that the data are iid. Changes in version 1.2-0 The following modifications were suggested by the reviewers of JSS. These changes will not be noticed by users. They are intended to make it easier for developpers to contribute. o The new method specTest computes the specification tests from objects created by "gmm" or "gel" o The structure of the package has been modified in order to make it more flexible. The changes include: o The new method "getModel" prepares the estimation. It creates what is required by the method "momentEstim". o The method "momentEstim" estimates the model defined by "getModel" o The method "FinRes" finalizes the estimation when it is needed. For now, the method is used only by "gmm" to compute the final resutls. o Some bugs are fixed o The Vignette has been modified a little and the errors in the manual corrected. Changes in version 1.1-0 This is a major upgrade of the package. It follows great suggestions from two anonymous reviewers from JSS. The package as been modified so that it is now very close to lm and glm. o The name of the vector of parameters is now "coefficients" instead of "par" o The following methods are now available (see lm): fitted, residuals, vcov, coef, confint o There are now print methods for gmm, gel and summary.gel and summary.gmm o These modifications allows to use linear.hypothesis from the car package so the function lintest is no longer needed and as been removed o The following are now available from gmm and gel objects when g is a formula: residuals, fitted.values, model.frame, terms, model, the response and model matrix o Because the presence of the confint method, the option "interval" as been removed from the summary methods o A new plot method is available for both gmm and gel objects. It is a beta version. Comments and suggestions are welcome. o If there is only one instrument it can be provided as a vector. It does not need to be a matrix anymore. o It is now possible to select nlminb as optimizer. It allows to put restriction on the parameter space. o The package no longer depends on mvtnorm. It is a suggested package as it is only required for examples. o The packages car and fBasics (and therefore MASS, timeDate and timeSeries) are now suggested for examples in the vignette. o The new function charStable has been added. It computes the characteristic function of a stable distribution. An example is shown in the vignette o There was a bug when trying to estimate a model by ETEL with gel and numerical computation of lambda. It is fixed. Thanks to Márcio Laurini. o The vignette as beed rewritten. Changes in Version 1.0-7 o Modified some functions to remove dependencies on tseries and sandwich packages o Convert de Finance Data to data frame format Changes in Version 1.0-6 o Some bugs fixed. Thanks to Justinas Brazys Changes in Version 1.0-4 o documentation enhancements o Added finance data for applied examples Changes in Version 1.0-3 o Some bugs fixed Changes in Version 1.0-2 o Added a new example for better understanding. gmm/R/0000755000176200001440000000000014017531111011225 5ustar liggesusersgmm/R/Methods.sysGmm.R0000644000176200001440000001074612623157321014251 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ summary.sysGmm <- function(object, ...) { z <- object se <- sqrt(diag(z$vcov)) k <- attr(z$dat, "k") if (attr(z$dat, "sysInfo")$commonCoef) { seList <- rep(list(se), length(z$dat)) } else { seList <- list() for (i in 1:length(z$dat)) { seList[[i]] <- se[1:k[[i]]] se <- se[-(1:k[[i]])] } } par <- z$coefficients tval <- lapply(1:length(z$dat), function(i) par[[i]]/seList[[i]]) ans <- list(met=z$met,kernel=z$kernel,algo=z$algo,call=z$call) names(ans$met) <- "GMM method" names(ans$kernel) <- "kernel for cov matrix" ans$coefficients <- lapply(1:length(z$dat), function(i) cbind(par[[i]],seList[[i]], tval[[i]], 2 * pnorm(abs(tval[[i]]), lower.tail = FALSE))) ans$stest <- specTest(z) ans$algoInfo <- z$algoInfo ans$initTheta <- object$initTheta for (i in 1:length(z$dat)) { dimnames(ans$coefficients[[i]]) <- list(names(z$coefficients[[i]]), c("Estimate", "Std. Error", "t value", "Pr(>|t|)")) names(ans$initTheta[[i]]) <- names(z$coefficients[[i]]) } ans$specMod <- object$specMod ans$bw <- attr(object$w0,"Spec")$bw ans$weights <- attr(object$w0,"Spec")$weights ans$Sysnames <- names(z$dat) ans$met <- attr(object$dat, "sysInfo")$typeDesc if(object$infVcov != "HAC") ans$kernel <- NULL class(ans) <- "summary.sysGmm" ans } print.summary.sysGmm <- function(x, digits = 5, ...) { cat("\nCall:\n") cat(paste(deparse(x$call), sep="\n", collapse = "\n"), "\n\n", sep="") cat("Method\n", x$met,"\n\n") cat("\n") if( !is.null(x$kernel)) { cat("Kernel: ", x$kernel) if (!is.null(x$bw)) cat("(with bw = ", round(x$bw,5),")\n\n") else cat("\n\n") } cat("Coefficients:\n") m <- length(x$coefficients) for (i in 1:m) { cat(x$Sysnames[i], "\n") cat("#########\n") #print.default(format(x$coefficients[[i]], digits=digits), # print.gap = 2, quote = FALSE) printCoefmat(x$coefficients[[i]], digits=digits, ...) cat("\n") } cat(x$stest$ntest,"\n") print.default(format(x$stest$test, digits=digits), print.gap = 2, quote = FALSE) cat("\n") if(!is.null(x$initTheta)) { cat("Initial values of the coefficients\n") for (i in 1:m) { cat(x$Sysnames[i], "\n") print(x$initTheta[[i]]) } cat("\n") } cat(x$specMod) if(!is.null(x$algoInfo)) { cat("#############\n") cat("Information related to the numerical optimization\n") } if(!is.null(x$algoInfo$convergence)) cat("Convergence code = ", x$algoInfo$convergence,"\n") if(!is.null(x$algoInfo$counts)) { cat("Function eval. = ",x$algoInfo$counts[1],"\n") cat("Gradian eval. = ",x$algoInfo$counts[2],"\n") } if(!is.null(x$algoInfo$message)) cat("Message: ",x$algoInfo$message,"\n") invisible(x) } print.sysGmm <- function(x, digits=5, ...) { cat("Method\n", attr(x$dat, "sysInfo")$typeDesc,"\n\n") cat("Objective function value: ",x$objective,"\n\n") for (i in 1:length(x$coefficients)) { cat(names(x$dat)[[i]], ": \n") print.default(format(coef(x)[[i]], digits=digits), print.gap = 2, quote = FALSE) } cat("\n") if(!is.null(x$algoInfo$convergence)) cat("Convergence code = ", x$algoInfo$convergence,"\n") cat(x$specMod) invisible(x) } gmm/R/gel.R0000644000176200001440000005103513571757641012151 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ .rho <- function(x, lamb, derive = 0, type = c("EL", "ET", "CUE", "HD", "ETEL", "ETHD", "RCUE"), k = 1) { type <- match.arg(type) if (type == "RCUE") type <- "CUE" gml <- x%*%c(lamb)*k if (derive == 0) { if (type == "EL") { if (any(gml>=1)) stop("Computation of Lambda fails because NAs produced by log(1-gt*l)") rhomat <- log(1 - gml) } if (type == "ET") rhomat <- -exp(gml) if (type == "CUE") rhomat <- -gml -0.5*gml^2 if (type == "HD") rhomat <- -2/(1-gml/2) if (type == "ETEL") { w <- -exp(gml) w <- w/sum(w) rhomat <- -log(w*NROW(gml)) } if (type == "ETHD") { w <- -exp(gml) w <- w/sum(w) rhomat <- (sqrt(w)-1/sqrt(NROW(gml)))^2 } } if (derive==1) { if (type == "EL") rhomat <- -1/(1 - gml) if (type == "ET") rhomat <- -exp(gml) if (type == "CUE") rhomat <- -1 - gml if (type == "HD") rhomat <- -1/((1-gml/2)^2) if (any(type == c("ETEL","ETHD"))) rhomat <- NULL } if (derive==2) { if (type == "EL") rhomat <- -1/(1 - gml)^2 if (type == "ET") rhomat <- -exp(gml) if (type == "CUE") rhomat <- -rep(1,nrow(x)) if (type == "HD") rhomat <- -1/((1-gml/2)^3) if (any(type == c("ETEL","ETHD"))) rhomat <- NULL } return(c(rhomat)) } .getCgelLam <- function(gt, l0, type = c('EL', 'ET', 'CUE', "HD"), method = c("nlminb", "optim", "constrOptim"), control=list(), k = 1, alpha = 0.01) { type <- match.arg(type) method <- match.arg(method) fct <- function(l, X) { r1 <- colMeans(.rho(gt,l,derive=1,type=type,k=k)*X) crossprod(r1) + alpha*crossprod(l) } Dfct <- function(l, X) { r2 <- .rho(X,l,derive=2,type=type,k=k) r1 <- .rho(X,l,derive=1,type=type,k=k) H <- t(X*r2)%*%X/nrow(X) 2*H%*%colMeans(r1*X) + 2*alpha*l } if (method == "nlminb") res <- nlminb(l0, fct, Dfct, X = gt, control = control) if (method == "optim") res <- optim(l0, fct, Dfct, X = gt, method="BFGS", control = control) if (method == "constrOptim") { ci <- -rep(1,nrow(gt)) res <- constrOptim(rep(0,ncol(gt)),fct,Dfct,-gt,ci,control=control,X=gt) } if (method == "optim") { conv <- list(convergence = res$convergence, counts = res$counts, message = res$message) } else { conv <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) } return(list(lambda = res$par, convergence = conv, obj = mean(.rho(gt,res$par, derive=0,type=type,k=k)) - .rho(1,0, derive=0,type=type,k=k))) } .Wu <- function(gt, tol_lam = 1e-8, maxiterlam = 50, K=1) { u <- as.matrix(gt) n=length(nrow(u)) M=rep(0,ncol(u)) dif=1 tol=tol_lam k=0 while(dif>tol & k<=maxiterlam) { D1=t(u)%*%((1/(1+u%*%M))*rep(1,n)) DD=-t(u)%*%(c((1/(1+u%*%M)^2))*u) D2=solve(DD,D1,tol=1e-40) dif=max(abs(D2)) rule=1 while(rule>0) { rule=0 if(min(1+t(M-D2)%*%t(u))<=0) rule=rule+1 if(rule>0) D2=D2/2 } M=M-D2 k=k+1 } if(k>=maxiterlam) { M=rep(0,ncol(u)) conv = list(convergence=1) } else { conv = list(convergence=0) } return(list(lambda = c(-M), convergence = conv, obj = mean(.rho(gt,-M,derive=0,type="EL",k=K)))) } .Wu2 <- function(gt, tol_lam = 1e-8, maxiter = 50, K = 1) { gt <- as.matrix(gt) res <- .Fortran(F_wu, as.double(gt), as.double(tol_lam), as.integer(maxiter), as.integer(nrow(gt)), as.integer(ncol(gt)), as.double(K), conv=integer(1), obj=double(1), lambda=double(ncol(gt))) list(lambda=res$lambda, convergence=list(convergence=res$conv), obj = res$obj) } .CUE_lam <- function(gt, K=1) { q <- qr(gt) n <- nrow(gt) l0 <- -qr.coef(q, rep(1,n)) conv <- list(convergence=0) list(lambda = l0, convergence = conv, obj = mean(.rho(gt,l0,derive=0,type="CUE",k=K))) } .CUE_lamPos <- function(gt, K=1, control=list()) { getpt <- function(gt,lam) { gl <- c(gt%*%lam) pt <- 1 + gl pt/sum(pt) } maxit <- ifelse("maxit" %in% names(control), control$maxit, 50) res <-.CUE_lam(gt, K) n <- nrow(gt) i <- 1 pt <- getpt(gt, res$lambda) w <- pt<0 while (TRUE) { gt2 <- gt[!w,] n1 <- nrow(gt2) if (n1 == n) break res <- try(.CUE_lam(gt2), silent=TRUE) if (i > maxit) return(list(lambda=rep(0,ncol(gt)), obj=0, pt=rep(1/n,n), convergence=list(convergence=1))) if (any(class(res) == "try-error")) return(list(lambda=rep(0,ncol(gt)), obj=0, pt=rep(1/n,n), convergence=list(convergence=2))) pt[!w] <- getpt(gt2, res$lambda) pt[w] <- 0 if (all(pt>=0)) break i <- i+1 w[!w] <- pt[!w]<0 } n0 <- n-n1 res$obj <- res$obj*n1/n + n0/(2*n) res } .CUE_lamPos2 <- function(gt, K=1, control=list()) { gt <- as.matrix(gt) n <- nrow(gt) q <- ncol(gt) maxit <- ifelse("maxit" %in% names(control), control$maxit, 50) res <- try(.Fortran(F_lamcuep, as.double(gt), as.integer(n), as.integer(q), as.double(K), as.integer(maxit),conv=integer(1), lam=double(q),pt=double(n), obj=double(1) ), silent=TRUE) if (any(class(res) == "try-error")) return(list(lambda=rep(0,q), obj=0, pt=rep(1/n,n), convergence=list(convergence=3))) list(lambda=res$lam, obj=res$obj, pt=res$pt, convergence=list(convergence=res$conv)) } getLamb <- function(gt, l0, type = c('EL', 'ET', 'CUE', "ETEL", "HD", "ETHD", "RCUE"), tol_lam = 1e-7, maxiterlam = 100, tol_obj = 1e-7, k = 1, method = c("nlminb", "optim", "iter", "Wu"), control=list()) { method <- match.arg(method) type <- match.arg(type) gt <- as.matrix(gt) if (method == "Wu" & type != "EL") stop("Wu (2005) method to compute Lambda is for EL only") if (method == "Wu") return(.Wu2(gt, tol_lam, maxiterlam, k)) if (type == "CUE") return(.CUE_lam(gt, k)) if (type == "RCUE") return(.CUE_lamPos2(gt, k, control)) if (method == "iter") { if ((type == "ETEL") | (type == "ETHD")) type = "ET" for (i in 1:maxiterlam) { r1 <- .rho(gt,l0,derive=1,type=type,k=k) r2 <- .rho(gt,l0,derive=2,type=type,k=k) F <- -colMeans(r1*gt) J <- crossprod(r2*gt,gt) if (sum(abs(F))= 2) { w <- c(w[rt:2], w) w <- w / sum(w) w <- kernel(w[rt:length(w)]) } else { w <- kernel(1) } } else { if (class(w)[1] != "tskernel") stop("Provided weights must be a numeric vector or an object of class 'tskernel'") } if (length(w$coef)>1) x <- kernapply(x, w) sx <- list("smoothx" = x, "kern_weights" = w, bw = bw) return(sx) } gel <- function(g, x, tet0 = NULL, gradv = NULL, smooth = FALSE, type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD","RCUE"), kernel = c("Truncated", "Bartlett"), bw = bwAndrews, approx = c("AR(1)", "ARMA(1,1)"), prewhite = 1, ar.method = "ols", tol_weights = 1e-7, tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, constraint = FALSE, optfct = c("optim", "optimize", "nlminb"), optlam = c("nlminb", "optim", "iter", "Wu"), data, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, TypeGel = "baseGel", alpha = NULL, eqConst = NULL, eqConstFullVcov = FALSE, onlyCoefficients=FALSE, ...) { type <- match.arg(type) optfct <- match.arg(optfct) optlam <- match.arg(optlam) weights <- weightsAndrews approx <- match.arg(approx) kernel <- match.arg(kernel) if (!is.null(eqConst)) TypeGel <- "constGel" if(missing(data)) data<-NULL all_args <- list(g = g, x = x, tet0 = tet0, gradv = gradv, smooth = smooth, type = type, kernel = kernel, bw = bw, approx = approx, prewhite = prewhite, ar.method = ar.method, tol_weights = tol_weights, tol_lam = tol_lam, tol_obj = tol_obj, tol_mom = tol_mom, maxiterlam = maxiterlam, constraint = constraint, optfct = optfct, weights = weights, optlam = optlam, model = model, X = X, Y = Y, TypeGel = TypeGel, call = match.call(), Lambdacontrol = Lambdacontrol, alpha = alpha, data = data, eqConst = eqConst, eqConstFullVcov = eqConstFullVcov, onlyCoefficients=onlyCoefficients) class(all_args)<-TypeGel Model_info<-getModel(all_args) z <- momentEstim(Model_info, ...) if (!onlyCoefficients) class(z) <- "gel" return(z) } evalGel <- function(g, x, tet0, gradv = NULL, smooth = FALSE, type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD","RCUE"), kernel = c("Truncated", "Bartlett"), bw = bwAndrews, approx = c("AR(1)", "ARMA(1,1)"), prewhite = 1, ar.method = "ols", tol_weights = 1e-7, tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, optlam = c("nlminb", "optim", "iter", "Wu"), data, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, alpha = NULL, ...) { type <- match.arg(type) optlam <- match.arg(optlam) weights <- weightsAndrews approx <- match.arg(approx) kernel <- match.arg(kernel) TypeGel <- "baseGel" if(missing(data)) data<-NULL all_args <- list(g = g, x = x, tet0 = tet0, gradv = gradv, smooth = smooth, type = type, kernel = kernel, bw = bw, approx = approx, prewhite = prewhite, ar.method = ar.method, tol_weights = tol_weights, tol_lam = tol_lam, tol_obj = tol_obj, tol_mom = tol_mom, maxiterlam = maxiterlam, weights = weights, optlam = optlam, model = model, X = X, Y = Y, call = match.call(), Lambdacontrol = Lambdacontrol, alpha = alpha, data = data, optfct="optim") class(all_args)<-TypeGel Model_info<-getModel(all_args) class(Model_info) <- "baseGel.eval" z <- momentEstim(Model_info, ...) class(z) <- "gel" return(z) } .thetf <- function(tet, P, output=c("obj","all"), l0Env) { output <- match.arg(output) gt <- P$g(tet, P$x) l0 <- get("l0",envir=l0Env) if (((P$type=="ETEL")|(P$type=="ETHD"))&(!is.null(P$CGEL))) { P$CGEL <- NULL warning("CGEL not implemented for ETEL or for ETHD") } if (is.null(P$CGEL)) { if (P$optlam != "optim" & P$type == "EL") { lamb <- try(getLamb(gt, l0, type = P$type, tol_lam = P$tol_lam, maxiterlam = P$maxiterlam, tol_obj = P$tol_obj, k = P$k1/P$k2, control = P$Lambdacontrol, method = P$optlam), silent = TRUE) if(any(class(lamb) == "try-error")) lamb <- getLamb(gt, l0, type = P$type, tol_lam = P$tol_lam, maxiterlam = P$maxiterlam, tol_obj = P$tol_obj, k = P$k1/P$k2, control = P$Lambdacontrol, method = "optim") } else { lamb <- getLamb(gt, l0, type = P$type, tol_lam = P$tol_lam, maxiterlam = P$maxiterlam, tol_obj = P$tol_obj, k = P$k1/P$k2, control = P$Lambdacontrol, method = P$optlam) } } else { lamb <- try(.getCgelLam(gt, l0, type = P$type, method = "nlminb", control=P$Lambdacontrol, k = P$k1/P$k2, alpha = P$CGEL),silent=TRUE) if (any(class(lamb) == "try-error")) lamb <- try(.getCgelLam(gt, l0, type = P$type, method = "constrOptim", control=P$Lambdacontrol, k = P$k1/P$k2, alpha = P$CGEL),silent=TRUE) } if (P$type == "ETEL") obj <- mean(.rho(gt, lamb$lambda, type = P$type, derive = 0, k = P$k1/P$k2) - .rho(1, 0, type = P$type, derive = 0, k = P$k1/P$k2)) else if (P$type == "ETHD") obj <- sum(.rho(gt, lamb$lambda, type = P$type, derive = 0, k = P$k1/P$k2) - .rho(1, 0, type = P$type, derive = 0, k = P$k1/P$k2)) else obj <- lamb$obj assign("l0",lamb$lambda,envir=l0Env) if(output == "obj") return(obj) else return(list(obj = obj, lambda = lamb, gt = gt)) } .getImpProb <- function(gt, lambda, type, k1, k2) { if ((type == "ETEL")|(type=="ETHD")) type <- "ET" n <- NROW(gt) pt <- -.rho(gt, lambda, type = type, derive = 1, k = k1/k2)/n # Making sure pt>0 if (type=="CUE") { eps <- -length(pt)*min(min(pt),0) pt <- (pt+eps/length(pt))/(1+eps) } if (type=="RCUE") pt[pt<0] <- 0 ################### conv_moment <- colSums(pt*gt) conv_pt <- sum(as.numeric(pt)) pt <- pt/sum(pt) attr(pt, "conv_moment") <- conv_moment attr(pt, "conv_pt") <- conv_pt pt } .vcovGel <- function(gt, G, k1, k2, bw, pt=NULL,tol=1e-16) { q <- NCOL(gt) n <- NROW(gt) if (is.null(pt)) pt <- 1/n G <- G/k1 gt <- gt*sqrt(pt*bw/k2) qrGt <- qr(gt) piv <- sort.int(qrGt$pivot, index.return=TRUE)$ix R <- qr.R(qrGt)[,piv] X <- forwardsolve(t(R), G) Y <- forwardsolve(t(R), diag(q)) res <- lm.fit(X,Y) u <- res$residuals Sigma <- chol2inv(res$qr$qr)/n diag(Sigma)[diag(Sigma)<0] <- tol if (q==ncol(G)) { SigmaLam <- matrix(0, q, q) } else { SigmaLam <- backsolve(R, u)/n*bw^2 diag(SigmaLam)[diag(SigmaLam)<0] <- tol } khat <- crossprod(R) list(vcov_par=Sigma, vcov_lambda=SigmaLam,khat=khat) } gmm/R/PlotMethods.R0000644000176200001440000000611311331045226013616 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ plot.gmm <- function (x, which = c(1L:3), main = list("Residuals vs Fitted values", "Normal Q-Q", "Response variable and fitted values"), panel = if(add.smooth) panel.smooth else points, ask = prod(par("mfcol")) < length(which) && dev.interactive(), ..., add.smooth = getOption("add.smooth")) { if (!inherits(x, "gmm")) stop("use only with \"gmm\" objects") if (!inherits(x, "gel")) { if(!is.numeric(which) || any(which < 1) || any(which > 3)) stop("'which' must be in 1L:3") show <- rep(FALSE, 3) } else { if(!is.numeric(which) || any(which < 1) || any(which > 4)) stop("'which' must be in 1L:4") show <- rep(FALSE, 4) } show[which] <- TRUE r <- residuals(x) if(ncol(r)>1) stop("plot.gmm is not yet implemented for system of equations") yh <- fitted(x) n <- length(r) if (ask) { oask <- devAskNewPage(TRUE) on.exit(devAskNewPage(oask)) } ##---------- Do the individual plots : ---------- if (show[1L]) { ylim <- range(r, na.rm=TRUE) ylim <- extendrange(r= ylim, f = 0.08) plot(yh, r, xlab = "Fitted", ylab = "Residuals", main = main[1L], ylim = ylim, type = "n", ...) panel(yh, r) abline(h = 0, lty = 3, col = "gray") } if (show[2L]) { ## Normal rs <- (r-mean(r))/sd(r) ylim <- range(rs, na.rm=TRUE) ylim[2L] <- ylim[2L] + diff(ylim) * 0.075 qq <- qqnorm(rs, main = main[2L], ylab = "stand. residuals", ylim = ylim, ...) qqline(rs, lty = 3, col = "gray50") } if (show[3L]) { y <- as.matrix(model.response(x$model, "numeric")) ylim <- range(yh, na.rm=TRUE) ylim <- extendrange(r= ylim, f = 0.08) plot(y, main = main[3L], ylim = ylim, ...) lines(yh,col=2) } if (inherits(x, "gel")) { if (show[4L]) { pt <- x$pt plot(pt, type='l',main = main[4L],ylab="Implied Prob.", ...) emp_pt <- rep(1/length(pt),length(pt)) lines(emp_pt,col=2) legend("topleft",c("Imp. Prob.","Empirical (1/T)"),col=1:2,lty=c(1,1)) } } invisible() } plot.gel <- function (x, which = c(1L:4), main = list("Residuals vs Fitted values", "Normal Q-Q", "Response variable and fitted values","Implied probabilities"), panel = if(add.smooth) panel.smooth else points, ask = prod(par("mfcol")) < length(which) && dev.interactive(), ..., add.smooth = getOption("add.smooth")) { if (!inherits(x, "gel")) stop("use only with \"gel\" objects") class(x) <- c("gmm","gel") plot(x,which=which,main=main,panel=panel, ask=ask, ..., add.smooth=add.smooth) invisible() } gmm/R/Methods.gmm.R0000644000176200001440000002171613571757641013567 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ summary.gmm <- function(object, ...) { z <- object se <- sqrt(diag(z$vcov)) par <- z$coefficients tval <- par/se ans <- list(met=z$met,kernel=z$kernel,algo=z$algo,call=z$call) names(ans$met) <- "GMM method" names(ans$kernel) <- "kernel for cov matrix" ans$coefficients <- cbind(par,se, tval, 2 * pnorm(abs(tval), lower.tail = FALSE)) dimnames(ans$coefficients) <- list(names(z$coefficients), c("Estimate", "Std. Error", "t value", "Pr(>|t|)")) ans$stest <- specTest(z) ans$algoInfo <- z$algoInfo if(z$met=="cue") ans$cue <- object$cue if (!is.null(object$initTheta)) { ans$initTheta <- object$initTheta names(ans$initTheta) <- names(z$coefficients) } ans$specMod <- object$specMod ans$bw <- attr(object$w0,"Spec")$bw ans$weights <- attr(object$w0,"Spec")$weights if(object$infVcov == "iid") ans$kernel <- NULL class(ans) <- "summary.gmm" ans } summary.tsls <- function(object, vcov = NULL, ...) { if (!is.null(vcov)) object$vcov=vcov ans <- summary.gmm(object) ans$met <- paste(ans$met, "(Meat type = ", attr(object$vcov, "vcovType"), ")",sep="") k <- object$dat$k if (!is.null(object$fsRes)) { fstat <- vector() fsRes <- object$fsRes if (class(fsRes)[1] == "listof") { nendo <- length(fsRes) } else { nendo <- 1 } if (nendo == 1) { fstat[1] <- fsRes$fstatistic[1] df1 <- fsRes$fstatistic[2] df2 <- fsRes$fstatistic[3] } else { fstat[1] <- fsRes[[1]]$fstatistic[1] df1 <- fsRes[[1]]$fstatistic[2] df2 <- fsRes[[1]]$fstatistic[3] } if (nendo > 1){ for (i in 2:nendo) fstat[i] <- fsRes[[i]]$fstatistic[1] } pvfstat <- 1-pf(fstat,df1, df2) names(fstat) <- attr(fsRes,"Endo") ans$fstatistic <- list(fstat = fstat, pvfstat = pvfstat, df1 = df1, df2 = df2) } ans$specMod <- object$specMod class(ans) <- "summary.tsls" return(ans) } print.summary.tsls <- function(x, digits = 5, ...) { print.summary.gmm(x,digits) if (!is.null(x$fstatistic)) { cat("\n First stage F-statistics: \n") if(names(x$fstatistic$fstat)[1]=="(Intercept)") start=2 else start=1 for (i in start:length(x$fstatistic$fstat)) cat(names(x$fstatistic$fstat)[i], ": F(",x$fstatistic$df1,", ",x$fstatistic$df2,") = ",x$fstatistic$fstat[i], " (P-Vavue = ",x$fstatistic$pvfstat[i],")\n") } else { cat("\n No first stage F-statistics (just identified model)\n") } } print.summary.gmm <- function(x, digits = 5, ...) { cat("\nCall:\n") cat(paste(deparse(x$call), sep="\n", collapse = "\n"), "\n\n", sep="") cat("\nMethod: ", x$met,"\n") if (x$met=="cue") { if (!is.null(x$cue$message)) { cat(" (",x$cue$message,")\n\n") } } else { cat("\n") } if( !is.null(x$kernel)) { cat("Kernel: ", x$kernel) if (!is.null(x$bw)) cat("(with bw = ", round(x$bw,5),")\n\n") else cat("\n\n") } cat("Coefficients:\n") print.default(format(x$coefficients, digits=digits), print.gap = 2, quote = FALSE) cat("\n") cat(x$stest$ntest,"\n") print.default(format(x$stest$test, digits=digits), print.gap = 2, quote = FALSE) cat("\n") if(!is.null(x$initTheta)) { cat("Initial values of the coefficients\n") print(x$initTheta) cat("\n") } cat(x$specMod) if(!is.null(x$algoInfo)) { cat("#############\n") cat("Information related to the numerical optimization\n") } if(!is.null(x$algoInfo$convergence)) cat("Convergence code = ", x$algoInfo$convergence,"\n") if(!is.null(x$algoInfo$counts)) { cat("Function eval. = ",x$algoInfo$counts[1],"\n") cat("Gradian eval. = ",x$algoInfo$counts[2],"\n") } if(!is.null(x$algoInfo$message)) cat("Message: ",x$algoInfo$message,"\n") invisible(x) } formula.gmm <- function(x, ...) { if(is.null(x$terms)) stop("The gmm object was not created by a formula") else formula(x$terms) } confint.gmm <- function(object, parm, level=0.95, ...) { ntest <- "Wald type confidence interval" z <- object se <- sqrt(diag(z$vcov)) par <- z$coefficients zs <- qnorm((1-level)/2,lower.tail=FALSE) ch <- zs*se ans <- cbind(par-ch,par+ch) dimnames(ans) <- list(names(par),c((1-level)/2,0.5+level/2)) if(!missing(parm)) ans <- ans[parm,] ans <- list(test=ans, ntest=ntest) class(ans) <- "confint" ans } residuals.gmm <- function(object,...) { if(is.null(object$model)) stop("The residuals method is valid only for g=formula") object$residuals } fitted.gmm <- function(object,...) { if(is.null(object$model)) stop("The residuals method is valid only for g=formula") object$fitted.value } print.gmm <- function(x, digits=5, ...) { cat("Method\n", x$met,"\n\n") cat("Objective function value: ",x$objective,"\n\n") print.default(format(coef(x), digits=digits), print.gap = 2, quote = FALSE) cat("\n") if(!is.null(x$algoInfo$convergence)) cat("Convergence code = ", x$algoInfo$convergence,"\n") cat(x$specMod) invisible(x) } coef.gmm <- function(object,...) object$coefficients vcov.gmm <- function(object,...) object$vcov estfun.gmmFct <- function(x, y = NULL, theta = NULL, ...) { if (is(x, "function")) { gmat <- x(theta, y) return(gmat) } else return(x) } estfun.tsls <- function(x, ...) { model.matrix(x)*c(residuals(x)) } model.matrix.tsls <- function(object, ...) { dat <- object$dat ny <- dat$ny nh <- dat$nh k <- dat$k x <- dat$x n <- nrow(x) hm <- as.matrix(x[,(ny+k+1):(ny+k+nh)]) xm <- as.matrix(x[,(ny+1):(ny+k)]) xhat <- lm(xm~hm-1)$fitted assign <- 1:ncol(xhat) if (attr(object$terms,"intercept")==1) assign <- assign-1 attr(xhat,"assign") <- assign xhat } vcov.tsls <- function(object, type=c("Classical","HC0","HC1","HAC"), hacProp = list(), ...) { type <- match.arg(type) if (type == "Classical") { sig <- sum(c(residuals(object))^2)/(nrow(object$dat$x)-object$dat$k) ny <- object$dat$ny nh <- object$dat$nh k <- object$dat$k n <- nrow(object$dat$x) hm <- as.matrix(object$dat$x[,(ny+k+1):(ny+k+nh)]) Omega <- crossprod(hm)*sig/nrow(object$dat$x) vcovType <- "Classical" V <- solve(crossprod(object$G,solve(Omega,object$G)))/nrow(object$dat$x) } else if (strtrim(type,2) == "HC") { meat <- meatHC(object, type) bread <- bread(object) vcovType <- paste("HCCM: ", type, sep="") V <- crossprod(bread, meat%*%bread)/nrow(object$dat$x) } else { object$centeredVcov <- TRUE gt <- model.matrix(object)*c(residuals(object)) gt <- lm(gt~1) arg <- c(list(x=gt,sandwich=FALSE),hacProp) meat <- do.call(kernHAC, arg) KType <- ifelse(is.null(hacProp$kernel), formals(kernHAC)$kernel[[2]], hacProp$kernel) vcovType <- paste("HAC: ", KType, sep="") bread <- bread(object) V <- crossprod(bread, meat%*%bread)/nrow(object$dat$x) } attr(V, "vcovType") <- vcovType return(V) } estfun.gmm <- function(x, ...) { foc <- x$gt %*% x$w %*% x$G return(foc) } bread.gmm <- function(x, ...) { GWG <- crossprod(x$G, x$w %*% x$G) b <- try(solve(GWG), silent = TRUE) if (any(class(b) == "try-error")) stop("The bread matrix is singular") return(b) } bread.tsls <- function(x, ...) { dat <- x$dat ny <- dat$ny nh <- dat$nh k <- dat$k x <- dat$x n <- nrow(x) hm <- as.matrix(x[,(ny+k+1):(ny+k+nh)]) xm <- as.matrix(x[,(ny+1):(ny+k)]) xhat <- lm(xm~hm-1)$fitted solve(crossprod(xhat)/n) } gmm/R/getModel.R0000644000176200001440000006505613571757641013152 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ getModel <- function(object, ...) { UseMethod("getModel") } getModel.tsls <- function(object, ...) { obj <- getModel.baseGmm(object, ...) return(obj) } getModel.sysGmm <- function(object, ...) { if (object$commonCoef & !is.null(object$crossEquConst)) { object$commonCoef <- FALSE warning("When crossEquConst is not NULL, commonCoef=TRUE is ignore and set to FALSE") } object$allArg <- c(object, ...) object$formula <- list(g=object$g, h=object$h) if (!is.list(object$g)) stop("g must be list of formulas") if (length(object$g) == 1) stop("For single equation GMM, use the function gmm()") if (!all(sapply(1:length(object$g), function(i) is(object$g[[i]], "formula")))) stop("g must be a list of formulas") if (!is.list(object$h)) { if(!is(object$h, "formula")) stop("h is either a list of formulas or a formula") else object$h <- list(object$h) } else { if (!all(sapply(1:length(object$h), function(i) is(object$h[[i]], "formula")))) stop("h is either a list of formulas or a formula") } if (length(object$h) == 1) { object$h <- rep(object$h, length(object$g)) typeDesc <- "System Gmm with common instruments" typeInst <- "Common" } else { if (length(object$h) != length(object$g)) stop("The number of formulas in h should be the same as the number of formulas in g, \nunless the instruments are the same in each equation, \nin which case the number of equations in h should be 1") typeDesc <- "System Gmm" typeInst <- "nonCommon" } if (object$commonCoef) typeDesc <- paste(typeDesc, " (Common Coefficients)") dat <- lapply(1:length(object$g), function(i) try(getDat(object$g[[i]], object$h[[i]], data = object$data, error=!object$commonCoef), silent=TRUE)) chk <- sapply(1:length(dat), function(i) any(class(dat[[i]]) == "try-error")) chk <- which(chk) mess <- vector() for (i in chk) { mess <- paste(mess, "\nSystem of equations:", i, "\n###############\n", sep="") mess <- paste(mess, attr(dat[[i]], "condition")[[1]]) } if (length(chk)>0) stop(mess) if (is.null(names(object$g))) names(dat) <- paste("System_", 1:length(dat), sep="") else names(dat) <- names(object$g) object$gradv <- .DmomentFct_Sys object$formula <- list(g=object$g, h=object$h) if (!all(sapply(1:length(dat), function(i) dat[[i]]$ny == 1))) stop("The number of dependent variable must be one in every equation") clname <- "sysGmm.twoStep.formula" object$x <- dat namex <- lapply(1:length(dat), function(i) colnames(dat[[i]]$x[,2:(1+dat[[i]]$k), drop=FALSE])) nameh <- lapply(1:length(dat), function(i) colnames(dat[[i]]$x[,(2+dat[[i]]$k):(1+dat[[i]]$k+dat[[i]]$nh), drop=FALSE])) namey <- lapply(1:length(dat), function(i) colnames(dat[[i]]$x[,1, drop=FALSE])) object$namesCoef <- namex namesgt <- lapply(1:length(dat), function(i) paste(namey[[i]], "_", nameh[[i]], sep="")) object$namesgt <- namesgt object$namesy <- namey attr(object$x,"ModelType") <- "linear" #for (i in 1:length(object$x)) # attr(object$x[[i]], c("linear")) <- attr(object$x, "modelType") attr(object$x, "k") <- lapply(1:length(dat), function(i) length(object$namesCoef[[i]])) attr(object$x, "q") <- lapply(1:length(dat), function(i) length(object$namesgt[[i]])) attr(object$x, "n") <- lapply(1:length(dat), function(i) nrow(object$x[[i]]$x)) object$TypeGmm <- class(object) attr(object$x, "weight") <- list(w=object$weightsMatrix, centeredVcov=object$centeredVcov) attr(object$x, "weight")$WSpec <- list() attr(object$x, "weight")$WSpec$sandwich <- list(kernel = object$kernel, bw = object$bw, prewhite = object$prewhite, ar.method = object$ar.method, approx = object$approx, tol = object$tol) attr(object$x, "weight")$vcov <- object$vcov k <- lapply(1:length(dat), function(i) dat[[i]]$k) q <- lapply(1:length(dat), function(i) dat[[i]]$nh) df <- lapply(1:length(dat), function(i) dat[[i]]$nh-dat[[i]]$k) k2 <- do.call(c,k) if (object$commonCoef | !is.null(object$crossEquConst)) { if (!all(k2==k2[1])) stop("In a common coefficient model the number of regressors is the same in each equation") if (object$commonCoef) totK <- k2[1] else totK <- length(dat)*(k2[1]-length(object$crossEquConst)) + length(object$crossEquConst) if (sum(do.call(c,q))=length(object$t0)) stop("Too many constraints; use evalGmm() if all coefficients are fixed") if (is.character(object$eqConst)) { if (is.null(names(object$t0))) stop("t0 must be a named vector if you want eqConst to be names") if (any(!(object$eqConst %in% names(object$t0)))) stop("Wrong coefficient names in eqConst") object$eqConst <- sort(match(object$eqConst,names(object$t0))) } restTet <- object$t0[object$eqConst] obj$t0 <- object$t0[-object$eqConst] object$eqConst <- cbind(object$eqConst,restTet) } else { if (is.null(dim(object$eqConst))) stop("When t0 is not provided, eqConst must be a 2xq matrix") } attr(obj$x, "eqConst") <- list(eqConst = object$eqConst) rownames(attr(obj$x, "eqConst")$eqConst) <- obj$namesCoef[object$eqConst[,1]] object$eqConst <- attr(obj$x, "eqConst")$eqConst if(is(object$g, "formula")) { if (obj$x$ny>1) stop("Constrained GMM not implemented yet for system of equations") if (obj$x$k<=0) stop("Nothing to estimate") } obj$eqConst <- object$eqConst attr(obj$x, "k") <- attr(obj$x, "k")-nrow(object$eqConst) obj$namesCoef <- obj$namesCoef[-object$eqConst[,1]] obj$type <- paste(obj$type,"(with equality constraints)",sep=" ") mess <- paste(rownames(object$eqConst), " = " , object$eqConst[,2], "\n",collapse="") mess <- paste("#### Equality constraints ####\n",mess,"##############################\n\n",sep="") obj$specMod <- mess return(obj) } getModel.baseGmm <- function(object, ...) { object$allArg <- c(object, list(...)) if(is(object$g, "formula")) { object$gradv <- .DmomentFct object$gradvf <- FALSE dat <- getDat(object$g, object$x, data = object$data) if(is.null(object$weightsMatrix)) { clname <- paste(class(object), ".", object$type, ".formula", sep = "") } else { clname <- "fixedW.formula" object$type <- "One step GMM with fixed W" } object$x <- dat object$gform<-object$g namex <- colnames(dat$x[,(dat$ny+1):(dat$ny+dat$k), drop=FALSE]) nameh <- colnames(dat$x[,(dat$ny+dat$k+1):(dat$ny+dat$k+dat$nh), drop=FALSE]) if (dat$ny > 1) { namey <- colnames(dat$x[,1:dat$ny, drop=FALSE]) object$namesCoef <- paste(rep(namey, dat$k), "_", rep(namex, rep(dat$ny, dat$k)), sep = "") object$namesgt <- paste(rep(namey, dat$nh), "_", rep(nameh, rep(dat$ny, dat$nh)), sep = "") } else { object$namesCoef <- namex object$namesgt <- nameh } attr(object$x,"ModelType") <- "linear" attr(object$x, "k") <- object$x$k attr(object$x, "q") <- object$x$ny*object$x$nh attr(object$x, "n") <- NROW(object$x$x) attr(object$x, "namesgt") <- object$namesgt } else { attr(object$x,"ModelType") <- "nonlinear" attr(object$x, "momentfct") <- object$g if (object$optfct == "optimize") attr(object$x, "k") <- 1 else attr(object$x, "k") <- length(object$t0) attr(object$x, "q") <- NCOL(object$g(object$t0, object$x)) attr(object$x, "n") <- NROW(object$x) if (object$optfct == "optimize") { object$namesCoef <- "Theta1" } else { if(is.null(names(object$t0))) object$namesCoef <- paste("Theta[" ,1:attr(object$x, "k"), "]", sep = "") else object$namesCoef <- names(object$t0) } if(is.null(object$weightsMatrix)) { clname <- paste(class(object), "." ,object$type, sep = "") } else { clname <- "fixedW" object$type <- "One step GMM with fixed W" attr(object$x, "weight")$w <- object$weightsMatrix } if (!is.function(object$gradv)) { object$gradvf <- FALSE } else { attr(object$x, "gradv") <- object$gradv object$gradvf <- TRUE } object$gradv <- .DmomentFct } object$TypeGmm <- class(object) attr(object$x, "weight") <- list(w=object$weightsMatrix, centeredVcov=object$centeredVcov) attr(object$x, "weight")$WSpec <- list() attr(object$x, "weight")$WSpec$sandwich <- list(kernel = object$kernel, bw = object$bw, prewhite = object$prewhite, ar.method = object$ar.method, approx = object$approx, tol = object$tol) attr(object$x, "weight")$vcov <- object$vcov attr(object$x, "mustar") <- object$mustar object$g <- .momentFct class(object) <- clname return(object) } getModel.constGel <- function(object, ...) { class(object) <- "baseGel" obj <- getModel(object) if (!is.null(dim(object$eqConst))) stop("eqConst must be a vector which indicates which parameters to fix") if (length(object$eqConst)>=length(object$tet0)) stop("Too many constraints; use evalGel() if all coefficients are fixed") if (is.character(object$eqConst)) { if (is.null(names(object$tet0))) stop("tet0 must be a named vector if you want eqConst to be names") if (any(!(object$eqConst %in% names(object$tet0)))) stop("Wrong coefficient names in eqConst") object$eqConst <- sort(match(object$eqConst,names(object$tet0))) } restTet <- object$tet0[object$eqConst] obj$tet0 <- object$tet0[-object$eqConst] object$eqConst <- cbind(object$eqConst,restTet) attr(obj$x, "eqConst") <- list(eqConst = object$eqConst) rownames(attr(obj$x, "eqConst")$eqConst) <- obj$namesCoef[object$eqConst[,1]] object$eqConst <- attr(obj$x, "eqConst")$eqConst if(is(object$g, "formula")) { if (obj$x$ny>1) stop("Constrained GMM not implemented yet for system of equations") } obj$eqConst <- object$eqConst attr(obj$x, "k") <- attr(obj$x, "k")-nrow(object$eqConst) obj$namesCoef <- obj$namesCoef[-object$eqConst[,1]] obj$typeDesc <- paste(obj$typeDesc,"(with equality constraints)",sep=" ") mess <- paste(rownames(object$eqConst), " = " , object$eqConst[,2], "\n",collapse="") mess <- paste("#### Equality constraints ####\n",mess,"##############################\n\n",sep="") obj$specMod <- mess return(obj) } getModel.baseGel <- function(object, ...) { object$allArg <- c(object, list(...)) object$allArg$weights <- NULL object$allArg$call <- NULL if(is(object$g, "formula")) { dat <- getDat(object$g, object$x, data = object$data) k <- dat$k if (is.null(object$tet0)) { if (!is.null(object$eqConst)) stop("You have to provide tet0 with equality constrains") if (object$optfct == "optimize") stop("For optimize, you must provide the 2x1 vector tet0") res0 <- gmm(object$g, object$x, data=object$data) object$tet0 <- res0$coefficients if (object$smooth) gt <- res0$gt } else { if (object$optfct == "optimize") { if (k != 1) stop("optimize() is for univariate optimization") if (length(object$tet0) != 2) stop("For optimize(), tet0 must be a 2x1 vector") } else { if (k != length(object$tet0)) stop("The number of starting values does not correspond to the number of regressors") } if (object$smooth) gt <- gmm(object$g, object$x, data=object$data)$gt } clname <- paste(class(object), ".modFormula", sep = "") object$gradv <- .DmomentFct object$gradvf <- FALSE object$x <- dat object$gform<-object$g namex <- colnames(dat$x[,(dat$ny+1):(dat$ny+dat$k), drop=FALSE]) nameh <- colnames(dat$x[,(dat$ny+dat$k+1):(dat$ny+dat$k+dat$nh), drop=FALSE]) if (dat$ny > 1) { namey <- colnames(dat$x[,1:dat$ny]) namesCoef <- paste(rep(namey, dat$k), "_", rep(namex, rep(dat$ny, dat$k)), sep = "") object$namesgt <- paste(rep(namey, dat$nh), "_", rep(nameh, rep(dat$ny, dat$nh)), sep = "") } else { namesCoef <- namex object$namesgt <- nameh } if (is.null(names(object$tet0))) object$namesCoef <- namesCoef else object$namesCoef <- names(object$tet0) attr(object$x,"ModelType") <- "linear" attr(object$x, "k") <- k attr(object$x, "q") <- object$x$ny*object$x$nh attr(object$x, "n") <- NROW(object$x$x) } else { if (is.null(object$tet0)) stop("You must provide the starting values tet0 for nonlinear moments") if(any(object$optfct == c("optim", "nlminb"))) k <- length(object$tet0) else k <- 1 attr(object$x,"ModelType") <- "nonlinear" attr(object$x, "momentfct") <- object$g attr(object$x, "k") <- k attr(object$x, "q") <- NCOL(object$g(object$tet0, object$x)) attr(object$x, "n") <- NROW(object$x) if(is.null(names(object$tet0))) object$namesCoef <- paste("Theta[" ,1:attr(object$x, "k"), "]", sep = "") else object$namesCoef <- names(object$tet0) if (!is.function(object$gradv) | object$smooth) { object$gradvf <- FALSE } else { attr(object$x, "gradv") <- object$gradv object$gradvf <- TRUE } object$gradv <- .DmomentFct if (object$smooth) gt <- gmm(object$g, object$x, object$tet0, wmatrix = "ident", ...)$gt clname <- paste(class(object), ".mod", sep = "") } if (object$smooth) { if (is.function(object$gradv)) warning("The provided gradv is not used when smooth=TRUE", call. = FALSE) if(object$kernel == "Truncated") { object$wkernel <- "Bartlett" object$k1 <- 2 object$k2 <- 2 } if(object$kernel == "Bartlett") { object$wkernel <- "Parzen" object$k1 <- 1 object$k2 <- 2/3 } gt <- scale(gt, scale=FALSE) class(gt) <- "gmmFct" if (is.function(object$bw)) object$bwVal <- object$bw(gt, kernel = object$wkernel, prewhite = object$prewhite, ar.method = object$ar.method, approx = object$approx) else object$bwVal <- object$bw object$w <- smoothG(gt, bw = object$bwVal, kernel = object$kernel, tol = object$tol_weights)$kern_weights attr(object$x,"smooth") <- list(bw=object$bwVal, w=object$w, kernel=object$kernel) } else { object$k1 <- 1 object$k2 <- 1 object$w <- kernel(1) object$bwVal <- 1 } object$g <- .momentFct object$CGEL <- object$alpha object$typeDesc <- object$type class(object) <- clname return(object) } getModel.ateGel <- function(object, ...) { object$allArg <- c(object, list(...)) object$allArg$weights <- NULL object$allArg$call <- NULL if(is(object$g, "formula")) { dat <- getDat(object$g, object$x, data = object$data) if (!is.null(object$w)) if (is(object$w, "formula")) { dat$w <- model.matrix(object$w, object$data)[,-1,drop=FALSE] } else { stop("w must be a formula") } if (dat$ny>1 | dat$ny==0) stop("You need one and only one dependent variable") k <- dat$k if (k>2 & object$momType=="ATT") stop("Cannot compute ATT with multiple treatments") if (attr(dat$mt, "intercept")!=1) stop("An intercept is needed to compute the treatment effect") if (!all(dat$x[,3:(k+1)] %in% c(0,1))) stop("The treatment indicators can only take values 0 or 1") if (colnames(dat$x)[k+2] == "(Intercept)") { dat$x <- dat$x[,-(k+2)] dat$nh <- dat$nh-1 } if (!is.null(object$popMom)) { if (length(object$popMom)!=dat$nh) stop("The dim. of the population moments does not match the dim. of the vector of covariates") } if (is.null(object$tet0)) { if (is.null(dat$w)) { tet0 <- lm(dat$x[,1]~dat$x[,3:(k+1)])$coef } else { tet0 <- lm(dat$x[,1]~dat$x[,3:(k+1)]+dat$w)$coef } tet0 <- c(tet0, colMeans(dat$x[,3:(k+1),drop=FALSE])) names(tet0) <- NULL object$tet0 <- tet0 } else { ntet0 <- 2*k-1 + ifelse(is.null(dat$w), 0, ncol(dat$w)) if (length(object$tet0) != ntet0) stop("tet0 should have a length equal to 2x(number of treatments)+1+number of w's if any") } if (object$family != "linear") { if (any(!(dat$x[,1]%in%c(0,1)))) stop("For logit or probit, Y can only take 0s and 1s") family <- binomial(link=object$family) if (object$family == "logit") family$mu.eta2 <- function(x, family) family$mu.eta(x)*(1-2*family$linkinv(x)) else family$mu.eta2 <- function(x, family) -x*family$mu.eta(x) } else { family <- NULL } q <- dat$nh + 2*k+1 if (object$momType != "bal" | !is.null(object$popMom)) q <- q+dat$nh if (!is.null(dat$w)) { q <- q+ncol(dat$w) namew <- colnames(dat$w) } else { namew <- NULL } object$gradv <- .DmomentFctATE object$x <- dat object$gradvf <- FALSE object$gform<-object$g namex <- colnames(dat$x[,2:(k+1)]) nameh <- colnames(dat$x[,(k+2):ncol(dat$x), drop=FALSE]) namesCoef <- c(namex, namew, paste("TreatProb", 1:(k-1), sep="")) namesgt <- paste(rep(paste("Treat", 1:(k-1), sep=""), rep(dat$nh, k-1)), "_", rep(nameh, k-1), sep="") object$namesgt <- c(namesCoef,namesgt) if (object$momType != "bal" | !is.null(object$popMom)) object$namesgt <- c(object$namesgt, paste("bal_", nameh, sep="")) if (is.null(names(object$tet0))) object$namesCoef <- namesCoef else object$namesCoef <- names(object$tet0) attr(object$x,"ModelType") <- "linear" attr(object$x, "k") <- k attr(object$x, "q") <- q attr(object$x, "n") <- nrow(dat$x) attr(object$x, "momType") <- object$momType attr(object$x, "popMom") <- object$popMom attr(object$x, "family") <- family } else { stop("Not implemented yet for nonlinear regression") } if (object$momType == "ATT") metname <- "ATT" else metname <- "ATE" if (!is.null(object$popMom)) { metname2 <- " with balancing based on population moments" } else { if (object$momType == "balSample") metname2 <- " with balancing based on the sample moments" else metname2 <- " with unrestricted balancing" } metname3 <- paste("\nMethod: ", object$type, sep="") if (!is.null(family)) metname3 <- paste(metname3, ", Family: Binomial with ", family$link, " link", sep="") clname <- "baseGel.mod" object$k1 <- 1 object$k2 <- 1 object$w <- kernel(1) object$bwVal <- 1 object$g <- .momentFctATE object$CGEL <- object$alpha object$typeDesc <- paste(metname, metname2, metname3, sep="") class(object) <- clname return(object) } gmm/R/charStable.R0000644000176200001440000000401211741057114013424 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ charStable <- function(theta, tau, pm = 0) { # pm is the type parametrization as described by Nolan(2009) # It takes the value 0 or 1 # const can fixe parameters. It is NULL for no constraint or # a matrix in which case the constraint is theta[const[,1]]=const[,2] a <- theta[1] b <- theta[2] g <- theta[3] d <- theta[4] if(pm == 0) { if(a == 1) { if(g == 0) { the_car <- exp(complex(imaginary = d*tau)) } else { re_p <- -g * abs(tau) im_p <- d * tau im_p[tau!=0] <- im_p[tau != 0] + re_p[tau != 0]*2/pi*b*sign(tau[tau != 0])*log(g*abs(tau[tau != 0])) the_car <- exp(complex(real = re_p, imaginary = im_p)) } } else { if(g == 0) { the_car <- exp(complex(imaginary = d*tau)) } else { phi <- tan(pi*a/2) re_p <- -g^a*abs(tau)^a im_p <- d*tau*1i im_p[tau != 0] <- im_p[tau != 0] + re_p*( b*phi*sign(tau[tau != 0])*(abs(g*tau[tau != 0])^(1-a) - 1) ) the_car <- exp(complex(real = re_p, imaginary = im_p)) } } } if(pm == 1) { if(a == 1) { re_p <- -g*abs(tau) im_p <- d*tau im_p[tau!=0] <- im_p[tau != 0]+re_p*(b*2/pi*sign(tau[tau != 0])*log(abs(tau[tau!=0]))) the_car <- exp(complex(real = re_p, imaginary = im_p)) } else { phi <- tan(pi*a/2) re_p <- -g^a*abs(tau)^a im_p <- re_p*(-phi*b*sign(tau)) + d*tau the_car <- exp(complex(real = re_p, imaginary = im_p)) } } return(the_car) } gmm/R/ategel.R0000644000176200001440000003442513467070535012641 0ustar liggesusersATEgel <- function(g, balm, w=NULL, y=NULL, treat=NULL, tet0=NULL, momType=c("bal","balSample","ATT"), popMom = NULL, family=c("linear","logit", "probit"), type = c("EL", "ET", "CUE", "ETEL", "HD", "ETHD","RCUE"), tol_lam = 1e-9, tol_obj = 1e-9, tol_mom = 1e-9, maxiterlam = 100, optfct = c("optim", "nlminb"), optlam = c("nlminb", "optim", "iter", "Wu"), data=NULL, Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, ...) { type <- match.arg(type) optfct <- match.arg(optfct) optlam <- match.arg(optlam) momType <- match.arg(momType) family <- match.arg(family) TypeGel <- "ateGel" all_args <- list(g = g, x = balm, w=w, y=y, treat=treat, tet0 = tet0, type = type, tol_lam = tol_lam, tol_obj = tol_obj, tol_mom = tol_mom, maxiterlam = maxiterlam, optfct = optfct, optlam = optlam, model = model, X = X, Y = Y, TypeGel = TypeGel, call = match.call(), Lambdacontrol = Lambdacontrol, data = data, constraint=FALSE, kernel = "Truncated", bw = bwAndrews, approx = "AR(1)", prewhite = 1, ar.method = "ols", tol_weights = 1e-7, alpha = NULL, eqConst = NULL, eqConstFullVcov = FALSE, momType=momType, popMom=popMom, family=family, onlyCoefficients=FALSE) class(all_args)<-TypeGel Model_info<-getModel(all_args) z <- momentEstim(Model_info, ...) class(z) <- c("ategel", "gel") res <- try(.vcovate(z), silent=TRUE) if (any(class(res)=="try-error")) { warning("Could not compute the robust-to misspecification standard errors") z$robVcov <- FALSE } else { z$vcov_par <- res$vcov_par z$vcov_lambda <- res$vcov_lambda z$robVcov <- TRUE } return(z) } .momentFctATE <- function(tet, dat) { x <- dat$x k <- dat$k if (is.null(dat$w)) { Z <- x[,2:(k+1),drop=FALSE] } else { Z <- cbind(x[,2:(k+1)], dat$w) } tetz <- tet[1:ncol(Z)] tetb <- tail(tet, k-1) ZT <- c(Z%*%tetz) if (is.null(attr(dat, "family"))) e <- x[,1] - ZT else e <- x[,1] - attr(dat, "family")$linkinv(ZT) gt1 <- e * Z gt2 <- sweep(x[,3:(k+1),drop=FALSE],2,tetb,"-") gt3 <- lapply(1:(k-1), function(i) gt2[,i]*x[,-(1:(k+1))]) gt3 <- do.call(cbind, gt3) gt <- cbind(gt1,gt2,gt3) if (is.null(attr(dat, "popMom"))) { if (attr(dat, "momType") == "balSample") { momB <- scale(x[,-(1:(k+1)),drop=FALSE], scale=FALSE) gt <- cbind(gt, momB) } if (attr(dat, "momType") == "ATT") { momB <- sweep(x[,-(1:3), drop=FALSE], 2, colMeans(x[x[,3]==1,-(1:3), drop=FALSE]), FUN="-") gt <- cbind(gt, momB) } } else { momB <- sweep(x[,-(1:(k+1)), drop=FALSE], 2, attr(dat, "popMom"), "-") gt <- cbind(gt, momB) } return(as.matrix(gt)) } .DmomentFctATE <- function(tet, dat, pt=NULL) { if (is.null(pt)) pt <- rep(1/nrow(dat$x), nrow(dat$x)) x <- dat$x k <- dat$k q <- dat$nh*(k-1)+2*k-1 if (is.null(dat$w)) { Z <- x[,2:(k+1),drop=FALSE] } else { Z <- cbind(x[,2:(k+1)], dat$w) q <- q+ncol(dat$w) } l <- ncol(Z) ntet <- length(tet) ZT <- c(Z%*%tet[1:l]) G <- matrix(0, q, ntet) if (is.null(attr(dat, "family"))) { tau <- rep(1, nrow(x)) } else { tau <- attr(dat, "family")$mu.eta(ZT) } G11 <- lapply(1:l, function(i) -colSums(pt*Z[,i]*tau*Z)) G[1:l, 1:l] <- do.call(rbind, G11) G[(l+1):ntet, (l+1):ntet] <- -sum(pt)*diag(k-1) uK <- colSums(pt*x[,-(1:(k+1)),drop=FALSE]) G[(l+k):q, (l+1):ntet] <- -kronecker(diag(k-1), uK) if (attr(dat, "momType") != "bal" | !is.null(attr(dat, "popMom"))) G <- rbind(G, matrix(0, dat$nh, ntet)) return(G) } .DmomentFctATE2 <- function(tet, dat, pt=NULL) { G <- .DmomentFctATE(tet, dat, pt) #k <- attr(dat, "k") k <- dat$k q <- nrow(G)-dat$nh if (is.null(pt)) pt <- rep(1/nrow(dat$x), nrow(dat$x)) if (attr(dat, "momType") != "bal" & is.null(attr(dat, "popMom"))) G <- cbind(G, rbind(matrix(0,q, dat$nh), -sum(pt)*diag(dat$nh))) return(G) } .psiGam <- function(object) { n <- nrow(object$dat$x) nh <- object$dat$nh lam <- object$lambda q <- length(lam) k <- attr(object$dat, "k") theta <- object$coefficients gt <- object$gt rho1 <- .rho(x=gt, lamb=lam, derive=1, type=object$type) rho2 <- .rho(x=gt, lamb=lam, derive=2, type=object$type) if (is.null(object$dat$w)) { Z <- object$dat$x[,2:(k+1)] } else { Z <- cbind(object$dat$x[,2:(k+1)], object$dat$w) } l <- ncol(Z) ZT <- c(Z%*%theta[1:l]) X <- object$dat$x[,-(1:(k+1)), drop=FALSE] family <- attr(object$dat, "family") momType <- attr(object$dat, "momType") popMom <- attr(object$dat, "popMom") if (is.null(family)) { tau1 <- rep(1, n) } else { tau1 <- family$mu.eta(ZT) tau2 <- family$mu.eta2(ZT, family) } lG1 <- sapply(1:l, function(i) -(tau1*Z[,i]*Z)%*%lam[1:l]) q2 <- nh*(k-1)+l+k-1 lamM <- matrix(lam[(l+k):q2], ncol=(k-1)) lG2 <- sapply(1:(k-1), function(i) -lam[l+i]-X%*%lamM[,i]) lG <- cbind(lG1, lG2) G <- .DmomentFctATE2(theta, object$dat, rho1) G22 <- crossprod(rho2*gt, gt)/n if (momType == "bal" | !is.null(popMom)) { Psi <- cbind(rho1*lG, rho1*gt) G11 <- crossprod(rho2*lG, lG)/n G12 <- t(G)/n + crossprod(rho2*lG, gt)/n if (!is.null(family)) { G12tmp <- lapply(1:l, function(i) colSums(-rho1*tau2*Z[,i]*c(Z%*%lam[1:l])*Z)) G12.2 <- matrix(0, nrow(G12), ncol(G12)) G12.2[1:l,1:l] <- do.call(rbind, G12tmp) G12 <- G12 + G12.2/n } Gamma <- rbind(cbind(G11, G12), cbind(t(G12), G22)) addPar <- 0 } else { lG <- cbind(lG, matrix(-tail(lam, nh), n, nh, byrow=TRUE)) G11 <- crossprod(rho2*lG, lG)/n G12 <- t(G)/n + crossprod(rho2*lG, gt)/n if (!is.null(family)) { G12tmp <- lapply(1:l, function(i) colSums(-rho1*tau2*Z[,i]*c(Z%*%lam[1:l])*Z)) G12.2 <- matrix(0, nrow(G12), ncol(G12)) G12.2[1:l,1:l] <- do.call(rbind, G12tmp) G12 <- G12 + G12.2/n } if (momType == "balSample") Xi <- rep(1,n) else Xi <- Z[,(2:k)] nj <- sum(Xi) lam2 <- -sum(rho1)*tail(lam,nh)/nj theta4 <- colSums(Xi*X)/nj G13 <- rbind(matrix(0, l+k-1, nh), -nj/n*diag(nh)) G23 <- matrix(0,q, nh) G33 <- matrix(0, nh, nh) Psi <- cbind(rho1*lG, rho1*gt, Xi*sweep(X, 2, theta4, "-")) Psi[,(l+k):(l+k+nh-1)] <- Psi[,(l+k):(l+k+nh-1)]-Xi%*%t(lam2) Gamma <- rbind(cbind(G11, G12, G13), cbind(t(G12), G22, G23), cbind(t(G13), t(G23), G33)) addPar <- nh } list(Psi=Psi, Gamma=Gamma, k=length(theta), q=q, addPar=addPar, n=n) } .vcovate <- function (object) { res <- .psiGam(object) k <- res$k q <- res$q addPar <- res$addPar qrPsi <- qr(res$Psi/sqrt(res$n)) piv <- sort.int(qrPsi$pivot, index.return=TRUE)$ix R <- qr.R(qrPsi)[,piv] T1 <- solve(res$Gamma, t(R)) V <- T1%*%t(T1)/res$n allV <- list() allV$vcov_par <- V[1:k, 1:k] allV$vcov_lambda <- V[(k+addPar+1):(k+addPar+q), (k+addPar+1):(k+addPar+q)] if (addPar > 0) { allV$vcov_Allpar <- V[1:(k+addPar), 1:(k+addPar)] allV$vcov_Alllambda <- V[-(1:(k+addPar)), -(1:(k+addPar))] } allV } vcov.ategel <- function(object, lambda = FALSE, robToMiss=TRUE, ...) { if (robToMiss) { return(vcov.gel(object, lambda)) } else { object$lambda <- rep(0, length(object$lambda)) res <- .vcovate(object) object$vcov_par <- res$vcov_par object$vcov_lambda <- res$vcov_lambda return(vcov.gel(object, lambda)) } } summary.ategel <- function(object, robToMiss=TRUE, ...) { if (robToMiss) { ans <- summary.gel(object) ans$typeDesc = paste(ans$typeDesc, "\n(S.E. are robust to misspecification)", sep="") } else { object$vcov_par <- vcov(object, robToMiss=FALSE) object$vcov_lambda <- vcov(object, TRUE, robToMiss=FALSE) ans <- summary.gel(object) ans$typeDesc = paste(ans$typeDesc, "\n(S.E. are not robust to misspecification", sep="") } ans } confint.ategel <- function (object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, robToMiss=TRUE, ...) { type <- match.arg(type) if (type=="Wald") { if (!robToMiss) { object$vcov_par <- vcov(object, robToMiss=FALSE) object$vcov_lambda <- vcov(object, TRUE, robToMiss=FALSE) } return(confint.gel(object, parm, level, lambda, type, fact, corr, ...)) } object$allArg$g <- .momentFctATE object$allArg$y <- NULL object$allArg$w <- NULL object$allArg$treat <- NULL object$allArg$popMom <- NULL object$allArg$momType <- NULL object$allArg$family <- NULL object$allArg$x <- object$dat return(confint.gel(object, parm, level, lambda, type, fact, corr, ...)) } marginal <- function(object, ...) UseMethod("marginal") marginal.ategel <- function(object, ...) { family <- attr(object$dat, "family") if (is.null(family)) return(summary(object)$coef) k <- attr(object$dat, "k") p0 <- family$linkinv(object$coef[1]) p1 <- family$linkinv(object$coef[1]+object$coef[2:k]) p01 <- family$mu.eta(object$coef[1]) p11 <- family$mu.eta(object$coef[1]+object$coef[2:k]) A <- cbind(p11-p01, p11) V <- vcov(object, ...)[1:k,1:k] sd0 <- p01*sqrt(V[1,1]) sdd <- sapply(1:(k-1), function(i) sqrt(c(t(A[i,])%*%V[c(1,i+1), c(1, i+1)]%*%A[i,]))) coef <- cbind(c(p0,p1-p0), c(sd0, sdd)) coef <- cbind(coef, coef[,1]/coef[,2]) coef <- cbind(coef, 2*pnorm(-abs(coef[,3]))) colnames(coef) <- c("Estimate", "Std. Error", "t value", "Pr(>|t|)") rownames(coef) <- c("Control", paste("Treat", 1:(k-1) , " versus Control", sep="")) coef } checkConv <- function(obj, tolConv=1e-4,verbose=TRUE, ...) { if (!any(class(obj)=="ategel")) stop("The function is for ategel objects produced by ATEgel()") momType <- obj$allArg$momType popMom <- obj$allArg$popMom conv <- c(Lambda=obj$conv_lambda$convergence==0, Coef= obj$conv_par == 0) dat <- obj$dat$x nZ <- attr(obj$dat, "k")-1 z <- dat[,3:(2+nZ),drop=FALSE] x <- dat[,-(1:(2+nZ)),drop=FALSE] pt <- getImpProb(obj, ...) pt1 <- lapply(1:nZ, function(i) pt[z[,i]==1]/sum(pt[z[,i]==1])) pt0 <- pt[rowSums(z)==0]/sum(pt[rowSums(z)==0]) m0 <- colSums(x[rowSums(z)==0,,drop=FALSE]*pt0) m1 <- sapply(1:nZ, function(i) colSums(x[z[,i]==1,,drop=FALSE]*pt1[[i]])) mAll <- cbind(m0, m1) n0 <- paste(paste(colnames(z),collapse="=", sep=""),"=0",sep="") colnames(mAll) <- c(n0, paste(colnames(z),"=1",sep="")) if (!is.null(popMom)) { m <- popMom } else { m <- switch(momType, bal=m0, balSample=colMeans(x), ATT=c(m1)) } chk <- all(abs(mAll-m)0,pchisq(j,x$df, lower.tail = FALSE), "*******"))) } J_test <- noquote(paste("J-Test: degrees of freedom is ",x$df,sep="")) dimnames(j) <- list("Test E(g)=0: ", c("J-test", "P-value")) ans<-list(ntest=J_test, test = j) class(ans) <- "specTest" ans } print.specTest <- function(x, digits=5, ...) { cat("\n","## ",x$ntest," ##","\n\n") print.default(format(x$test, digits=digits), print.gap = 2, quote = FALSE) cat("\n") invisible(x) } specTest.gel <- function(x, ...) { n <- nrow(x$gt) khat <- x$khat gbar <- colMeans(x$gt) LR_test <- 2*x$objective*n*x$k2/(x$bwVal*x$k1^2) if (x$type == "ETHD") LR_test <- LR_test*2 LM_test <- n*crossprod(x$lambda, crossprod(khat, x$lambda))/(x$bwVal^2) J_test <- n*crossprod(gbar, solve(khat, gbar))/(x$k1^2) test <- c(LR_test, LM_test, J_test) df <- x$df ntest <- noquote(paste("Over-identifying restrictions tests: degrees of freedom is ", df, sep = "")) vptest <- pchisq(test,df,lower.tail = FALSE) if (df == 0) vptest[] <- "###" test <- cbind(test,vptest) dimnames(test) <- list(c("LR test", "LM test", "J test"), c("statistics", "p-value")) ans <- list(test = test, ntest = ntest) class(ans) <- "specTest" ans } gmm/R/gmm.R0000644000176200001440000005473713571757641012176 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ gmm <- function(g,x,t0=NULL,gradv=NULL, type=c("twoStep","cue","iterative"), wmatrix = c("optimal","ident"), vcov=c("HAC","MDS","iid","TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"),crit=10e-7,bw = bwAndrews, prewhite = 1, ar.method = "ols", approx="AR(1)",tol = 1e-7, itermax=100,optfct=c("optim","optimize","nlminb", "constrOptim"), model=TRUE, X=FALSE, Y=FALSE, TypeGmm = "baseGmm", centeredVcov = TRUE, weightsMatrix = NULL, traceIter = FALSE, data, eqConst = NULL, eqConstFullVcov = FALSE, mustar = NULL, onlyCoefficients=FALSE, ...) { type <- match.arg(type) kernel <- match.arg(kernel) vcov <- match.arg(vcov) wmatrix <- match.arg(wmatrix) optfct <- match.arg(optfct) if (!is.null(eqConst)) TypeGmm <- "constGmm" if(vcov=="TrueFixed" & is.null(weightsMatrix)) stop("TrueFixed vcov only for fixed weighting matrix") if(!is.null(weightsMatrix)) wmatrix <- "optimal" if(missing(data)) data<-NULL all_args<-list(data = data, g = g, x = x, t0 = t0, gradv = gradv, type = type, wmatrix = wmatrix, vcov = vcov, kernel = kernel, crit = crit, bw = bw, prewhite = prewhite, ar.method = ar.method, approx = approx, weightsMatrix = weightsMatrix, centeredVcov = centeredVcov, tol = tol, itermax = itermax, optfct = optfct, model = model, X = X, Y = Y, call = match.call(), traceIter = traceIter, eqConst = eqConst, eqConstFullVcov = eqConstFullVcov, mustar = mustar) class(all_args)<-TypeGmm Model_info<-getModel(all_args, ...) z <- momentEstim(Model_info, ...) if (onlyCoefficients) return(z[c("coefficients","objective")]) z <- FinRes(z, Model_info) z } evalGmm <- function(g, x, t0, tetw=NULL, gradv=NULL, wmatrix = c("optimal","ident"), vcov=c("HAC","iid","TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"),crit=10e-7,bw = bwAndrews, prewhite = FALSE, ar.method = "ols", approx="AR(1)",tol = 1e-7, model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE, weightsMatrix = NULL, data, mustar = NULL) { TypeGmm = "baseGmm" type <- "eval" kernel <- match.arg(kernel) vcov <- match.arg(vcov) wmatrix <- match.arg(wmatrix) if (is.null(tetw) & is.null(weightsMatrix)) stop("If the weighting matrix is not provided, you need to provide the vector of parameters tetw") if(vcov=="TrueFixed" & is.null(weightsMatrix)) stop("TrueFixed vcov only for fixed weighting matrix") if(!is.null(weightsMatrix)) wmatrix <- "optimal" if(missing(data)) data<-NULL all_args<-list(data = data, g = g, x = x, t0 = t0, tetw = tetw, gradv = gradv, type = type, wmatrix = wmatrix, vcov = vcov, kernel = kernel, crit = crit, bw = bw, prewhite = prewhite, ar.method = ar.method, approx = approx, weightsMatrix = weightsMatrix, centeredVcov = centeredVcov, tol = tol, itermax = 100, model = model, X = X, Y = Y, call = match.call(), traceIter = NULL, optfct="optim", eqConst = NULL, eqConstFullVcov = FALSE, mustar = mustar) class(all_args)<-TypeGmm Model_info<-getModel(all_args) class(Model_info) <- "baseGmm.eval" z <- momentEstim(Model_info) z <- FinRes(z, Model_info) z } tsls <- function(g,x,data) { if(class(g)[1] != "formula") stop("2SLS is for linear models expressed as formula only") ans <- gmm(g,x,data=data,vcov="iid", TypeGmm="tsls") ans$met <- "Two Stage Least Squares" ans$call <- match.call() class(ans) <- c("tsls","gmm") ans$vcov <- vcov(ans, type="Classical") return(ans) } .myKernHAC <- function(gmat, obj) { gmat <- as.matrix(gmat) if(obj$centeredVcov) gmat <- scale(gmat, scale=FALSE) class(gmat) <- "gmmFct" AllArg <- obj$WSpec$sandwich AllArg$x <- gmat if (is.function(AllArg$bw)) { if (identical(AllArg$bw, bwWilhelm)) { G <- .DmomentFct(obj$tet, obj$dat) obj <- list(gt=gmat, G=G) class(obj) <- "gmm" } else { obj <- gmat } AllArg$bw <- AllArg$bw(obj, order.by = AllArg$order.by, kernel = AllArg$kernel, prewhite = AllArg$prewhite, ar.method = AllArg$ar.method, approx=AllArg$approx) } weights <- weightsAndrews(x=gmat, bw=AllArg$bw, kernel=AllArg$kernel, prewhite=AllArg$prewhite, tol=AllArg$tol) w <- vcovHAC(x=gmat, order.by=AllArg$order.by, weights=weights, prewhite=AllArg$prewhite, sandwich=FALSE, ar.method=AllArg$ar.method, adjust=FALSE) attr(w,"Spec") <- list(weights = weights, bw = AllArg$bw, kernel = AllArg$kernel) w } getDat <- function (formula, h, data, error=TRUE) { cl <- match.call() mf <- match.call(expand.dots = FALSE) m <- match(c("formula", "data"), names(mf), 0L) mf <- mf[c(1L, m)] mf$drop.unused.levels <- TRUE mf$na.action <- "na.pass" mf[[1L]] <- as.name("model.frame") mf <- eval(mf, parent.frame()) mt <- attr(mf, "terms") y <- as.matrix(model.response(mf, "numeric")) namey <- as.character(formula)[2] if (ncol(y)>1) namey <- paste(namey, ".", 1:ncol(y), sep="") xt <- as.matrix(model.matrix(mt, mf, NULL)) n <- NROW(y) if (inherits(h,'formula')) { tmp <- as.character(formula) termsh <- terms(h) h <- paste(tmp[2], "~", as.character(h)[2], sep="") h <- as.formula(h) mfh <- match.call(expand.dots = FALSE) mh <- match(c("h", "data"), names(mfh), 0L) mfh <- mfh[c(1L, mh)] mfh$formula <- h mfh$h <- NULL mfh$drop.unused.levels <- TRUE mfh$na.action <- "na.pass" mfh[[1L]] <- as.name("model.frame") mfh <- eval(mfh, parent.frame()) mth <- attr(mfh, "terms") h <- as.matrix(model.matrix(mth, mfh, NULL)) } else { h <- as.matrix(h) chkInt <- sapply(1:ncol(h), function(i) all(h[,i]/mean(h[,i]) == 1)) if (sum(chkInt) > 1) stop("Too many intercept in the matrix h") if (any(chkInt)) { h <- h[,!chkInt, drop=FALSE] h <- cbind(1,h) intercept_h <- TRUE } else { if (attr(mt,"intercept")==1) { h <- cbind(1, h) intercept_h <- TRUE } else { intercept_h <- FALSE } } if(is.null(colnames(h))) { if (intercept_h) colnames(h) <- c("(Intercept)",paste("h",1:(ncol(h)-1),sep="")) else colnames(h) <- paste("h",1:ncol(h),sep="") } else { if (intercept_h) colnames(h)[1] <- "(Intercept)" coln_h <- colnames(h) coln_h <- gsub(" ", "", coln_h) chk <- which(coln_h == "") if (length(chk) >0) coln_h[chk] <- paste("h", 1:length(chk), sep="") if (any(duplicated(coln_h))) stop("colnames of the matrix h must be unique") colnames(h) <- coln_h } if (!intercept_h) { hFormula <- paste(colnames(h), collapse="+") hFormula <- as.formula(paste("~", hFormula, "-1", sep="")) } else { hFormula <- paste(colnames(h)[-1], collapse="+") hFormula <- as.formula(paste("~", hFormula, sep="")) } termsh <- terms(hFormula) } ny <- ncol(y) k <- ncol(xt) nh <- ncol(h) if (nh 0) { endo <- xm[, -includeExo, drop = FALSE] endoName <- colnames(endo) if (ncol(endo) != 0) { if (attr(dat$termsh, "intercept") == 1) restsls <- lm(endo~hm[,-1]) else restsls <- lm(endo~hm-1) fsls <- xm fsls[, -includeExo] <- restsls$fitted } else { fsls <- xm restsls <- NULL } } else { if (attr(dat$termsh, "intercept") == 1) restsls <- lm(xm~hm[,-1]) else restsls <- lm(xm~hm-1) fsls <- restsls$fitted endoName <- colnames(xm) } par <- lm.fit(as.matrix(fsls), ym)$coefficients if (ny == 1) { e2sls <- ym-xm%*%par v2sls <- lm.fit(as.matrix(hm), e2sls)$fitted value <- sum(v2sls^2)/sum(e2sls^2) } else { par <- c(t(par)) g2sls <- g(par, dat) w <- crossprod(g2sls)/n gb <- matrix(colMeans(g2sls), ncol = 1) value <- crossprod(gb, solve(w, gb)) } } } else { if (ny>1) { if (inv) { whx <- solve(w, (crossprod(hm, xm) %x% diag(ny))) wvecyh <- solve(w, matrix(crossprod(ym, hm), ncol = 1)) } else { whx <- w%*% (crossprod(hm, xm) %x% diag(ny)) wvecyh <- w%*%matrix(crossprod(ym, hm), ncol = 1) } dg <- gradv(NULL, dat) xx <- crossprod(dg, whx) par <- solve(xx, crossprod(dg, wvecyh)) } else { if (nh>k) { if(inv) xzwz <- crossprod(xm,hm)%*%solve(w,t(hm)) else xzwz <- crossprod(xm,hm)%*%w%*%t(hm) par <- solve(xzwz%*%xm,xzwz%*%ym) } else { par <- solve(crossprod(hm,xm),crossprod(hm,ym)) } } gb <- matrix(colSums(g(par, dat))/n, ncol = 1) if(inv) value <- crossprod(gb, solve(w, gb)) else value <- crossprod(gb, w%*%gb) } res <- list(par = par, value = value) if (!is.null(mustar)) { if (!is.null(type)) { w <- crossprod(hm)/NROW(hm) attr(w, "inv") <- TRUE } res <- .mustarLin(par, xm, hm, w, dat, mustar) } if (!is.null(type)) { if (type == "2sls") res$firstStageReg <- restsls if (!is.null(restsls)) { chk <- .chkPerfectFit(restsls) res$fsRes <- suppressWarnings(summary(restsls))[!chk] attr(res$fsRes, "Endo") <- endoName[!chk] } } return(res) } .mustarLin <- function(par, xm, hm, w, dat, mustar) { if (ncol(xm) == ncol(hm)) { par <- par-solve(crossprod(hm,xm),mustar) } else { hmxm <- crossprod(hm,xm) if (attr(w, "inv")) T1 <- solve(w, hmxm) else T1 <- w%*%hmxm par <- par-solve(crossprod(hmxm, T1), crossprod(T1, mustar)) } gb <- colSums(.momentFct(par, dat))/NROW(xm) if(attr(w, "inv")) value <- crossprod(gb, solve(w, gb)) else value <- crossprod(gb, w%*%gb) list(value=value, par=par) } .obj1 <- function(thet, x, w) { gt <- .momentFct(thet, x) gbar <- as.vector(colMeans(gt)) INV <- attr(w, "inv") if (INV) obj <- crossprod(gbar, solve(w, gbar)) else obj <- crossprod(gbar,w)%*%gbar return(obj) } .Gf <- function(thet, x, pt = NULL) { myenv <- new.env() assign('x', x, envir = myenv) assign('thet', thet, envir = myenv) barg <- function(x, thet) { gt <- .momentFct(thet, x) if (is.null(pt)) gbar <- as.vector(colMeans(gt)) else gbar <- as.vector(colSums(c(pt)*gt)) return(gbar) } G <- attr(numericDeriv(quote(barg(x, thet)), "thet", myenv), "gradient") return(G) } .objCue <- function(thet, x, type=c("HAC", "MDS", "iid", "ident", "fct", "fixed")) { type <- match.arg(type) gt <- .momentFct(thet, x) gbar <- as.vector(colMeans(gt)) w <- .weightFct(thet, x, type) if (attr(w, "inv")) obj <- crossprod(gbar,solve(w,gbar)) else obj <- crossprod(gbar,w%*%gbar) return(obj) } .momentFct <- function(tet, dat) { if (!is.null(attr(dat, "eqConst"))) { resTet <- attr(dat,"eqConst")$eqConst tet2 <- vector(length=length(tet)+nrow(resTet)) tet2[resTet[,1]] <- resTet[,2] tet2[-resTet[,1]] <- tet } else { tet2 <- tet } if (attr(dat, "ModelType") == "linear") { x <- dat$x ny <- dat$ny nh <- dat$nh k <- dat$k tet2 <- matrix(tet2, ncol = k) e <- x[,1:ny] - x[,(ny+1):(ny+k)] %*% t(tet2) gt <- e * x[, ny+k+1] if(nh > 1) for (i in 2:nh) gt <- cbind(gt, e*x[, (ny+k+i)]) } else { gt <- attr(dat,"momentfct")(tet2, dat) } if (!is.null(attr(dat, "smooth"))) { bw <- attr(dat, "smooth")$bw w <- attr(dat, "smooth")$w gt <- smoothG(gt, bw = bw, weights = w)$smoothx } gt <- as.matrix(gt) if (!is.null(attr(dat, "mustar"))) { if (length(attr(dat, "mustar")) != ncol(gt)) stop("dim of mustar must match the number of moment conditions") gt <- sweep(gt, 2, attr(dat, "mustar"), "-") } return(gt) } .DmomentFct <- function(tet, dat, pt=NULL) { if (!is.null(attr(dat, "eqConst"))) { resTet <- attr(dat,"eqConst")$eqConst tet2 <- vector(length=length(tet)+nrow(resTet)) tet2[resTet[,1]] <- resTet[,2] tet2[-resTet[,1]] <- tet } else { tet2 <- tet } if ((attr(dat,"ModelType") == "linear") & (is.null(attr(dat, "smooth")))) { x <- dat$x ny <- dat$ny nh <- dat$nh k <- dat$k dgb <- -(t(x[,(ny+k+1):(ny+k+nh)]) %*% x[,(ny+1):(ny+k)]) %x% diag(rep(1,ny))/nrow(x) if (!is.null(attr(dat, "eqConst"))) dgb <- dgb[,-attr(dat,"eqConst")$eqConst[,1], drop=FALSE] } else { if (is.null(attr(dat,"gradv"))) { dgb <- .Gf(tet, dat, pt) } else { dgb <- attr(dat,"gradv")(tet2, dat) if (!is.null(attr(dat, "eqConst"))) dgb <- dgb[,-attr(dat,"eqConst")$eqConst[,1], drop=FALSE] } } return(as.matrix(dgb)) } .weightFct <- function(tet, dat, type=c("HAC", "MDS", "iid", "ident", "fct", "fixed")) { type <- match.arg(type) if (type == "fixed") { w <- attr(dat, "weight")$w attr(w, "inv") <- FALSE } else if (type == "ident") { w <- diag(attr(dat, "q")) attr(w, "inv") <- FALSE } else { gt <- .momentFct(tet,dat) if (!is.null(attr(dat, "namesgt"))) colnames(gt) <- attr(dat, "namesgt") if(attr(dat, "weight")$centeredVcov) gt <- scale(gt, scale=FALSE) n <- NROW(gt) } if (type == "HAC") { obj <- attr(dat, "weight") obj$centeredVcov <- FALSE obj$tet <- tet obj$dat <- dat w <- .myKernHAC(gt, obj) attr(w, "inv") <- TRUE } if (type == "MDS") { w <- crossprod(gt)/n attr(w, "inv") <- TRUE } if (type == "iid") { if (attr(dat, "ModelType") == "linear") { if (dat$ny == 1) { e <- .residuals(tet, dat)$residuals sig <- mean(scale(e,scale=FALSE)^2) z <- dat$x[,(1+dat$ny+dat$k):ncol(dat$x)] w <- sig*crossprod(z)/length(e) } else { w <- crossprod(gt)/n } } else { w <- crossprod(gt)/n } attr(w, "inv") <- TRUE } if (type == "fct") { w <- attr(dat, "weight")$fct(gt, attr(dat, "weight")$fctArg) attr(w, "inv") <- TRUE } return(w) } .residuals <- function(tet, dat) { if (!is.null(attr(dat, "eqConst"))) { resTet <- attr(dat,"eqConst")$eqConst tet2 <- vector(length=length(tet)+nrow(resTet)) tet2[resTet[,1]] <- resTet[,2] tet2[-resTet[,1]] <- tet } else { tet2 <- tet } tet2 <- t(matrix(tet2, nrow = dat$ny)) y <- as.matrix(dat$x[,1:dat$ny]) x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) yhat <- x%*%tet2 e <- y-yhat return(list(residuals=e, yhat=yhat)) } gmm/R/FinRes.R0000644000176200001440000001776313571757641012602 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ FinRes <- function(z, object, ...) { # object is computed by the getModel method # UseMethod("FinRes") } FinRes.baseGmm.res <- function(z, object, ...) { P <- object x <- z$dat n <- ifelse(is.null(nrow(z$gt)),length(z$gt),nrow(z$gt)) if (!is.null(attr(x,"eqConst")) & P$allArg$eqConstFullVcov) { eqConst <- attr(x,"eqConst")$eqConst coef <- rep(0,length(eqConst[,1])+length(z$coefficients)) ncoef <- rep("",length(eqConst[,1])+length(z$coefficients)) coef[-eqConst[,1]] <- z$coefficients ncoef[-eqConst[,1]] <- names(z$coefficients) coef[eqConst[,1]] <- eqConst[,2] ncoef[eqConst[,1]] <- rownames(eqConst) names(coef) <- ncoef z$coefficients <- coef if (!is.null(z$initTheta)) { initTheta <- rep(0,length(z$coefficients)) initTheta[eqConst[,1]] <- eqConst[,2] initTheta[-eqConst[,1]] <- z$initTheta z$initTheta <- initTheta } z$k <- z$k+nrow(eqConst) z$k2 <- z$k2+nrow(eqConst) attr(x, "eqConst") <- NULL z$specMod <- paste(z$specMod, "** Note: Covariance matrix computed for all coefficients based on restricted values \n Tests non-valid**\n\n") } z$G <- z$gradv(z$coefficients, x) G <- z$G if (P$vcov == "TrueFixed") v <- .weightFct(z$coefficient, x, "fixed") else v <- .weightFct(z$coefficient, x, P$vcov) z$v <- v if (P$vcov == "TrueFixed") { z$vcov=try(solve(crossprod(G, P$weightsMatrix) %*% G)/n, silent = TRUE) if(any(class(z$vcov) == "try-error")) { z$vcov <- matrix(Inf,length(z$coef),length(z$coef)) warning("The covariance matrix of the coefficients is singular") } } else if ( (is.null(P$weightsMatrix)) & (P$wmatrix != "ident") ) { if (dim(G)[1] == dim(G)[2]) { T1 <- try(solve(G), silent=TRUE) z$vcov <- try(T1%*%v%*%t(T1)/n, silent=TRUE) } else { z$vcov <- try(solve(crossprod(G, solve(v, G)))/n, silent = TRUE) } if(any(class(z$vcov) == "try-error")) { z$vcov <- matrix(Inf,length(z$coef),length(z$coef)) warning("The covariance matrix of the coefficients is singular") } } else { if (is.null(P$weightsMatrix)) w <- diag(ncol(z$gt)) else w <- P$weightsMatrix if (dim(G)[1] == dim(G)[2]){ T1 <- try(solve(G), silent=TRUE) } else { T1 <- try(solve(t(G)%*%w%*%G,t(G)%*%w), silent = TRUE) } if(any(class(T1) == "try-error")) { z$vcov <- matrix(Inf,length(z$coef),length(z$coef)) warning("The covariance matrix of the coefficients is singular") } else { z$vcov <- T1%*%v%*%t(T1)/n } } dimnames(z$vcov) <- list(names(z$coefficients), names(z$coefficients)) z$call <- P$call if(is.null(P$weightsMatrix)) { if(P$wmatrix == "ident") { z$w <- diag(ncol(z$gt)) } else { z$w <- try(solve(v), silent = TRUE) if(any(class(z$w) == "try-error")) warning("The covariance matrix of the moment function is singular") } } else { z$w <- P$weightsMatrix } z$weightsMatrix <- P$weightsMatrix z$infVcov <- P$vcov z$infWmatrix <- P$wmatrix z$allArg <- P$allArg if (P$wmatrix=="ident") z$met <- "One step GMM with W = identity" else z$met <- P$type z$kernel <- P$kernel z$coefficients <- c(z$coefficients) class(z) <- "gmm" return(z) } FinRes.sysGmm.res <- function(z, object, ...) { P <- object x <- z$dat z$G <- z$gradv(z$coefficients, x) n <- z$n G <- z$G v <- .weightFct_Sys(z$coefficient, x, P$vcov) nk <- z$k z$v <- v if (P$vcov == "TrueFixed") { z$vcov=try(solve(crossprod(G, P$weightsMatrix) %*% G)/n, silent = TRUE) if(any(class(z$vcov) == "try-error")) { z$vcov <- matrix(Inf,nk,nk) warning("The covariance matrix of the coefficients is singular") } } else if ( (is.null(P$weightsMatrix)) & (P$wmatrix != "ident") ) { z$vcov <- try(solve(crossprod(G, solve(v, G)))/n, silent = TRUE) if(any(class(z$vcov) == "try-error")) { z$vcov <- matrix(Inf,nk,nk) warning("The covariance matrix of the coefficients is singular") } } else { if (is.null(P$weightsMatrix)) w <- .weightFct_Sys(z$coefficients, x, "ident") else w <- P$weightsMatrix if (dim(G)[1] == dim(G)[2]){ T1 <- try(solve(G), silent=TRUE) } else { T1 <- try(solve(t(G)%*%w%*%G,t(G)%*%w), silent = TRUE) } if(any(class(T1) == "try-error")) { z$vcov <- matrix(Inf, nk, nk) warning("The covariance matrix of the coefficients is singular") } else { z$vcov <- T1%*%v%*%t(T1)/n } } if (attr(x, "sysInfo")$commonCoef) dimnames(z$vcov) <- list(P$namesCoef[[1]], P$namesCoef[[1]]) else dimnames(z$vcov) <- list(do.call(c, P$namesCoef), do.call(c, P$namesCoef)) z$call <- P$call if(is.null(P$weightsMatrix)) { if(P$wmatrix == "ident") { z$w <- .weightFct_Sys(z$coefficient, x, "ident") } else { z$w <- try(solve(v), silent = TRUE) if(any(class(z$w) == "try-error")) warning("The covariance matrix of the moment function is singular") } } else { z$w <- P$weightsMatrix } for (i in 1:length(z$coefficients)) names(z$coefficients[[i]]) <- P$namesCoef[[i]] z$weightsMatrix <- P$weightsMatrix z$infVcov <- P$vcov z$infWmatrix <- P$wmatrix z$allArg <- P$allArg z$met <- paste("System of Equations: ", P$type, sep="") z$kernel <- P$kernel z$coefficients <- c(z$coefficients) class(z) <- c("sysGmm", "gmm") return(z) } gmm/R/gmmTests.R0000644000176200001440000001354413571757641013210 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ # This function compute what is needed for the K statistics of KleiBergen (2005) ##################################################################################### .BigCov <- function(obj,theta0) { insertRC <- function(A,w,v) { NewA <- matrix(ncol=ncol(A)+length(w),nrow=nrow(A)+length(w)) NewA[-w,-w] <- A NewA[w,] <- v NewA[,w] <- v NewA } dg <- function(obj) { dat <- obj$dat x <- dat$x[,(dat$ny+1):(dat$ny+dat$k)] k <- ncol(x) h <- dat$x[,(dat$ny+dat$k+1):ncol(dat$x)] qt <- array(dim=c(dim(obj$gt),k)) for (i in 1:k) qt[,,i] <- -x[,i]*h qt} if (attr(obj$dat,"ModelType") == "nonlinear") { Myenv <- new.env() assign("obj", obj, envir=Myenv) assign("theta", theta0, envir=Myenv) gFunct <- obj$g assign("g",gFunct,envir=Myenv) res <- numericDeriv(quote(g(theta,obj$dat)),"theta",Myenv) qT <- attr(res,"gradient") } else { qT <- dg(obj)} qTmat <- apply(qT,3,colSums) qT <- matrix(qT,nrow=dim(qT)[1]) gt <- obj$g(theta0,obj$dat) fT <- colSums(gt) n <- nrow(gt) q <- ncol(gt) All <- cbind(gt,qT) All <- sweep(All,2,colMeans(All),FUN="-") f <- function(x) all(abs(x)<1e-7) w <- which(apply(All,2,f)) if (length(w) != 0) All <- All[,-w] if (dim(All)[2] >= dim(All)[1]) stop("Too many moment conditions. Cannot estimate V") if (attr(obj$dat, "weight")$vcov == "iid") { V <- crossprod(All)/nrow(All) } else { class(All) <- "gmmFct" argSand <- attr(obj$dat, "weight")$WSpec$sandwich argSand$x <- All argSand$sandwich <- FALSE V <- do.call(kernHAC,argSand) } if (length(w) != 0) V <- insertRC(V,w,0) Vff <- V[1:q,1:q] Vthetaf <- V[(q+1):nrow(V),1:q] list(Vff=Vff,Vthetaf=Vthetaf,qT=qTmat,fT=fT,n=n,q=q) } KTest <- function(obj, theta0=NULL, alphaK = 0.04, alphaJ = 0.01) { if (class(obj)[1] != "gmm") stop("KTest is only for gmm type objects") if (!is.null(attr(obj$dat,"eqConst"))) { if (!is.null(theta0)) warning("setting a value for theta0 has no effect when the gmm is already constrained") resTet <- attr(obj$dat,"eqConst")$eqConst tet <- obj$coefficients theta0 <- vector(length=length(tet)+nrow(resTet)) theta0[resTet[,1]] <- resTet[,2] theta0[-resTet[,1]] <- tet testName <- paste(rownames(resTet), " = ", resTet[,2], collapse="\n") if (attr(obj$dat,"ModelType") == "linear") { obj$dat$k <- length(theta0) } dfK <- nrow(resTet) which <- resTet[,1] attr(obj$dat, "eqConst") <- NULL } else { if (is.null(theta0)) stop("You must either estimate a restricted model first or set theta0 under H0") if (length(theta0) != length(obj$coef)) stop("theta0 is only for tests on the whole vector theta when obj is an unrestricted GMM") dfK <- length(theta0) testName <- paste(names(obj$coef), " = ", theta0, collapse="\n") which <- 1:length(theta0) } V <- .BigCov(obj, theta0) Vff <- V$Vff Vtf <- V$Vthetaf qT <- V$qT fT <- V$fT dfJ <- V$q-length(theta0) # the following is vec(D) D <- c(qT)-Vtf%*%solve(Vff,fT) D <- matrix(D,ncol=length(theta0)) meat <- t(D)%*%solve(Vff,D) bread <- t(fT)%*%solve(Vff,D) K <- bread%*%solve(meat,t(bread))/V$n pv <- 1-pchisq(K,dfK) S <- t(fT)%*%solve(Vff,fT)/V$n J <- S-K dfS <- dfK + dfJ pvJ <- 1-pchisq(J,dfJ) type <- c("K statistics","J statistics", "S statistics") test <- c(K,J,S) pvS <- 1-pchisq(S,dfS) test <- cbind(test,c(pv,pvJ,pvS),c(dfK,dfJ,dfS)) dist <- paste("Chi_sq with ", c(dfK,dfJ,dfS), " degrees of freedom", sep="") if(dfJ>0) ans <- list(test=test,dist=dist,type=type,testName=testName) else ans <- list(test=matrix(test[1,],nrow=1),dist=dist[1],type=type[1],testName=testName) if (pvJ=length(obj$coefficients)) stop("Too many constraints") if (is.character(which)) { if (any(!(which %in% names(obj$coefficients)))) stop("Wrong coefficient names in eqConst") if (attr(obj$dat,"ModelType") == "linear") which <- match(which,names(obj$coefficients)) } if (!is.null(argCall$t0)) { argCall$t0 <- obj$coefficients argCall$t0[which] <- value } if (attr(obj$dat,"ModelType") == "nonlinear") { eqConst <- which } else { eqConst <- cbind(which,value) } argCall$eqConst <- eqConst res <- do.call(gmm,argCall) res$call <- match.call() return(res) } gmm/R/bandwidth.R0000644000176200001440000001232412717151771013335 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ bwWilhelm <- function(x, order.by = NULL, kernel = c("Quadratic Spectral", "Bartlett", "Parzen", "Tukey-Hanning"), approx = c("AR(1)", "ARMA(1,1)"), weights = NULL, prewhite = 1, ar.method = "ols", data=list()) { ## ensure that NAs are omitted if(is.list(x) && !is.null(x$na.action)) class(x$na.action) <- "omit" # ensure that x is a gmm object stopifnot(attr(x,"class") == "gmm") kernel <- match.arg(kernel) approx <- match.arg(approx) prewhite <- as.integer(prewhite) # the following line applies only to the case class(res)=="gmm" umat <- x$gt n <- nrow(umat) k <- ncol(umat) # the following line applies only to the case class(res)=="gmm" G <- x$G p <- ncol(G) # return Andrews' bandwidth in the just-identified case if (p==k) { class(umat) <- "gmmFct" return(bwAndrews(umat, order.by=order.by, kernel=kernel, approx=approx, weights=weights, prewhite=prewhite, ar.method=ar.method, data=data)) } if(!is.null(order.by)) { if(inherits(order.by, "formula")) { z <- model.matrix(order.by, data = data) z <- as.vector(z[,ncol(z)]) } else { z <- order.by } index <- order(z) } else { index <- 1:n } umat <- umat[index, , drop = FALSE] ## compute weights (w_a, see p. 834) ## (try to set the intercept weight to 0) #### could be ignored by using: weights = 1 if(is.null(weights)) { weights <- rep(1, p) unames <- colnames(umat) if(!is.null(unames) && "(Intercept)" %in% unames) weights[which(unames == "(Intercept)")] <- 0 else { res <- try(as.vector(rowMeans(estfun(x)/model.matrix(x), na.rm = TRUE)), silent = TRUE) if(inherits(res, "try-error")) res <- try(residuals(x), silent = TRUE) if(!inherits(res, "try-error")) weights[which(colSums((umat - res)^2) < 1e-16)] <- 0 } if(isTRUE(all.equal(weights, rep(0, p)))) weights <- rep(1, p) } else { weights <- rep(weights, length.out = p) } if(length(weights) < 2) weights <- 1 ## pre-whitening if(prewhite > 0) { var.fit <- ar(umat, order.max = prewhite, demean = FALSE, aic = FALSE, method = ar.method) if(inherits(var.fit, "try-error")) stop(sprintf("VAR(%i) prewhitening of estimating functions failed", prewhite)) umat <- as.matrix(na.omit(var.fit$resid)) n <- n - prewhite } # define kernel constants kernConst <- switch(kernel, "Bartlett" = list(q = 1, g_q = 1, mu1 = 1, mu2 = 2/3), "Parzen" = list(q = 2, g_q = 6, mu1 = 0.75, mu2 = 0.539286), "Tukey-Hanning" = list(q = 2, g_q = pi^2/4, mu1 = 1, mu2 = 0.75), "Quadratic Spectral" = list(q = 2, g_q = 1.421223, mu1 = 1.25003, mu2 = 0.999985)) # fit approximating model to moments if(approx == "AR(1)") { fitAR1 <- function(x) { rval <- ar(x, order.max = 1, aic = FALSE, method = "ols") rval <- c(rval$ar, sqrt(rval$var.pred)) names(rval) <- c("rho", "sigma") return(rval) } ar.coef <- apply(umat, 2, fitAR1) Omega0 <- ar.coef["sigma", ]^2 / (1-ar.coef["rho", ])^2 Omega_q <- if(kernConst$q == 1) { diag(2 * ar.coef["sigma", ]^2 * ar.coef["rho", ] / ((1 - ar.coef["rho", ])^3 * (1 + ar.coef["rho", ]))) } else { diag(2 * ar.coef["sigma", ]^2 * ar.coef["rho", ] / (1 - ar.coef["rho", ])^4) } } else { fitARMA11 <- function(x) { rval <- arima(x, order = c(1, 0, 1), include.mean = FALSE) rval <- c(rval$coef, sqrt(rval$sigma2)) names(rval) <- c("rho", "psi", "sigma") return(rval) } arma.coef <- apply(umat, 2, fitARMA11) Omega0 <- (1 + ar.coef["psi", ])^2 * ar.coef["sigma", ]^2 / (1 - ar.coef["rho", ])^2 Omega_q <- if(kernConst$q == 1) { diag(2 * (1 + ar.coef["psi", ] * ar.coef["rho", ]) * (ar.coef["psi", ] + ar.coef["rho", ]) * ar.coef["sigma", ]^2 / ((1 - ar.coef["rho", ])^3 * (1 + ar.coef["rho", ]))) } else { diag(2 * (1 + ar.coef["psi", ] * ar.coef["rho", ]) * (ar.coef["psi", ] + ar.coef["rho", ]) * ar.coef["sigma", ]^2 / (1 - ar.coef["rho", ])^4) } } # compute remaining bandwidth components W <- diag(weights) Omega0inv <- diag(1/Omega0) Sigma0 <- solve( crossprod(Omega0inv%*%G, G) ) H0 <- Sigma0 %*% t(G) %*% Omega0inv P0 <- Omega0inv - Omega0inv %*% G %*% H0 # compute optimal bandwidth nu2 <- (2 * kernConst$mu1 + kernConst$mu2) * (k - p) * sum(diag(Sigma0 %*% W)) nu3 <- kernConst$g_q^2 * sum(diag(t(Omega_q) %*% t(H0) %*% W %*% H0 %*% Omega_q %*% P0)) c0 <- if(nu2 * nu3 > 0) 2 * kernConst$q else -1 rval <- (c0 * nu3/nu2 * n)^(1/(1 + 2 * kernConst$q)) return(rval) } gmm/R/momentEstim.R0000644000176200001440000013227413571757641013710 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ momentEstim <- function(object, ...) { UseMethod("momentEstim") } momentEstim.sysGmm.twoStep.formula <- function(object, ...) { dat <- object$x y <- lapply(1:length(dat), function(i) dat[[i]]$x[,1]) y <- do.call(c, y) z <- lapply(1:length(dat), function(i) dat[[i]]$x[,(2+dat[[i]]$k):ncol(dat[[i]]$x), drop=FALSE]) z <- .diagMatrix(z) x <- lapply(1:length(dat), function(i) dat[[i]]$x[,2:(dat[[i]]$k+1), drop=FALSE]) if (attr(dat, "sysInfo")$commonCoef) { x <- do.call(rbind, x) } else if (!is.null(attr(dat, "sysInfo")$crossEquConst)) { k <- attr(dat, "k")[[1]] x <- .diagMatrix(x, (1:k)[-attr(dat, "sysInfo")$crossEquConst]) } else { x <- .diagMatrix(x) } names(y) <- rownames(x) <- rownames(z) <- 1:length(y) data <- list(y=y, x=x, z=z) dat2 <- getDat(y~x-1, ~z-1, data=data) attr(dat2, "ModelType") <- "linear" res0 <- .tetlin(dat2, 1, "2sls") tet0 <- .getThetaList(res0$par, dat) fsRes <- res0$fsRes w <- .weightFct_Sys(tet=tet0, dat=dat, type=object$vcov) #return(list(w=w,tet0=tet0,dat=dat,dat2=dat2)) res <- .tetlin(dat2, w) par <- .getThetaList(res$par, dat) names(par) <- names(dat) df <- ncol(z) - ncol(x) k <- ncol(x) q <- ncol(z) n <- nrow(x) df.residuals <- n - k z = list(coefficients = par, objective = res$value, dat=dat, k=k, q=q, df=df, df.residual=df.residual, n=n) z$gt <- object$g(z$coefficients, dat) z$initTheta <- tet0 tmp <- lapply(1:length(dat), function(i) .residuals(z$coefficients[[i]], dat[[i]])) z$fitted.values <- lapply(1:length(dat), function(i) tmp[[i]]$yhat) z$residuals <- lapply(1:length(dat), function(i) tmp[[i]]$residuals) z$terms <- lapply(1:length(dat), function(i) dat[[i]]$mt) if(object$model) z$model <- lapply(1:length(dat), function(i) dat[[i]]$mf) if(object$X) z$x <- lapply(1:length(dat), function(i) as.matrix(dat[[i]]$x[,(dat[[i]]$ny+1):(dat[[i]]$ny+dat[[i]]$k)])) if(object$Y) z$y <- lapply(1:length(dat), function(i) as.matrix(dat[[i]]$x[,1:dat[[i]]$ny])) z$gradv <- object$gradv z$g <- object$g z$WSpec <- object$WSpec z$w0 <- w colnames(z$gt) <- do.call(c, object$namesgt) z$fsRes <- fsRes class(z) <- "sysGmm.res" z$specMod <- object$specMod return(z) } momentEstim.baseGmm.eval <- function(object, ...) { P <- object x <- P$x n <- attr(x, "n") q <- attr(x, "q") k <- attr(x, "k") k2 <- k df <- q - k if (is.null(object$tetw)) object$tetw <- object$t0 if (is.null(attr(x, "weight")$w)) w <- .weightFct(object$tetw, x, P$vcov) else w <- .weightFct(object$tetw, x, "fixed") fct <- .obj1(object$t0, x, w) if (attr(x,"ModelType") == "linear") { z = list(coefficients = object$t0, objective = fct, dat = x, k = k, k2 = k2, n = n, q = q, df = df, df.residual = (n-k)) b <- z$coefficients tmp <- .residuals(b, x) z$fitted.values <- tmp$yhat z$residuals <- tmp$rediduals z$terms <- x$mt if(P$model) z$model <- x$mf if(P$X) z$x <- as.matrix(x$x[,(x$ny+1):(x$ny+x$k)]) if(P$Y) z$y <- as.matrix(x$x[,1:x$ny]) } else { attr z = list(coefficients = object$t0, objective = fct, k=k, k2=k2, n=n, q=q, df=df, initTheta = object$t0, dat=x) } z$gt <- .momentFct(z$coefficients, x) if (is.null(colnames(z$gt))) colnames(z$gt) <- paste("gt",1:ncol(z$gt),sep="") else colnames(z$gt) <- P$namesgt z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec z$w0 <- w names(z$coefficients) <- P$namesCoef class(z) <- paste(P$TypeGmm,".res",sep="") z$specMod <- P$specMod return(z) } momentEstim.baseGmm.twoStep <- function(object, ...) { P <- object x <- P$x n <- attr(x, "n") q <- attr(x, "q") k <- attr(x, "k") k2 <- k df <- q - k w = .weightFct(NULL, x, "ident") chkOptim <- any(P$optfct == c("optim", "constrOptim")) if (chkOptim) { if (P$gradvf) { gr2 <- function(thet, x, w) { gt <- .momentFct(thet, x) Gbar <- .DmomentFct(thet, x) gbar <- as.vector(colMeans(gt)) INV <- attr(w, "inv") if (INV) obj <- crossprod(Gbar, solve(w, gbar)) else obj <- crossprod(Gbar,w)%*%gbar return(obj*2) } } else { gr2 <- NULL } } if (P$optfct == "optim") { argDots <- list(...) allArgOptim <- list(par = P$t0, fn = .obj1, gr = gr2, x = x, w = w) allArgOptim <- c(allArgOptim,argDots) res <- do.call(optim,allArgOptim) } if (P$optfct == "constrOptim") { if (!any(c("ui","ci") %in% names(list(...)))) stop("You must specify ui and ci when optfct is set to constrOptim") argDots <- list(...) ui <- argDots$ui ci <- argDots$ci argDots$ui <- NULL argDots$ci <- NULL allArgOptim <- list(theta = P$t0, f = .obj1, grad = gr2, ui = ui, ci = ci, x = x, w = w) allArgOptim <- c(allArgOptim,argDots) res <- do.call(constrOptim,allArgOptim) } if (P$optfct == "nlminb") { res <- nlminb(P$t0, .obj1, x = x, w = w, ...) res$value <- res$objective } if (P$optfct == "optimize") { res <- optimize(.obj1, P$t0, x = x, w = w, ...) res$par <- res$minimum res$value <- res$objective } if (q == k2 | P$wmatrix == "ident") { z = list(coefficients = res$par, objective = res$value, k=k, k2=k2, n=n, q=q, df=df) if (chkOptim) z$algoInfo <- list(convergence = res$convergence, counts = res$counts, message = res$message) else if(P$optfct == "nlminb") z$algoInfo <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) } else { w <- .weightFct(res$par, P$x, P$vcov) if (chkOptim) { allArgOptim$w <- w res2 <- do.call(get(object$optfct),allArgOptim) } if (P$optfct == "nlminb") { res2 <- nlminb(P$t0, .obj1, x = x, w = w, ...) res2$value <- res2$objective } if (P$optfct == "optimize") { res2 <- optimize(.obj1, P$t0, x = x, w = w, ...) res2$par <- res2$minimum res2$value <- res2$objective } z = list(coefficients = res2$par, objective = res2$value, k=k, k2=k2, n=n, q=q, df=df, initTheta = res$par) if (chkOptim) { z$algoInfo <- list(convergence = res2$convergence, counts = res2$counts, message = res2$message) z$InitialAlgoInfo <- list(convergence = res$convergence, counts = res$counts, message = res$message) } else if(P$optfct == "nlminb") { z$algoInfo <- list(convergence = res2$convergence, counts = res2$evaluations, message = res2$message) z$InitialAlgoInfo <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) } } z$dat <- P$x z$gt <- P$g(z$coefficients, P$x) z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec names(z$coefficients) <- P$namesCoef if (is.null(colnames(z$gt))) colnames(z$gt) <- paste("gt",1:ncol(z$gt),sep="") class(z) <- paste(P$TypeGmm,".res",sep="") z$specMod <- P$specMod z$w0 <- w return(z) } momentEstim.tsls.twoStep.formula <- function(object, ...) { P <- object g <- P$g dat <- P$x if (dat$ny > 1) stop("tsls is for one dimentional dependent variable") n <- attr(dat, "n") q <- attr(dat, "q") k <- attr(dat, "k") k2 <- k*dat$ny df <- q-k*dat$ny w = .weightFct(NULL, dat, "ident") if (q == k2) { res2 <- .tetlin(dat, w) z = list(coefficients = res2$par, objective = res2$value, dat = dat, k = k, k2 = k2, n = n, q = q, df = df, df.residual = (n-k)) } else { res2 <- .tetlin(dat, w, type="2sls") z = list(coefficients = res2$par, objective = res2$value, dat=dat, k=k, k2=k2, n=n, q=q, df=df, df.residual = (n-k)) } z$gt <- g(z$coefficients, dat) tmp <- .residuals(z$coefficients, dat) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$terms <- dat$mt if(P$model) z$model <- dat$mf if(P$X) z$x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) if(P$Y) z$y <- as.matrix(dat$x[,1:dat$ny]) z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec z$w0 <- w names(z$coefficients) <- P$namesCoef colnames(z$gt) <- P$namesgt z$fsRes <- res2$fsRes class(z) <- "baseGmm.res" z$specMod <- P$specMod return(z) } momentEstim.baseGmm.twoStep.formula <- function(object, ...) { P <- object g <- P$g dat <- P$x n <- attr(dat, "n") q <- attr(dat, "q") k <- attr(dat, "k") k2 <- k*dat$ny df <- q-k*dat$ny w = .weightFct(NULL, dat, "ident") if (q == k2 | P$wmatrix == "ident") { res2 <- .tetlin(dat, w) z = list(coefficients = res2$par, objective = res2$value, dat = dat, k = k, k2 = k2, n = n, q = q, df = df, df.residual = (n-k)) } else { res1 <- .tetlin(dat, w, type="2sls") initTheta <- res1$par w <- .weightFct(res1$par, dat, P$vcov) res2 <- .tetlin(dat, w) res2$firstStageReg <- res1$firstStageReg res2$fsRes <- res1$fsRes z = list(coefficients = res2$par, objective = res2$value, dat=dat, k=k, k2=k2, n=n, q=q, df=df, initTheta = initTheta, df.residual = (n-k)) } z$gt <- g(z$coefficients, dat) b <- z$coefficients tmp <- .residuals(b, dat) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$terms <- dat$mt if(P$model) z$model <- dat$mf if(P$X) z$x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) if(P$Y) z$y <- as.matrix(dat$x[,1:dat$ny]) z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec z$w0 <- w names(z$coefficients) <- P$namesCoef colnames(z$gt) <- P$namesgt if (P$vcov == "iid" & P$wmatrix != "ident") z$fsRes <- res2$fsRes class(z) <- paste(P$TypeGmm,".res",sep="") z$specMod <- P$specMod return(z) } momentEstim.baseGmm.iterative.formula <- function(object, ...) { P <- object g <- P$g dat <- P$x n <- attr(dat, "n") q <- attr(dat, "q") k <- attr(dat, "k") k2 <- k*dat$ny df <- q-k*dat$ny w = .weightFct(NULL, dat, "ident") if (q == k2 | P$wmatrix == "ident") { res <- .tetlin(dat, w) z = list(coefficients = res$par, objective = res$value, dat = dat, k = k, k2 = k2, n = n, q = q, df = df, df.residual = (n-k)) } else { res <- .tetlin(dat, w, type="2sls") fsRes <- res$fsRes initTheta <- res$par ch <- 100000 j <- 1 while(ch > P$crit) { tet <- res$par w <- .weightFct(tet, dat, P$vcov) res <- .tetlin(dat, w) ch <- crossprod(abs(tet- res$par)/tet)^.5 if (j>P$itermax) { cat("No convergence after ", P$itermax, " iterations") ch <- P$crit } if(P$traceIter) cat("Iter :",j,": value=",res$value,", Coef = ", res$par,"\n") j <- j+1 } z = list(coefficients = res$par, objective = res$value, dat=dat, k=k, k2=k2, n=n, q=q, df=df, initTheta=initTheta, df.residual = (n-k)) } z$gt <- g(z$coefficients, dat) tmp <- .residuals(z$coefficients, dat) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$terms <- dat$mt if(P$model) z$model <- dat$mf if(P$X) z$x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) if(P$Y) z$y <- as.matrix(dat$x[,1:dat$ny]) z$terms <- dat$mt if(P$model) z$model <- dat$mf z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec z$w0 <- w names(z$coefficients) <- P$namesCoef colnames(z$gt) <- P$namesgt if (P$vcov == "iid" & P$wmatrix != "ident") z$fsRes <- fsRes class(z) <- paste(P$TypeGmm,".res",sep="") z$specMod <- P$specMod return(z) } momentEstim.baseGmm.iterative <- function(object, ...) { P <- object x <- P$x n <- attr(x, "n") q <- attr(x, "q") k <- attr(x, "k") k2 <- k df <- q - k w = .weightFct(NULL, x, "ident") chkOptim <- any(P$optfct == c("optim", "constrOptim")) if (chkOptim) { if (P$gradvf) { gr2 <- function(thet, x, w) { gt <- .momentFct(thet, x) Gbar <- .DmomentFct(thet, x) gbar <- as.vector(colMeans(gt)) INV <- attr(w, "inv") if (INV) obj <- crossprod(Gbar, solve(w, gbar)) else obj <- crossprod(Gbar,w)%*%gbar return(obj*2) } } else { gr2 <- NULL } } if (P$optfct == "optim") { argDots <- list(...) allArgOptim <- list(par = P$t0, fn = .obj1, gr = gr2, x = x, w = w) allArgOptim <- c(allArgOptim,argDots) res <- do.call(optim,allArgOptim) } if (P$optfct == "constrOptim") { if (!any(c("ui","ci") %in% names(list(...)))) stop("You must specify ui and ci when optfct is set to constrOptim") argDots <- list(...) ui <- argDots$ui ci <- argDots$ci argDots$ui <- NULL argDots$ci <- NULL allArgOptim <- list(theta = P$t0, f = .obj1, grad = gr2, ui = ui, ci = ci, x = x, w = w) allArgOptim <- c(allArgOptim,argDots) res <- do.call(constrOptim,allArgOptim) } if (P$optfct == "nlminb") { res <- nlminb(P$t0, .obj1, x = x, w = w, ...) res$value <- res$objective } if (P$optfct == "optimize") { res <- optimize(.obj1, P$t0, x = x, w = w, ...) res$par <- res$minimum res$value <- res$objective } if (q == k2 | P$wmatrix == "ident") { z <- list(coefficients = res$par, objective = res$value, k=k, k2=k2, n=n, q=q, df=df) if (chkOptim) z$algoInfo <- list(convergence = res$convergence, counts = res$counts, message = res$message) else if(P$optfct == "nlminb") z$algoInfo <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) } else { initTheta = res$par z <- list() if (chkOptim) z$initialAlgoInfo <- list(convergence = res$convergence, counts = res$counts, message = res$message) else if(P$optfct == "nlminb") z$initialAlgoInfo <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) ch <- 100000 j <- 1 while(ch > P$crit) { tet <- res$par w <- .weightFct(tet, P$x, P$vcov) if (chkOptim) { allArgOptim$w <- w allArgOptim[[1]] <- tet res <- do.call(get(P$optfct),allArgOptim) } if (P$optfct == "nlminb") { res <- nlminb(tet, .obj1, x = P$x, w = w, ...) res$value <- res$objective } if (P$optfct == "optimize") { res <- optimize(.obj1, P$t0, x = P$x, w = w, ...) res$par <- res$minimum res$value <- res$objective } ch <- crossprod(tet-res$par)^.5/(1+crossprod(tet)^.5) if (j>P$itermax) { cat("No convergence after ", P$itermax, " iterations") ch <- P$crit } if(P$traceIter) cat("Iter :",j,": value=",res$value,", Coef = ", res$par,"\n") j <- j+1 } z2 = list(coefficients = res$par, objective = res$value,k=k, k2=k2, n=n, q=q, df=df, initTheta=initTheta) z <- c(z, z2) if (chkOptim) z$algoInfo <- list(convergence = res$convergence, counts = res$counts, message = res$message) else if(P$optfct == "nlminb") z$algoInfo <- list(convergence = res$convergence, counts = res$evaluations, message = res$message) } z$dat <- P$x z$gt <- P$g(z$coefficients, P$x) z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec z$w0 <- w names(z$coefficients) <- P$namesCoef if (is.null(colnames(z$gt))) colnames(z$gt) <- paste("gt",1:ncol(z$gt),sep="") z$specMod <- P$specMod class(z) <- paste(P$TypeGmm,".res",sep="") return(z) } momentEstim.baseGmm.cue.formula <- function(object, ...) { P <- object g <- P$g dat <- P$x n <- attr(dat, "n") q <- attr(dat, "q") k <- attr(dat, "k") k2 <- k*dat$ny df <- q-k*dat$ny w <- .weightFct(NULL, dat, "ident") if (q == k2 | P$wmatrix == "ident") { res <- .tetlin(dat, w) z = list(coefficients = res$par, objective = res$value, dat = dat, k = k, k2 = k2, n = n, q = q, df = df, df.residual = (n-k)) P$weightMessage <- "No CUE needed because the model is just identified" } else { if (is.null(P$t0)) { P$t0 <- .tetlin(dat, w, type="2sls")$par initTheta <- P$t0 } else { initTheta <- P$t0 } if (P$vcov == "HAC") { w <- .weightFct(P$t0, dat, P$vcov) attr(dat, "weight")$WSpec$sandwich$bw <- attr(w,"Spec")$bw P$weightMessage <- "Weights for kernel estimate of the covariance are fixed and based on the first step estimate of Theta" } if (P$optfct == "optim") res2 <- optim(P$t0,.objCue, x = dat, type = P$vcov, ...) if (P$optfct == "constrOptim") { if (!any(c("ui","ci") %in% names(list(...)))) stop("You must specify ui and ci when optfct is set to constrOptim") argDots <- list(...) ui <- argDots$ui ci <- argDots$ci argDots$ui <- NULL argDots$ci <- NULL allArgOptim <- list(theta = P$t0, f = .objCue, grad = NULL, ui = ui, ci = ci, x = dat, type = P$vcov) allArgOptim <- c(allArgOptim,argDots) res2 <- do.call(constrOptim,allArgOptim) } if (P$optfct == "nlminb") { res2 <- nlminb(P$t0,.objCue, x = dat, type = P$vcov, ...) res2$value <- res2$objective } if (P$optfct == "optimize") { res2 <- optimize(.objCue,P$t0, x = dat, type = P$vcov, ...) res2$par <- res2$minimum res2$value <- res2$objective } z = list(coefficients = res2$par, objective = res2$value, dat = dat, k = k, k2 = k2, n = n, q = q, df = df, initTheta=initTheta, df.residual = (n-k)) if (any(P$optfct == c("optim", "constrOptim"))) z$algoInfo <- list(convergence = res2$convergence, counts = res2$counts, message = res2$message) else if(P$optfct == "nlminb") z$algoInfo <- list(convergence = res2$convergence, counts = res2$evaluations, message = res2$message) } z$gt <- g(z$coefficients, dat) tmp <- .residuals(z$coefficients, dat) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$terms <- dat$mt if(P$model) z$model <- dat$mf if(P$X) z$x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) if(P$Y) z$y <- as.matrix(dat$x[,1:dat$ny]) z$dat <- dat z$terms <- dat$mt if(P$model) z$model <- dat$mf z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$specMod <- P$specMod z$cue <- list(weights=P$fixedKernW,message=P$weightMessage) z$WSpec <- P$WSpec z$w0 <- .weightFct(z$coefficients, dat, P$vcov) names(z$coefficients) <- P$namesCoef colnames(z$gt) <- P$namesgt class(z) <- paste(P$TypeGmm,".res",sep="") return(z) } momentEstim.baseGmm.cue <- function(object, ...) { P <- object x <- P$x n <- attr(x, "n") q <- attr(x, "q") k <- attr(x, "k") k2 <- k df <- q - k if (q == k2 | P$wmatrix == "ident") { w = .weightFct(NULL, x, "ident") res <- gmm(P$allArg$g,P$allArg$x,P$t0,wmatrix="ident",optfct=P$optfct, ...) z <- list(coefficients = res$coef, objective = res$objective, algoInfo = res$algoInfo, k=k, k2=k2, n=n, q=q, df=df, initTheta=P$t0) P$weightMessage <- "No CUE needed because the model if just identified or you set wmatrix=identity" } else { w <- .weightFct(P$t0, x, P$vcov) initTheta <- P$t0 if (P$vcov == "HAC") { res <- try(gmm(P$allArg$g,P$allArg$x,P$t0,wmatrix="ident", optfct=P$optfct, ...)) if(any(class(res)=="try-error")) stop("Cannot get a first step estimate to compute the weights for the Kernel estimate of the covariance matrix; try different starting values") w <- .weightFct(res$coefficients, x, P$vcov) attr(x, "weight")$WSpec$sandwich$bw <- attr(w,"Spec")$bw P$weightMessage <- "Weights for kernel estimate of the covariance are fixed and based on the first step estimate of Theta" } else { res <- list() } if (P$optfct == "optim") { res2 <- optim(P$t0, .objCue, x = x, type = P$vcov, ...) } if (P$optfct == "constrOptim") { if (!any(c("ui","ci") %in% names(list(...)))) stop("You must specify ui and ci when optfct is set to constrOptim") argDots <- list(...) ui <- argDots$ui ci <- argDots$ci argDots$ui <- NULL argDots$ci <- NULL allArgOptim <- list(theta = P$t0, f = .objCue, grad = NULL, ui = ui, ci = ci, x = x, type = P$vcov) allArgOptim <- c(allArgOptim,argDots) res2 <- do.call(constrOptim,allArgOptim) } if (P$optfct == "nlminb") { res2 <- nlminb(P$t0, .objCue, x = x, type = P$vcov, ...) res2$value <- res2$objective } if (P$optfct == "optimize") { res2 <- optimize(.objCue,P$t0, x = x, type = P$vcov, ...) res2$par <- res2$minimum res2$value <- res2$objective } z = list(coefficients=res2$par,objective=res2$value, k=k, k2=k2, n=n, q=q, df=df, initTheta=initTheta) if (any(P$optfct == c("optim", "constrOptim"))) { z$algoInfo <- list(convergence = res2$convergence, counts = res2$counts, message = res2$message) z$InitialAlgoInfo <- list(convergence = res$algoInfo$convergence, counts = res$algoInfo$counts, message = res$algoInfo$message) } else if (P$optfct == "nlminb") { z$algoInfo <- list(convergence = res2$convergence, counts = res2$evaluations, message = res2$message) z$InitialAlgoInfo <- list(convergence = res$algoInfo$convergence, counts = res$algoInfo$evaluations, message = res$algoInfo$message) } } z$dat <- x z$gradv <- P$gradv z$gt <- P$g(z$coefficients, x) z$w0 <- .weightFct(z$coefficients, x, P$vcov) z$iid <- P$iid z$g <- P$g z$cue <- list(weights=P$fixedKernW,message=P$weightMessage) names(z$coefficients) <- P$namesCoef if (is.null(colnames(z$gt))) colnames(z$gt) <- paste("gt",1:ncol(z$gt),sep="") z$WSpec <- P$WSpec z$specMod <- P$specMod class(z) <- paste(P$TypeGmm, ".res", sep = "") return(z) } momentEstim.baseGel.modFormula <- function(object, ...) { P <- object g <- P$g q <- attr(P$x, "q") n <- attr(P$x, "n") k <- attr(P$x, "k") df <- q-k*P$x$ny l0Env <- new.env() assign("l0",rep(0,q),envir=l0Env) if (!P$constraint) { if (P$optfct == "optim") res <- optim(P$tet0, .thetf, P = P, l0Env = l0Env, ...) if (P$optfct == "nlminb") res <- nlminb(P$tet0, .thetf, P = P, l0Env = l0Env, ...) if (P$optfct == "optimize") { res <- optimize(.thetf, P$tet0, P = P, l0Env = l0Env, ...) res$par <- res$minimum res$convergence <- "There is no convergence code for optimize" } } if(P$constraint) res <- constrOptim(P$tet0, .thetf, grad = NULL, P = P, l0Env = l0Env, ...) All <- .thetf(res$par, P, "all",l0Env = l0Env) gt <- All$gt rlamb <- All$lambda z <- list(coefficients = res$par, lambda = rlamb$lambda, conv_lambda = rlamb$conv, conv_par = res$convergence, dat=P$x) z$type <- P$type z$gt <- gt pt <- .getImpProb(z$gt, z$lambda, P$type, P$k1, P$k2) z$pt <- c(pt) z$conv_moment <- attr(pt, "conv_moment") z$conv_pt <- attr(pt, "conv_pt") z$objective <- All$obj z$call <- P$call z$k1 <- P$k1 z$k2 <- P$k2 z$CGEL <- P$CGEL z$typeDesc <- P$typeDesc z$specMod <- P$specMod z$df <- df names(z$coefficients) <- object$namesCoef if (P$onlyCoefficients) return(z[c("coefficients","lambda","conv_lambda","conv_par","objective")]) if (!is.null(object$namesgt)) { colnames(z$gt) <- object$namesgt } else { colnames(z$gt) <- paste("g",1:ncol(z$gt), sep="") } names(z$lambda) <- paste("Lam(", colnames(z$gt), ")", sep="") if (!is.null(attr(P$x,"eqConst")) & P$allArg$eqConstFullVcov) { eqConst <- attr(P$x,"eqConst")$eqConst coef <- rep(0,length(eqConst[,1])+length(z$coefficients)) ncoef <- rep("",length(eqConst[,1])+length(z$coefficients)) coef[-eqConst[,1]] <- z$coefficients ncoef[-eqConst[,1]] <- names(z$coefficients) coef[eqConst[,1]] <- eqConst[,2] ncoef[eqConst[,1]] <- rownames(eqConst) names(coef) <- ncoef z$coefficients <- coef attr(P$x, "k") <- attr(P$x, "k") + nrow(eqConst) z$df <- z$df - nrow(eqConst) attr(P$x,"eqConst") <- NULL z$specMod <- paste(z$specMod, "** Note: Covariance matrix computed for all coefficients based on restricted values \n Tests non-valid**\n\n") } if(P$gradvf) G <- P$gradv(z$coefficients, P$x) else G <- P$gradv(z$coefficients, P$x, z$pt) allVcov <- try(.vcovGel(gt, G, P$k1, P$k2, P$bwVal, z$pt), silent=TRUE) if (any(class(allVcov) == "try-error")) { z$vcov_par <- matrix(NA, length(z$coefficients), length(z$coefficients)) z$vcov_lambda <- matrix(NA, length(z$lambda), length(z$lambda)) z$khat <- NULL warning("Cannot compute the covariance matrices") } else { z <- c(z, allVcov) } z$weights <- P$w z$bwVal <- P$bwVal names(z$bwVal) <- "Bandwidth" dimnames(z$vcov_par) <- list(names(z$coefficients), names(z$coefficients)) dimnames(z$vcov_lambda) <- list(names(z$lambda), names(z$lambda)) tmp <- .residuals(z$coefficients, P$x) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$dat <- P$x z$terms <- P$x$mt if(P$model) z$model <- P$x$mf if(P$X) z$x <- as.matrix(P$x$x[,(P$x$ny+1):(P$x$ny+P$x$k)]) if(P$Y) z$y <- as.matrix(P$x$x[,1:P$x$ny]) class(z) <- paste(P$TypeGel, ".res", sep = "") z$allArg <- P$allArg return(z) } momentEstim.baseGel.mod <- function(object, ...) { P <- object x <- P$x q <- attr(x, "q") n <- attr(x, "n") df <- q - attr(x, "k") l0Env <- new.env() assign("l0",rep(0,q),envir=l0Env) if (!P$constraint) { if (P$optfct == "optim") res <- optim(P$tet0, .thetf, P = P, l0Env = l0Env, ...) if (P$optfct == "nlminb") res <- nlminb(P$tet0, .thetf, P = P, l0Env = l0Env, ...) if (P$optfct == "optimize") { res <- optimize(.thetf, P$tet0, P = P, l0Env = l0Env, ...) res$par <- res$minimum res$convergence <- "There is no convergence code for optimize" } } if(P$constraint) res <- constrOptim(P$tet0, .thetf, grad = NULL, P = P,l0Env = l0Env, ...) All <- .thetf(res$par, P, "all",l0Env = l0Env) gt <- All$gt rlamb <- All$lambda z <- list(coefficients = res$par, lambda = rlamb$lambda, conv_lambda = rlamb$conv, conv_par = res$convergence, dat=x) z$type <- P$type z$gt <- gt pt <- .getImpProb(z$gt, z$lambda, P$type, P$k1, P$k2) z$pt <- c(pt) z$conv_moment <- attr(pt, "conv_moment") z$conv_pt <- attr(pt, "conv_pt") z$objective <- All$obj names(z$coefficients) <- P$namesCoef z$specMod <- P$specMod z$df <- df if (!is.null(object$namesgt)) { colnames(z$gt) <- object$namesgt } else { colnames(z$gt) <- paste("g",1:ncol(z$gt), sep="") } names(z$lambda) <- paste("Lam(", colnames(z$gt), ")", sep="") if (!is.null(attr(x,"eqConst")) & P$allArg$eqConstFullVcov) { eqConst <- attr(x,"eqConst")$eqConst coef <- rep(0,length(eqConst[,1])+length(z$coefficients)) ncoef <- rep("",length(eqConst[,1])+length(z$coefficients)) coef[-eqConst[,1]] <- z$coefficients ncoef[-eqConst[,1]] <- names(z$coefficients) coef[eqConst[,1]] <- eqConst[,2] ncoef[eqConst[,1]] <- rownames(eqConst) names(coef) <- ncoef z$coefficients <- coef attr(x, "k") <- attr(x, "k") + nrow(eqConst) z$df <- z$df - nrow(eqConst) attr(x,"eqConst") <- NULL z$specMod <- paste(z$specMod, "** Note: Covariance matrix computed for all coefficients based on restricted values \n Tests non-valid**\n\n") } if (P$onlyCoefficients) return(z[c("coefficients", "lambda", "conv_lambda", "conv_par", "objective")]) if(P$gradvf) G <- P$gradv(z$coefficients, x) else G <- P$gradv(z$coefficients, x, z$pt) z$G <- G allVcov <- try(.vcovGel(gt, G, P$k1, P$k2, P$bwVal, z$pt), silent=TRUE) if (any(class(allVcov) == "try-error")) { z$vcov_par <- matrix(NA, length(z$coefficients), length(z$coefficients)) z$vcov_lambda <- matrix(NA, length(z$lambda), length(z$lambda)) z$khat <- NULL warning("Cannot compute the covariance matrices") } else { z <- c(z, allVcov) } z$weights <- P$w z$bwVal <- P$bwVal names(z$bwVal) <- "Bandwidth" dimnames(z$vcov_par) <- list(names(z$coefficients), names(z$coefficients)) dimnames(z$vcov_lambda) <- list(names(z$lambda), names(z$lambda)) if(P$X) z$x <- x z$call <- P$call z$k1 <- P$k1 z$k2 <- P$k2 z$CGEL <- P$CGEL z$typeDesc <- P$typeDesc z$allArg <- P$allArg class(z) <- paste(P$TypeGel, ".res", sep = "") return(z) } momentEstim.fixedW.formula <- function(object, ...) { P <- object g <- P$g dat <- P$x n <- attr(dat, "n") q <- attr(dat, "q") k <- attr(dat, "k") k2 <- k*dat$ny df <- q-k*dat$ny w = .weightFct(NULL, dat, "fixed") if(!all(dim(w) == c(q,q))) stop("The matrix of weights must be qxq") eigenW <- svd(w)$d if(!is.double(eigenW)) warning("The matrix of weights is not strictly positive definite") if(is.double(eigenW)) { if(any(eigenW<=0)) warning("The matrix of weights is not strictly positive definite") } res2 <- .tetlin(dat, w) z = list(coefficients = res2$par, objective = res2$value, dat=dat, k=k, k2=k2, n=n, q=q, df=df, df.residual = (n-k)) z$gt <- g(z$coefficients, dat) tmp <- .residuals(z$coefficients, dat) z$fitted.values <- tmp$yhat z$residuals <- tmp$rediduals z$terms <- dat$mt if(P$model) z$model <- dat$mf if(P$X) z$x <- as.matrix(dat$x[,(dat$ny+1):(dat$ny+dat$k)]) if(P$Y) z$y <- as.matrix(dat$x[,1:dat$ny]) z$dat <- dat z$terms <- dat$mt if(P$model) z$model <- dat$mf z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec names(z$coefficients) <- P$namesCoef colnames(z$gt) <- P$namesgt z$specMod <- P$specMod class(z) <- paste(P$TypeGmm,".res",sep="") return(z) } momentEstim.fixedW <- function(object, ...) { P <- object x <- P$x n <- attr(x, "n") q <- attr(x, "q") k <- attr(x, "k") k2 <- k df <- q - k w = .weightFct(NULL, x, "fixed") if(!all(dim(w) == c(q,q))) stop("The matrix of weights must be qxq") eigenW <- svd(w)$d if(!is.double(eigenW)) warning("The matrix of weights is not strictly positive definite") if(is.double(eigenW)) { if(any(eigenW<=0)) warning("The matrix of weights is not strictly positive definite") } chkOptim <- any(P$optfct == c("optim", "constrOptim")) if (chkOptim) { if (P$gradvf) { gr2 <- function(thet, x, w) { gt <- .momentFct(thet, x) Gbar <- .DmomentFct(thet, x) gbar <- as.vector(colMeans(gt)) INV <- attr(w, "inv") if (INV) obj <- crossprod(Gbar, solve(w, gbar)) else obj <- crossprod(Gbar,w)%*%gbar return(obj*2) } } else { gr2 <- NULL } } if (P$optfct == "optim") { argDots <- list(...) allArgOptim <- list(par = P$t0, fn = .obj1, gr = gr2, x = x, w = w) argDots$gr <- NULL allArgOptim <- c(allArgOptim,argDots) res2 <- do.call(optim,allArgOptim) } if (P$optfct == "constrOptim") { if (!any(c("ui","ci") %in% names(list(...)))) stop("You must specify ui and ci when optfct is set to constrOptim") argDots <- list(...) ui <- argDots$ui ci <- argDots$ci argDots$ui <- NULL argDots$ci <- NULL allArgOptim <- list(theta = P$t0, f = .obj1, grad = gr2, ui = ui, ci = ci, x = x, w = w) allArgOptim <- c(allArgOptim,argDots) res2 <- do.call(constrOptim,allArgOptim) } if (P$optfct == "nlminb") { res2 <- nlminb(P$t0, .obj1, x = P$x, w = w, ...) res2$value <- res2$objective } if (P$optfct == "optimize") { res2 <- optimize(.obj1, P$t0, x = P$x, w = w, ...) res2$par <- res2$minimum res2$value <- res2$objective } z = list(coefficients = res2$par, objective = res2$value, k=k, k2=k2, n=n, q=q, df=df) if (chkOptim) z$algoInfo <- list(convergence = res2$convergence, counts = res2$counts, message = res2$message) else if(P$optfct == "nlminb") z$algoInfo <- list(convergence = res2$convergence, counts = res2$evaluations, message = res2$message) z$dat <- P$x z$gt <- P$g(z$coefficients, P$x) z$gradv <- P$gradv z$iid <- P$iid z$g <- P$g z$WSpec <- P$WSpec names(z$coefficients) <- P$namesCoef if (is.null(colnames(z$gt))) colnames(z$gt) <- paste("gt",1:ncol(z$gt),sep="") z$specMod <- P$specMod class(z) <- paste(P$TypeGmm,".res",sep="") return(z) } momentEstim.baseGel.eval <- function(object, ...) { P <- object q <- attr(P$x, "q") n <- attr(P$x, "n") l0Env <- new.env() assign("l0",rep(0,q),envir=l0Env) All <- .thetf(P$tet0, P, "all",l0Env = l0Env) gt <- All$gt rlamb <- All$lambda z <- list(coefficients = P$tet0, lambda = rlamb$lambda, conv_lambda = rlamb$conv, conv_par = NULL, dat=P$x) z$type <- P$type z$gt <- gt pt <- .getImpProb(z$gt, z$lambda, P$type, P$k1, P$k2) z$pt <- c(pt) z$conv_moment <- attr(pt, "conv_moment") z$conv_pt <- attr(pt, "conv_pt") z$objective <- All$obj z$call <- P$call z$k1 <- P$k1 z$k2 <- P$k2 z$CGEL <- P$CGEL z$typeDesc <- paste(P$typeDesc, " (Eval only, tests non-valid) ", sep="") z$specMod <- P$specMod names(z$coefficients) <- P$namesCoef z$df <- length(z$lambda) - length(z$coefficients) if (!is.null(object$namesgt)) { colnames(z$gt) <- object$namesgt } else { colnames(z$gt) <- paste("g",1:ncol(z$gt), sep="") } names(z$lambda) <- paste("Lam(", colnames(z$gt), ")", sep="") if(P$gradvf) G <- P$gradv(z$coefficients, P$x) else G <- P$gradv(z$coefficients, P$x, z$pt) z$G <- G allVcov <- .vcovGel(gt, G, P$k1, P$k2, P$bwVal, z$pt) z <- c(z, allVcov) z$weights <- P$w z$bwVal <- P$bwVal names(z$bwVal) <- "Bandwidth" dimnames(z$vcov_par) <- list(names(z$coefficients), names(z$coefficients)) dimnames(z$vcov_lambda) <- list(names(z$lambda), names(z$lambda)) if (attr(P$x,"ModelType") == "linear") { tmp <- .residuals(z$coefficients, P$x) z$fitted.values <- tmp$yhat z$residuals <- tmp$residuals z$terms <- P$x$mt if(P$model) z$model <- P$x$mf if(P$X) z$x <- as.matrix(P$x$x[,(P$x$ny+1):(P$x$ny+P$x$k)]) if(P$Y) z$y <- as.matrix(P$x$x[,1:P$x$ny]) } return(z) } gmm/R/sysGmm.R0000644000176200001440000002725112634331531012645 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ five <- function(g, h, commonCoef = FALSE, data = NULL) { res <- sysGmm(g, h, wmatrix = "optimal", vcov = "CondHom", commonCoef=commonCoef, data=data) attr(res$dat, "sysInfo")$typeDesc <- "Full-Information IV (FIVE)" res$call <- match.call() res } threeSLS <- function(g, h, commonCoef = FALSE, data = NULL) { if (!is(h, "formula")) if (length(h) != 1) stop("In 3SLS, h is a single equation since the same instruments are used in each equation") res <- sysGmm(g, h, vcov = "CondHom", commonCoef=commonCoef, data=data) attr(res$dat, "sysInfo")$typeDesc <- "3-stage Least Squares" res$call <- match.call() res } sur <- function(g, commonCoef = FALSE, data = NULL) { if (!is.list(g)) stop("g must be list of formulas") if (length(g) == 1) stop("For single equation GMM, use the function gmm()") if (!all(sapply(1:length(g), function(i) is(g[[i]], "formula")))) stop("g must be a list of formulas") tm <- lapply(1:length(g), function(i) terms(g[[i]])) reg <- lapply(1:length(g), function(i) attr(tm[[i]], "term.labels")) reg <- unique(do.call(c,reg)) h <- paste(reg, collapse = "+") if (all(sapply(1:length(g), function(i) attr(tm[[i]], "intercept") == 0))) h <- paste(h, "-1") h <- as.formula(paste("~", h)) res <- sysGmm(g, h, vcov = "CondHom", commonCoef=commonCoef, data=data) attr(res$dat, "sysInfo")$typeDesc <- "Seemingly Unrelated Regression (SUR)" res$call <- match.call() res } randEffect <- function(g, data = NULL) { res <- sur(g, commonCoef = TRUE, data = data) attr(res$dat, "sysInfo")$typeDesc <- "Random Effect Estimator" res$call <- match.call() res } sysGmm <- function(g, h, wmatrix = c("optimal","ident"), vcov=c("MDS", "HAC", "CondHom", "TrueFixed"), kernel=c("Quadratic Spectral","Truncated", "Bartlett", "Parzen", "Tukey-Hanning"), crit=10e-7,bw = bwAndrews, prewhite = FALSE, ar.method = "ols", approx="AR(1)",tol = 1e-7, model=TRUE, X=FALSE, Y=FALSE, centeredVcov = TRUE, weightsMatrix = NULL, data, crossEquConst = NULL, commonCoef = FALSE) { kernel <- match.arg(kernel) vcov <- match.arg(vcov) wmatrix <- match.arg(wmatrix) TypeGmm = "sysGmm" if(vcov=="TrueFixed" & is.null(weightsMatrix)) stop("TrueFixed vcov only for fixed weighting matrix") if(!is.null(weightsMatrix)) wmatrix <- "optimal" if(missing(data)) data<-NULL all_args<-list(data = data, g = g, h = h, wmatrix = wmatrix, vcov = vcov, kernel = kernel, crit = crit, bw = bw, prewhite = prewhite, ar.method = ar.method, approx = approx, weightsMatrix = weightsMatrix, centeredVcov = centeredVcov, tol = tol, model = model, X = X, Y = Y, call = match.call(), commonCoef=commonCoef, crossEquConst = crossEquConst) class(all_args)<-TypeGmm Model_info<-getModel(all_args) z <- momentEstim(Model_info) z <- FinRes(z, Model_info) return(z) } .momentFct_Sys <- function(tet, dat) { q <- length(dat) f <- function(i, dat) { d <- dat[[i]] attr(d, "eqConst") <- attr(dat, "eqConst") attr(d, "ModelType") <- attr(dat, "ModelType") attr(d,"momentfct") <- attr(dat,"momentfct") attr(d, "smooth") <- attr(dat, "smooth") .momentFct(tet[[i]], d) } mom <- lapply(1:q, function(i) f(i, dat)) do.call(cbind, mom) } .DmomentFct_Sys <- function(tet, dat, pt=NULL) { q <- length(dat) f <- function(i, dat) { d <- dat[[i]] attr(d, "eqConst") <- attr(dat, "eqConst") attr(d, "ModelType") <- attr(dat, "ModelType") attr(d,"momentfct") <- attr(dat,"momentfct") attr(d, "smooth") <- attr(dat, "smooth") .DmomentFct(tet[[i]], d, pt) } dmom <- lapply(1:q, function(i) f(i, dat)) if (attr(dat, "sysInfo")$commonCoef) do.call(rbind,dmom) else .diagMatrix(dmom) } .weightFct_Sys<- function(tet, dat, type=c("MDS", "HAC", "CondHom", "ident", "fct", "fixed")) { type <- match.arg(type) if (type == "fixed") { w <- attr(dat, "weight")$w attr(w, "inv") <- FALSE } else if (type == "ident") { w <- diag(attr(dat, "q")) attr(w, "inv") <- FALSE } else { if (type == "HAC") { gt <- .momentFct_Sys(tet,dat) if(attr(dat, "weight")$centeredVcov) gt <- residuals(lm(gt~1)) n <- NROW(gt) obj <- attr(dat, "weight") obj$centeredVcov <- FALSE w <- .myKernHAC(gt, obj) attr(w, "inv") <- TRUE } if (type == "MDS") { gt <- .momentFct_Sys(tet,dat) n <- NROW(gt) if(attr(dat, "weight")$centeredVcov) gt <- scale(gt, scale=FALSE) w <- crossprod(gt)/n attr(w, "inv") <- TRUE } if (type == "CondHom") { e <- lapply(1:length(dat), function(i) .residuals(tet[[i]], dat[[i]])$residuals) e <- do.call(cbind, e) Sig <- crossprod(scale(e, scale=FALSE))/nrow(e) Z <- lapply(1:length(dat), function(i) dat[[i]]$x[,(2+dat[[i]]$k):ncol(dat[[i]]$x)]) Z <- do.call(cbind, Z) w <- crossprod(Z)/nrow(e) for (i in 1:length(dat)) for (j in 1:length(dat)) { s1 <- 1+(i-1)*dat[[i]]$nh e1 <- i*dat[[i]]$nh s2 <- 1+(j-1)*dat[[j]]$nh e2 <- j*dat[[j]]$nh w[s1:e1, s2:e2] <- w[s1:e1, s2:e2]*Sig[i,j] } attr(w, "inv") <- TRUE } } return(w) } .diagMatrix <- function(xlist, which=NULL) { # Create block diagonal matrix from matrices with the same number of rows. m <- length(xlist) n <- NROW(xlist[[1]]) l <- sapply(1:m, function(i) dim(as.matrix(xlist[[i]]))) if (!is.null(which)) { if (any(l[2,1]!=l[2,])) stop("diagMatrix with which given is for X with the same number of columns") which <- sort(which) if (length(which) == l[2,1]) { dimX <- rowSums(l) which <- NULL } else { dimX <- c(sum(l[1,]), length(which)*m + l[2,1]-length(which)) } } else { dimX <- rowSums(l) } X <- matrix(0, dimX[1], dimX[2]) if (is.null(which)) { for (i in 1:m) { s1 <- 1 + (i-1)*n e1 <- n*i s2 <- 1 + sum(l[2,][-(i:m)]) e2 <- sum(l[2,][1:i]) X[s1:e1, s2:e2] <- xlist[[i]] } } else { q <- match(1:l[2,1],which) q2 <- which(is.na(q)) wai <- 1 for (j in q2) { if (wai < j) { xlist2 <- lapply(1:length(xlist), function(i) xlist[[i]][,wai:(j-1), drop=FALSE]) k <- j-wai Xtmp <- .diagMatrix(xlist2) X[,wai:(wai+length(xlist)*k-1)] <- Xtmp wai <- wai + length(xlist)*k } X[,wai] <- do.call(c, lapply(1:length(xlist), function(i) xlist[[i]][,j])) wai <- wai+1 } if (max(q2) < l[2,1]) { xlist2 <- lapply(1:length(xlist), function(i) xlist[[i]][,-(1:max(q2))]) Xtmp <- .diagMatrix(xlist2) X[,wai:ncol(X)] <- Xtmp } } X } .getThetaList <- function(tet, dat) { neq <- length(dat) if (attr(dat, "sysInfo")$commonCoef) { k <- attr(dat, "k")[[1]] if (length(tet) != k) stop("Wrong length of tet") tet2 <- rep(list(c(tet)), neq) } else if (!is.null(attr(dat, "sysInfo")$crossEquConst)) { k <- attr(dat, "k")[[1]] cst <- attr(dat, "sysInfo")$crossEquConst nk <- (k-length(cst))*neq + length(cst) if (nk != length(tet)) stop("Wrong length of tet") tet2 <- matrix(NA, k, neq) wai <- 1 for (j in cst) { if (wai < j) { ind <- wai:(j-1) tet2[ind,] <- tet[1:(neq*length(ind))] tet <- tet[-(1:(neq*(j-wai)))] wai <- wai + length(ind) } tet2[wai,] <- tet[1] tet <- tet[-1] wai <- wai+1 } if (max(cst) < k) tet2[(max(cst)+1):k,] <- tet attr(dat, "sysInfo")$crossEquConst <- NULL tet2 <- .getThetaList(c(tet2), dat) } else { k2 <- do.call(c, attr(dat, "k")) if (sum(k2) != length(tet)) stop("Wrong length of tet") tet2 <- list() for (j in 1:length(k2)) { tet2[[j]] <- tet[1:k2[j]] tet <- tet[-(1:k2[j])] } } tet2 } .chkPerfectFit <- function(obj) { r <- as.matrix(obj$residuals) f <- as.matrix(obj$fitted.values) rdf <- obj$df.residual rss <- colSums(r^2) mf <- colMeans(f) mv <- apply(f, 2, var) resvar <- rss/rdf resvar < (mf^2 + mv) * 1e-30 } gmm/R/Methods.gel.R0000644000176200001440000002552713460072542013545 0ustar liggesusers# This program is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation; either version 2 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ .invTest <- function(object, which, level = 0.95, fact = 3, type=c("LR", "LM", "J"), corr=NULL) { type <- match.arg(type) argCall <- object$allArg if (length(which) > 1) stop("tests are inverted only for one parameter") if (is.character(which)) { which <- which(which == names(object$coefficients)) if (length(which) == 0) stop("Wrong parameter names") } wtest <- which(type==c("LR", "LM", "J")) if (object$df > 0) { test0 <- specTest(object)$test[wtest,] test0 <- test0[1] } else { test0 <- 0 } if (length(object$coefficients) == 1) { argCall$optfct <- NULL argCall$constraint <- NULL argCall$eqConst <- NULL argCall$eqConstFullVcov <- NULL fctCall <- "evalGel" } else { fctCall <- "gel" argCall$eqConst <- which } if (length(object$coefficients) == 2) { sdcoef <- sqrt(diag(vcov(object))[-which]) coef <- object$coefficients[-which] upper <- coef + fact*sdcoef lower <- coef - fact*sdcoef argCall$method <- "Brent" argCall$lower <- lower argCall$upper <- upper } sdcoef <- sqrt(diag(vcov(object))[which]) coef <- object$coefficients[which] int1 <- c(coef, coef + fact*sdcoef) int2 <- c(coef - fact*sdcoef, coef) fct <- function(coef, which, wtest, level, test0, corr=NULL) { argCall$tet0 <- object$coefficients argCall$tet0[which] <- coef obj <- do.call(get(fctCall), argCall) test <- as.numeric(specTest(obj)$test[wtest,1]) - test0 if (is.null(corr)) level - pchisq(test, 1) else level - pchisq(test/corr, 1) } res1 <- uniroot(fct, int1, which = which, wtest=wtest, level=level, test0=test0, corr=corr) res2 <- uniroot(fct, int2, which = which, wtest=wtest, level=level, test0=test0, corr=corr) sort(c(res1$root, res2$root)) } confint.gel <- function(object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, ...) { type <- match.arg(type) z <- object n <- nrow(z$gt) if (type == "Wald") { ntest <- "Direct Wald type confidence interval" se_par <- sqrt(diag(z$vcov_par)) par <- z$coefficients tval <- par/se_par se_parl <- sqrt(diag(z$vcov_lambda)) lamb <- z$lambda zs <- qnorm((1 - level)/2, lower.tail=FALSE) ch <- zs*se_par if(!lambda) { ans <- cbind(par-ch, par+ch) dimnames(ans) <- list(names(par), c((1 - level)/2, 0.5+level/2)) } if(lambda) { if (length(z$coefficients) == length(z$lambda)) { cat("\nNo confidence intervals for lambda when the model is just identified.\n") return(NULL) } else { chl <- zs*se_parl ans <- cbind(lamb - chl, lamb + chl) dimnames(ans) <- list(names(lamb), c((1 - level)/2, 0.5 + level/2)) } } if(!missing(parm)) ans <- ans[parm,] } else { if(missing(parm)) parm <- names(object$coefficients) type <- strsplit(type, "v")[[1]][2] ntest <- paste("Confidence interval based on the inversion of the ", type, " test", sep="") ans <- lapply(parm, function(w) .invTest(object, w, level = level, fact = fact, type=type, corr=corr)) ans <- do.call(rbind, ans) if (!is.character(parm)) parm <- names(object$coefficients)[parm] dimnames(ans) <- list(parm, c((1 - level)/2, 0.5+level/2)) } ans <- list(test=ans,ntest=ntest) class(ans) <- "confint" ans } print.confint <- function(x, digits = 5, ...) { cat("\n", x$ntest, sep="") cat("\n#######################################\n") print.default(format(x$test, digits = digits), print.gap = 2, quote = FALSE) invisible(x) } coef.gel <- function(object, lambda = FALSE, ...) { if(!lambda) object$coefficients else object$lambda } vcov.gel <- function(object, lambda = FALSE, ...) { if(!lambda) object$vcov_par else object$vcov_lambda } print.gel <- function(x, digits = 5, ...) { if (is.null(x$CGEL)) cat("Type de GEL: ", x$typeDesc, "\n") else cat("CGEL of type: ", x$typeDesc, " (alpha = ", x$CGEL, ")\n") if (!is.null(attr(x$dat,"smooth"))) { cat("Kernel: ", attr(x$dat,"smooth")$kernel," (bw=", attr(x$dat,"smooth")$bw,")\n\n") } else cat("\n") cat("Coefficients:\n") print.default(format(coef(x), digits = digits), print.gap = 2, quote = FALSE) cat("\n") cat("Lambdas:\n") print.default(format(coef(x, lambda = TRUE), digits = digits), print.gap = 2, quote = FALSE) cat("\n") cat("Convergence code for the coefficients: ", x$conv_par,"\n") cat("Convergence code for Lambda: ", x$conv_lambda$convergence,"\n") cat(x$specMod) invisible(x) } print.summary.gel <- function(x, digits = 5, ...) { cat("\nCall:\n") cat(paste(deparse(x$call), sep = "\n", collapse = "\n"), "\n\n", sep = "") if (is.null(x$CGEL)) cat("Type of GEL: ", x$typeDesc, "\n") else cat("CGEL of type: ", x$typeDesc, " (alpha = ", x$CGEL, ")\n") if (!is.null(x$smooth)) { cat("Kernel: ", x$smooth$kernel," (bw=", x$smooth$bw,")\n\n") }else { cat("\n") } cat("Coefficients:\n") print.default(format(x$coefficients, digits = digits), print.gap = 2, quote = FALSE) if (length(x$coefficients)|t|)")) dimnames(ans$lambda) <- list(names(z$lambda), c("Estimate", "Std. Error", "t value", "Pr(>|t|)")) ans$stest=specTest(z) if (z$type == "EL") ans$badrho <- z$badrho if (!is.null(z$weights)) { ans$weights <- z$weights } ans$conv_par <- z$conv_par ans$conv_pt <- z$conv_pt ans$conv_moment <- cbind(z$conv_moment) ans$conv_lambda <- z$conv_lambda ans$CGEL <- z$CGEL ans$typeDesc <- z$typeDesc ans$specMod <- z$specMod if (!is.null(attr(object$dat,"smooth"))) ans$smooth <- attr(object$dat,"smooth") names(ans$conv_pt) <- "Sum_of_pt" dimnames(ans$conv_moment) <- list(names(z$gt), "Sample_moment_with_pt") class(ans) <- "summary.gel" ans } residuals.gel <- function(object, ...) { if(is.null(object$model)) stop("The residuals method is valid only for g=formula") object$residuals } fitted.gel <- function(object, ...) { if(is.null(object$model)) stop("The residuals method is valid only for g=formula") object$fitted.value } formula.gel <- function(x, ...) { if(is.null(x$terms)) stop("The gel object was not created by a formula") else formula(x$terms) } estfun.gel <- function(x, ...) { stop("estfun is not yet available for gel objects") } bread.gel <- function(x, ...) { stop("Bread is not yet available for gel objects") } getImpProb <- function(object, ...) 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(x - tet[1])^2) m3 <- x^3-tet[1]*(tet[1]^2+3*tet[2]^2) f <- cbind(m1,m2,m3) return(f) } ## ----------------------------------------------------------------------------- Dg <- function(tet,x) { G <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2],-6*tet[1]*tet[2]), nrow=3,ncol=2) return(G) } ## ----------------------------------------------------------------------------- set.seed(123) n <- 200 x1 <- rnorm(n, mean = 4, sd = 2) ## ----------------------------------------------------------------------------- print(res <- gmm(g1,x1,c(mu = 0, sig = 0), grad = Dg)) ## ----------------------------------------------------------------------------- summary(res) ## ----------------------------------------------------------------------------- specTest(res) ## ----------------------------------------------------------------------------- sim_ex <- function(n,iter) { tet1 <- matrix(0,iter,2) tet2 <- tet1 for(i in 1:iter) { x1 <- rnorm(n, mean = 4, sd = 2) tet1[i,1] <- mean(x1) tet1[i,2] <- sqrt(var(x1)*(n-1)/n) tet2[i,] <- gmm(g1,x1,c(0,0),grad=Dg)$coefficients } bias <- cbind(rowMeans(t(tet1)-c(4,2)),rowMeans(t(tet2)-c(4,2))) dimnames(bias)<-list(c("mu","sigma"),c("ML","GMM")) Var <- cbind(diag(var(tet1)),diag(var(tet2))) dimnames(Var)<-list(c("mu","sigma"),c("ML","GMM")) MSE <- cbind(rowMeans((t(tet1)-c(4,2))^2),rowMeans((t(tet2)-c(4,2))^2)) dimnames(MSE)<-list(c("mu","sigma"),c("ML","GMM")) return(list(bias=bias,Variance=Var,MSE=MSE)) } ## ----------------------------------------------------------------------------- g2 <- function(theta,x) { tau <- seq(1,5,length.out=10) pm <- 1 x <- matrix(c(x),ncol=1) x_comp <- x%*%matrix(tau,nrow=1) x_comp <- matrix(complex(ima=x_comp),ncol=length(tau)) emp_car <- exp(x_comp) the_car <- charStable(theta,tau,pm) gt <- t(t(emp_car) - the_car) gt <- cbind(Im(gt),Re(gt)) return(gt) } ## ----------------------------------------------------------------------------- library(stabledist) set.seed(345) x2 <- rstable(500,1.5,.5,pm=1) t0 <- c(alpha = 2, beta = 0, gamma = sd(x2)/sqrt(2), delta = 0) print(res <- gmm(g2,x2,t0)) ## ----------------------------------------------------------------------------- summary(res) ## ----------------------------------------------------------------------------- res2 <- gmm(g2,x2,t0,optfct="nlminb",lower=c(0,-1,0,-Inf),upper=c(2,1,Inf,Inf)) summary(res2) ## ----warning=FALSE------------------------------------------------------------ data(Finance) x3 <- Finance[1:1500,"WMK"] t0<-c(alpha = 1.8, beta = 0.1, gamma = sd(x3)/sqrt(2),delta = 0) res3 <- gmm(g2,x3,t0,optfct="nlminb") summary(res3) ## ----------------------------------------------------------------------------- library(car) linearHypothesis(res3,cbind(diag(2),c(0,0),c(0,0)),c(2,0)) ## ----------------------------------------------------------------------------- library(mvtnorm) set.seed(112233) sig <- matrix(c(1,.5,.5,1),2,2) n <- 400 e <- rmvnorm(n,sigma=sig) x4 <- rnorm(n) w <- exp(-x4^2) + e[,1] y <- 0.1*w + e[,2] ## ----------------------------------------------------------------------------- h <- cbind(x4, x4^2, x4^3) g3 <- y~w ## ----------------------------------------------------------------------------- summary(res <- gmm(g3,x=h)) ## ----------------------------------------------------------------------------- res2 <- gmm(g3,x=h,type='iterative',crit=1e-8,itermax=200) coef(res2) ## ----------------------------------------------------------------------------- res3 <- gmm(g3,x=h,res2$coef,type='cue') coef(res3) ## ----------------------------------------------------------------------------- confint(res3,level=.90) ## ----------------------------------------------------------------------------- plot(w,y,main="LS vs GMM estimation") lines(w,fitted(res),col=2) lines(w,fitted(lm(y~w)),col=3,lty=2) lines(w,.1*w,col=4,lty=3) legend("topleft",c("Data","Fitted GMM","Fitted LS","True line"),pch=c(1,NA,NA,NA),col=1:3,lty=c(NA,1,2,3)) ## ----------------------------------------------------------------------------- t <- 400 set.seed(345) x5 <- arima.sim(n=t,list(ar=c(1.4,-0.6),ma=c(0.6,-0.3))) x5t<-cbind(x5) for (i in 1:6) x5t<-cbind(x5t,lag(x5,-i)) x5t<-na.omit(x5t) g4<-x5t[,1]~x5t[,2]+x5t[,3] res<-gmm(g4,x5t[,4:7]) summary(res) ## ----------------------------------------------------------------------------- res2 <- gmm(g4,x=x5t[,4:7],kernel="Truncated") coef(res2) res3 <- gmm(g4,x=x5t[,4:7],kernel="Bartlett") coef(res3) res4 <- gmm(g4,x=x5t[,4:7],kernel="Parzen") coef(res4) res5<- gmm(g4,x=x5t[,4:7],kernel="Tukey-Hanning") coef(res5) ## ----------------------------------------------------------------------------- diag(vcov(res2))^.5 diag(vcov(res3))^.5 diag(vcov(res4))^.5 diag(vcov(res5))^.5 ## ----------------------------------------------------------------------------- plot(res,which=2) ## ----------------------------------------------------------------------------- plot(res,which=3) ## ----------------------------------------------------------------------------- data(Finance) r <- Finance[1:500,1:5] rm <- Finance[1:500,"rm"] rf <- Finance[1:500,"rf"] z <- as.matrix(r-rf) zm <- as.matrix(rm-rf) res <- gmm(z~zm,x=zm) coef(res) R <- cbind(diag(5),matrix(0,5,5)) c <- rep(0,5) linearHypothesis(res,R,c,test = "Chisq") ## ----eval=FALSE--------------------------------------------------------------- # test <- paste(names(coef(res)[1:5])," = 0",sep="") # linearHypothesis(res,test) ## ----------------------------------------------------------------------------- res2<-gmm(z~zm-1,cbind(1,zm)) specTest(res2) ## ----------------------------------------------------------------------------- g5 <- function(tet, x) { gmat <- (tet[1] + tet[2] * (1 + c(x[, 1]))) * (1 + x[, 2:6]) - 1 return(gmat) } res_sdf <- gmm(g5, x = as.matrix(cbind(rm, r)), c(0, 0)) specTest(res_sdf) ## ----------------------------------------------------------------------------- g6 <- function(theta, x) { t <- length(x) et1 <- diff(x) - theta[1] - theta[2] * x[-t] ht <- et1^2 - theta[3] * x[-t]^(2 * theta[4]) g <- cbind(et1, et1 * x[-t], ht, ht * x[-t]) return(g) } ## ----------------------------------------------------------------------------- rf <- Finance[,"rf"] rf <- ((1 + rf/100)^(365) - 1) * 100 dr <- diff(rf) res_0 <- lm(dr ~ rf[-length(rf)]) tet0 <- c(res_0$coef, var(residuals(res_0)), 0) names(tet0) <- c("alpha", "beta", "sigma^2", "gamma") res_rf <- gmm(g6, rf, tet0, control = list(maxit = 1000, reltol = 1e-10)) coef(res_rf) ## ----eval=FALSE--------------------------------------------------------------- # y <- rbind(y1-mean(y1),y2-mean(y2),y3-mean(y3)) # x <- rbind(x1-mean(x1),x2-mean(x2),x3-mean(x3)) # res <- gmm(y~x,h) ## ----eval=FALSE--------------------------------------------------------------- # y <- rbind(y1,y2,y3) # x <- rbind(x1,x2,x3) # res <- gmm(y~x,h) ## ----eval=FALSE--------------------------------------------------------------- # gt <- g(t0, x) # V <- kernHAC(lm(gt~1),sandwich = FALSE) # W <- solve(V) ## ----------------------------------------------------------------------------- print(res<-gmm(g4,x5t[,4:7],wmatrix="ident")) diag(vcovHAC(res))^.5 ## ----------------------------------------------------------------------------- diag(vcov(res))^.5 ## ----------------------------------------------------------------------------- print(res<-gmm(g4,x5t[,4:7], weightsMatrix = diag(5))) ## ----------------------------------------------------------------------------- tet0 <- c(mu = mean(x1), sig = sd(x1)) res_el <- gel(g1,x1,tet0) summary(res_el) ## ----------------------------------------------------------------------------- res_et <- gel(g1,x1,tet0,type="ET") coef(res_et) ## ----------------------------------------------------------------------------- res_cue <- gel(g1,x1,tet0,type="CUE") coef(res_cue) ## ----------------------------------------------------------------------------- res_etel <- gel(g1,x1,c(mu=1,sig=1),type="ETEL") coef(res_etel) ## ----warning=FALSE------------------------------------------------------------ tet0 <- gmm(g4,x=x5t[,4:7],wmatrix="ident")$coef res <- gel(g4,x=x5t[,4:7],tet0,smooth=TRUE,kernel="Truncated") summary(res) ## ----------------------------------------------------------------------------- specTest(res) ## ----------------------------------------------------------------------------- plot(res,which=4) ## ----eval=FALSE--------------------------------------------------------------- # ui=cbind(0,-1,-1) # ci <- -1 ## ----eval=FALSE--------------------------------------------------------------- # res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE,kernel="Truncated", # constraint=TRUE, ui=ui,ci=ci) ## ----eval=FALSE--------------------------------------------------------------- # res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim") # res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim", # LambdaControl=list(trace=TRUE, parscale=rep(.1,5))) gmm/inst/doc/gmm_with_R.rnw0000644000176200001440000017532314011255710015407 0ustar liggesusers\documentclass[11pt,letterpaper]{article} \usepackage{amsthm} \usepackage[hmargin=2cm,vmargin=2.5cm]{geometry} \newtheorem{theorem}{Theorem} \newtheorem{col}{Corollary} \newtheorem{lem}{Lemma} \usepackage[utf8x]{inputenc} \newtheorem{ass}{Assumption} \usepackage{amsmath} \usepackage{verbatim} \usepackage[round]{natbib} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{graphicx} \usepackage{hyperref} \hypersetup{ colorlinks, citecolor=black, filecolor=black, linkcolor=black, urlcolor=black } \bibliographystyle{plainnat} \newcommand{\E}{\mathrm{E}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\Prob}{\mathrm{Pr}} \newcommand{\Var}{\mathrm{Var}} \let\proglang=\textsf \newcommand{\pkg}[1]{{\fontseries{m}\fontseries{b}\selectfont #1}} \newcommand{\Vect}{\mathrm{Vec}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\conP}{\overset{p}{\to}} \newcommand{\conD}{\overset{d}{\to}} \newcommand\Real{ \mathbb{R} } \newcommand\Complex{ \mathbb{C} } \newcommand\Natural{ \mathbb{N} } \newcommand\rv{{\cal R}} \newcommand\Q{\mathbb{Q}} \newcommand\PR{{\cal R}} \newcommand\T{{\cal T}} \newcommand\Hi{{\cal H}} \newcommand\La{{\cal L}} \newcommand\plim{plim} \renewcommand{\epsilon}{\varepsilon} \begin{document} \author{Pierre Chauss\'e} \title{Computing Generalized Method of Moments and Generalized Empirical Likelihood with \proglang{R}} \maketitle \abstract{This paper shows how to estimate models by the generalized method of moments and the generalized empirical likelihood using the \proglang{R} package \textbf{gmm}. A brief discussion is offered on the theoretical aspects of both methods and the functionality of the package is presented through several examples in economics and finance. It is a modified version of \cite{chausse10} published in the Journal of Statistical Software. It has been adapted to the version 1.4-0. \textbf{Notice that the maintenance of the package is converging to zero. The new \pkg{momentfit} package, available on CRAN, will soon replace the \pkg{gmm} package.}} %\VignetteIndexEntry{Computing Generalized Empirical Likelihood and Generalized Method of Moments with R} %\VignetteDepends{gmm,mvtnorm,stabledist, car, MASS, timeDate, timeSeries} %\VignetteKeywords{generalized empirical likelihood, generalized method of moments, empirical likelihood, continuous updated estimator, exponential tilting, exponentially tilted empirical likelihood} %\VignettePackage{gmm} %\VignetteEngine{knitr::knitr} <>= library(knitr) opts_chunk$set(size='footnotesize', fig.height=5, out.width='70%') @ \section{Introduction} The generalized method of moments (GMM) has become an important estimation procedure in many areas of applied economics and finance since \cite{hansen82} introduced the two step GMM (2SGMM). It can be seen as a generalization of many other estimation methods like least squares (LS), instrumental variables (IV) or maximum likelihood (ML). As a result, it is less likely to be misspecified. The properties of the estimators of LS depend on the exogeneity of the regressors and the circularity of the residuals, while those of ML depend on the choice of the likelihood function. GMM is much more flexible since it only requires some assumptions about moment conditions. In macroeconomics, for example, it allows to estimate a structural model equation by equation. In finance, most data such as stock returns are characterized by heavy-tailed and skewed distributions. Because it does not impose any restriction on the distribution of the data, GMM represents a good alternative in this area as well. As a result of its popularity, most statistical packages like \proglang{Matlab}, \proglang{Gauss} or \proglang{Stata} offer tool boxes to use the GMM procedure. It is now possible to easily use this method in \proglang{R} with the new \pkg{gmm} package. Although GMM has good potential theoretically, several applied studies have shown that the properties of the 2SGMM may in some cases be poor in small samples. In particular, the estimators may be strongly biased for certain choices of moment conditions. In response to this result, \cite{hansen-heaton-yaron96} proposed two other ways to compute GMM: the iterative GMM (ITGMM) and the continuous updated GMM (CUE)\footnote{See also \cite{hall05} for a detailed presentation of most recent developments regarding GMM.}. Furthermore, another family of estimation procedures inspired by \cite{owen01}, which also depends only on moment conditions, was introduced by \cite{smith97}. It is the generalized empirical likelihood (GEL). So far, this method has not reached the popularity of GMM and it was not included in any statistical package until \pkg{gmm} was developed for \proglang{R} which also includes a GEL procedure. Asymptotic properties of GMM and generalized empirical likelihood (GEL) are now well established in the econometric literature. \cite{newey-smith04} and \cite{anatolyev05} have compared their second order asymptotic properties. In particular, they show that the second order bias of the empirical likelihood (EL) estimator, which is a special case of GEL, is smaller than the bias of the estimators from the three GMM methods. Furthermore, as opposed to GMM, the bias does not increase with the number of moment conditions. Since the efficiency improves when the number of conditions goes up, this is a valuable property. However, these are only asymptotic results which do not necessarily hold in small sample as shown by \cite{guggenberger08}. In order to analyze small sample properties, we have to rely on Monte Carlo simulations. However, Monte Carlo studies on methods such as GMM or GEL depend on complicated algorithms which are often home made. Because of that, results from such studies are not easy to reproduce. The solution should be to use a common tool which can be tested and improved upon by the users. Because it is open source, \proglang{R} offers a perfect platform for such tool. The \pkg{gmm} package allows to estimate models using the three GMM methods, the empirical likelihood and the exponential tilting, which belong to the family of GEL methods, and the exponentially tilted empirical likelihood which was proposed by \cite{schennach07}, Also it offers several options to estimate the covariance matrix of the moment conditions. Users can also choose between \textit{optim}, if no restrictions are required on the coefficients of the model to be estimated, and either \textit{nlminb} or \textit{constrOptim} for constrained optimizations. The results are presented in such a way that \proglang{R} users who are familiar with \textit{lm} objects, find it natural. In fact, the same methods are available for \textit{gmm} and \textit{gel} objects produced by the estimation procedures. The paper is organized as follows. Section 2 presents the theoretical aspects of the GMM method along with several examples in economics and finance. Through these examples, the functionality of the \pkg{gmm} packages is presented in details. Section 3 presents the GEL method with some of the examples used in section 2. Section 4 concludes and Section 5 gives the computational details of the package. \section{Generalized method of moments} This section presents an overview of the GMM method. It is intended to help the users understand the options that the \pkg{gmm} package offers. For those who are not familiar with the method and require more details, see \cite{hansen82} and \cite{hansen-heaton-yaron96} for the method itself, \cite{newey-west94} and \cite{andrews91} for the choice of the covariance matrix or \cite{hamilton94}. We want to estimate a vector of parameters $\theta_0 \in \Real^p$ from a model based on the following $q\times 1$ vector of unconditional moment conditions: \begin{equation}\label{mcond} E[g(\theta_0,x_i)] = 0 , \end{equation} where $x_i$ is a vector of cross-sectional data, time series or both. In order for GMM to produce consistent estimates from the above conditions, $\theta_0$ has to be the unique solution to $E[g(\theta,x_i)]=0$ and be an element of a compact space. Some boundary assumptions on higher moments of $g(\theta,x_i)$ are also required. However, it does not impose any condition on the distribution of $x_i$, except for the degree of dependence of the observations when it is a vector of time series. Several estimation methods such as least squares (LS), maximum likelihood (ML) or instrumental variables (IV) can also be seen as being based on such moment conditions, which make them special cases of GMM. For example, the following linear model: \[ Y = X\beta + u , \] where $Y$ and $X$ are respectively $n\times 1$ and $n\times k$ matrices, can be estimated by LS. The estimate $\hat{\beta}$ is obtained by solving $\min_\beta \|u\|^2$ and is therefore the solution to the following first order condition: \[ \frac{1}{n}X'u(\beta) = 0 , \] which is the estimate of the moment condition $E(X_iu_i(\beta))=0$. The same model can be estimated by ML in which case the moment condition becomes: \[ E\left[\frac{d l_i(\beta)}{d\beta}\right]=0 , \] where $l_i(\beta)$ is the density of $u_i$. In presence of endogeneity of the explanatory variable $X$, which implies that $E(X_iu_i)\neq 0$, the IV method is often used. It solves the endogeneity problem by substituting $X$ by a matrix of instruments $H$, which is required to be correlated with $X$ and uncorrelated with $u$. These properties allow the model to be estimated by the conditional moment condition $E(u_i|H_i)=0$ or its implied unconditional moment condition $E(u_iH_i)=0$. In general we say that $u_i$ is orthogonal to an information set $I_i$ or that $E(u_i|I_i)=0$ in which case $H_i$ is a vector containing functions of any element of $I_i$. The model can therefore be estimated by solving \[ \frac{1}{T}H'u(\beta)=0 . \] When there is no assumption on the covariance matrix of $u$, the IV corresponds to GMM. If $E(X_iu_i)=0$ holds, generalized LS with no assumption on the covariance matrix of $u$ other than boundary ones is also a GMM method. For the ML procedure to be viewed as GMM, the assumption on the distribution of $u$ must be satisfied. If it is not, but $E(dl_i(\theta_0)/d\theta)=0$ holds, as it is the case for linear models with non normal error terms, the pseudo-ML which uses a robust covariance matrix can be seen as being a GMM method. Because GMM depends only on moment conditions, it is a reliable estimation procedure for many models in economics and finance. For example, general equilibrium models suffer from endogeneity problems because these are misspecified and they represent only a fragment of the economy. GMM with the right moment conditions is therefore more appropriate than ML. In finance, there is no satisfying parametric distribution which reproduces the properties of stock returns. The family of stable distributions is a good candidate but only the densities of the normal, Cauchy and Lévy distributions, which belong to this family, have a closed form expression. The distribution-free feature of GMM is therefore appealing in that case. Although GMM estimators are easily consistent, efficiency and bias depend on the choice of moment conditions. Bad instruments implies bad information and therefore low efficiency. The effects on finite sample properties are even more severe and are well documented in the literature on weak instruments. \cite{newey-smith04} show that the bias increases with the number of instruments but efficiency decreases. Therefore, users need to be careful when selecting the instruments. \cite{carrasco07} gives a good review of recent developments on how to choose instruments in her introduction. In general, the moment conditions $E(g(\theta_0,x_i))=0$ is a vector of nonlinear functions of $\theta_0$ and the number of conditions is not limited by the dimension of $\theta_0$. Since efficiency increases with the number of instruments $q$ is often greater than $p$, which implies that there is no solution to \[ \bar{g}(\theta) \equiv \frac{1}{n}\sum_{i=1}^n g(\theta,x_i)=0. \] The best we can do is to make it as close as possible to zero by minimizing the quadratic function $\bar{g}(\theta)'W\bar{g}(\theta)$, where $W$ is a positive definite and symmetric $q\times q$ matrix of weights. The optimal matrix $W$ which produces efficient estimators is defined as: \begin{equation}\label{optw} W^* = \left\{\lim_{n\rightarrow \infty} Var(\sqrt{n} \bar{g}(\theta_0)) \equiv \Omega(\theta_0) \right\}^{-1}. \end{equation} This optimal matrix can be estimated by an heteroskedasticity and auto-correlation consistent (HAC) matrix like the one proposed by \cite{newey-west87a}. The general form is: \begin{equation}\label{optw_hat} \hat{\Omega} = \sum_{s=-(n-1)}^{n-1} k_h(s) \hat{\Gamma}_s(\theta^*), \end{equation} where $k_h(s)$ is a kernel, $h$ is the bandwidth which can be chosen using the procedures proposed by \cite{newey-west87a} and \cite{andrews91}, \[ \hat{\Gamma}_s(\theta^*) = \frac{1}{n}\sum_i g(\theta^*,x_i)g(\theta^*,x_{i+s})' \] and $\theta^*$ is a convergent estimate of $\theta_0$. There are many choices for the HAC matrix. They depend on the kernel and bandwidth selection. Although the choice does not affect the asymptotic properties of GMM, very little is known about the impacts in finite samples. The GMM estimator $\hat{\theta}$ is therefore defined as: \begin{equation}\label{gmm} \hat{\theta} = \arg\min_{\theta} \bar{g}(\theta)'\hat{\Omega}(\theta^*)^{-1}\bar{g}(\theta) \end{equation} The original version of GMM proposed by \cite{hansen82} is called two-step GMM (2SGMM). It computes $\theta^*$ by minimizing $\bar{g}(\theta)'\bar{g}(\theta)$. The algorithm is therefore: \begin{itemize} \item[1-] Compute $\theta^* = \arg\min_\theta \bar{g}(\theta)'\bar{g}(\theta)$ \item[2-] Compute the HAC matrix $\hat{\Omega}(\theta^*)$ \item[3-] Compute the 2SGMM $\hat{\theta} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta^*)\big]^{-1}\bar{g}(\theta)$ \end{itemize} In order to improve the properties of 2SGMM, \cite{hansen-heaton-yaron96} suggest two other methods. The first one is the iterative version of 2SGMM (ITGMM) and can be computed as follows: \begin{itemize} \item[1-] Compute $\theta^{(0)} = \arg\min_\theta \bar{g}(\theta)'\bar{g}(\theta)$ \item[2-] Compute the HAC matrix $\hat{\Omega}(\theta^{(0)})$ \item[3-] Compute the $\theta^{(1)} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta^{(0)})\big]^{-1}\bar{g}(\theta)$ \item[4-] If $\| \theta^{(0)}-\theta^{(1)}\|< tol$ stops, else $\theta^{(0)}=\theta^{(1)}$ and go to 2- \item[5-] Define the ITGMM estimator $\hat{\theta}$ as $\theta^{(1)}$ \end{itemize} where $tol$ can be set as small as we want to increase the precision. In the other method, no preliminary estimate is used to obtain the HAC matrix. The latter is treated as a function of $\theta$ and is allowed to change when the optimization algorithm computes the numerical derivatives. It is therefore continuously updated as we move toward the minimum. For that, it is called the continuous updated estimator (CUE). This method is highly nonlinear. It is therefore crucial to choose a starting value that is not too far from the minimum. A good choice is the estimate from 2SGMM which is known to be root-n convergent. The algorithm is: \begin{itemize} \item[1-] Compute $\theta^*$ using 2SGMM \item[2-] Compute the CUE estimator defined as \[ \hat{\theta} = \arg\min_\theta \bar{g}(\theta)'\big[\hat{\Omega}(\theta)\big]^{-1}\bar{g}(\theta) \] using $\theta^*$ as starting value. \end{itemize} According to \cite{newey-smith04} and \cite{anatolyev05}, 2SGMM and ITGMM are second order asymptotically equivalent. On the other hand, they show that the second order asymptotic bias of CUE is smaller. The difference in the bias comes from the randomness of $\theta^*$ in $\Omega(\theta^*)$. Iterating only makes $\theta^*$ more efficient. These are second order asymptotic properties. They are informative but may not apply in finite samples. In most cases, we have to rely on numerical simulations to analyze the properties in small samples. Given some regularity conditions, the GMM estimator converges as $n$ goes to infinity to the following distribution: \[ \sqrt{n}(\hat{\theta}-\theta_0) \stackrel{L}{\rightarrow} N(0,V), \] where \[ V = E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right)'\Omega(\theta_0)^{-1}E\left(\frac{\partial g(\theta_0,x_i)}{\partial\theta}\right) \] Inference can therefore be performed on $\hat{\theta}$ using the assumption that it is approximately distributed as $N(\theta_0,\hat{V}/n)$. If $q>p$, we can perform a J-test to verify if the moment conditions hold. The null hypothesis and the statistics are respectively $H0:E[g(\theta,x_i)]=0$ and: \[ n\bar{g}(\hat{\theta})'[\hat{\Omega}(\theta^*)]^{-1}\bar{g}(\hat{\theta}) \stackrel{L}{\rightarrow} \chi^2_{q-p}. \] \section{GMM with R} The \pkg{gmm} package can be loaded the usual way. <<>>= library(gmm) @ The main function is \textit{gmm()} which creates an object of class \textit{gmm}. Many options are available but in many cases they can be set to their default values. They are explained in details below through examples. The main arguments are \textit{g} and \textit{x}. For a linear model, \textit{g} is a formula like \textit{y~z1+z2} and \textit{x} the matrix of instruments. In the nonlinear case, they are respectively the function $g(\theta,x_i)$ and its argument. The available methods are \textit{coef}, \textit{vcov}, \textit{summary}, \textit{residuals}, \textit{fitted.values}, \textit{plot}, \textit{confint}. The model and data in a \textit{data.frame} format can be extracted by the generic function \textit{model.frame}. \subsection{Estimating the parameters of a normal distribution} This example\footnote{Thanks to Dieter Rozenich for his suggestion.}, is not something we want to do in practice, but its simplicity allows us to understand how to implement the \textit{gmm()} procedure by providing the gradient of $g(\theta,x_i)$. It is also a good example of the weakness of GMM when the moment conditions are not sufficiently informative. In fact, the ML estimators of the mean and the variance of a normal distribution are more efficient because the likelihood carries more information than few moment conditions. For the two parameters of a normal distribution $(\mu,\sigma)$ we have the following vector of moment conditions: \[ E[g(\theta,x_i)] \equiv E\left[ \begin{array}{c} \mu - x_{i} \\ \sigma^2 - (x_{i}-\mu)^2 \\ x_{i}^{3} - \mu (\mu^2+3\sigma^{2}) \end{array} \right] = 0 , \] where the first two can be directly obtained by the definition of $(\mu,\sigma)$ and the last comes from the third derivative of the moment generating function evaluated at 0. We first need to create a function $g(\theta,x)$ which returns an $n\times 3$ matrix: <<>>= g1 <- function(tet,x) { m1 <- (tet[1]-x) m2 <- (tet[2]^2 - (x - tet[1])^2) m3 <- x^3-tet[1]*(tet[1]^2+3*tet[2]^2) f <- cbind(m1,m2,m3) return(f) } @ The following is the gradient of $\bar{g}(\theta)$: \[ G\equiv \frac{\partial \bar{g}(\theta)}{\partial \theta} = \left( \begin{array}{cc} 1 & 0\\ 2(\bar{x}-\mu) & 2\sigma\\ -3(\mu^{2}+\sigma^{2}) & -6\mu\sigma \end{array} \right). \] If provided, it will be used to compute the covariance matrix of $\hat{\theta}$. It can be created as follows: <<>>= Dg <- function(tet,x) { G <- matrix(c( 1, 2*(-tet[1]+mean(x)), -3*tet[1]^2-3*tet[2]^2,0, 2*tet[2],-6*tet[1]*tet[2]), nrow=3,ncol=2) return(G) } @ First we generate normally distributed random numbers: <<>>= set.seed(123) n <- 200 x1 <- rnorm(n, mean = 4, sd = 2) @ We then run \textit{gmm} using the starting values $(\mu_0,\sigma^2_0)=(0,0)$ <<>>= print(res <- gmm(g1,x1,c(mu = 0, sig = 0), grad = Dg)) @ The \textit{summary} method prints more results from the estimation: <<>>= summary(res) @ The section "Initial values of the coefficients" shows the first step estimates used to either compute the weighting matrix in the 2-step GMM or the fixed bandwidth in CUE or iterative GMM. The J-test of over-identifying restrictions can also be extracted by using the method \textit{specTest}: <<>>= specTest(res) @ A small simulation using the following function shows that ML produces estimators with smaller mean squared errors than GMM based on the above moment conditions. However, it is not GMM but the moment conditions that are not efficient, because ML is GMM with the likelihood derivatives as moment conditions. <<>>= sim_ex <- function(n,iter) { tet1 <- matrix(0,iter,2) tet2 <- tet1 for(i in 1:iter) { x1 <- rnorm(n, mean = 4, sd = 2) tet1[i,1] <- mean(x1) tet1[i,2] <- sqrt(var(x1)*(n-1)/n) tet2[i,] <- gmm(g1,x1,c(0,0),grad=Dg)$coefficients } bias <- cbind(rowMeans(t(tet1)-c(4,2)),rowMeans(t(tet2)-c(4,2))) dimnames(bias)<-list(c("mu","sigma"),c("ML","GMM")) Var <- cbind(diag(var(tet1)),diag(var(tet2))) dimnames(Var)<-list(c("mu","sigma"),c("ML","GMM")) MSE <- cbind(rowMeans((t(tet1)-c(4,2))^2),rowMeans((t(tet2)-c(4,2))^2)) dimnames(MSE)<-list(c("mu","sigma"),c("ML","GMM")) return(list(bias=bias,Variance=Var,MSE=MSE)) } @ The following results can be reproduced with $n=50$, $iter=2000$ and by setting \textit{set.seed(345)}: \begin{center} \begin{tabular}{|c|c|c|c||c|c|c|} \hline &\multicolumn{3}{c||}{$\mu$}&\multicolumn{3}{c}{$\sigma$} \\ \hline & Bias & Variance & MSE & Bias & Variance &MSE \\ \hline GMM& 0.0020 &0.0929 & 0.0928& -0,0838 & 0.0481& 0.0551\\ ML& 0.0021 &0.0823 & 0.0822 & -0.0349 & 0.0411& 0.0423 \\ \hline \end{tabular} \end{center} \subsection{Estimating the parameters of a stable distribution} The previous example showed that ML should be used when the true distribution is known. However, when the density does not have a closed form expression, we have to consider other alternatives. \cite{garcia-renault-veredas06} propose to use indirect inference and perform a numerical study to compare it with several other methods. One of them is GMM for a continuum of moment conditions and was suggested by \cite{carrasco-florens02}. It uses the fact that the characteristic function $E(e^{ix_i\tau})$, where $i$ is the imaginary number and $\tau\in\Real$, has a closed form expression (for more details on stable distribution, see \cite{nolan09}). The \pkg{gmm} package does not yet deal with continuum of moment conditions but we can choose a certain grid $\{\tau_1,..., \tau_q\}$ over a given interval and estimate the parameters using the following moment conditions: \[ E\left[e^{ix_i\tau_l}-\Psi(\theta;\tau_l) \right]=0~~\mbox{for}~~l=1,...,q~, \] where $\Psi(\theta;\tau_l)$ is the characteristic function. There is more than one way to define a stable distribution and it depends on the choice of parametrization. We will follow the notation of \cite{nolan09} and consider stable distributions $S(\alpha,\beta,\gamma,\delta;1)$, where $\alpha\in (0,2]$ is the characteristic exponent and $\beta\in[-1,1]$, $\gamma>0$ and $\delta\in\Real$ are respectively the skewness, the scale and the location parameters. The last argument defines which parametrization we use. The \pkg{stabledist} package of \cite{stabledist} offers a function to generate random variables from stable distributions and uses the same notation. This parametrization implies that: \[ \Psi(\theta;\tau_l) = \left\{ \begin{array}{lcr} \exp{(-\gamma^\alpha|\tau_l|^\alpha[1-i\beta(\tan{\frac{\pi\alpha}{2}})(\mathrm{sign}(\tau_l))] + i\delta \tau_l)} &\mbox{for}&\alpha\neq 1\\ \exp{(-\gamma|\tau_l|[1+i\beta\frac{2}{\pi}(\mathrm{sign}(\tau_l))\log{|\tau_l|}] + i\delta \tau_l)} &\mbox{for}&\alpha= 1\\ \end{array} \right. , \] The function \textit{charStable} included in the package computes the characteristic function and can be used to construct $g(\theta,x_i)$. To avoid dealing with complex numbers, it returns the imaginary and real parts in separate columns because both should have zero expectation. The function is: <<>>= g2 <- function(theta,x) { tau <- seq(1,5,length.out=10) pm <- 1 x <- matrix(c(x),ncol=1) x_comp <- x%*%matrix(tau,nrow=1) x_comp <- matrix(complex(ima=x_comp),ncol=length(tau)) emp_car <- exp(x_comp) the_car <- charStable(theta,tau,pm) gt <- t(t(emp_car) - the_car) gt <- cbind(Im(gt),Re(gt)) return(gt) } @ The parameters of a simulated random vector can be estimated as follows (by default, $\gamma$ and $\delta$ are set to $1$ and $0$ respectively in \textit{rstable}). For the example, the starting values are the ones of a normal distribution with mean 0 and variance equals to \textit{var(x)}: <<>>= library(stabledist) set.seed(345) x2 <- rstable(500,1.5,.5,pm=1) t0 <- c(alpha = 2, beta = 0, gamma = sd(x2)/sqrt(2), delta = 0) print(res <- gmm(g2,x2,t0)) @ The result is not very close to the true parameters. But we can see why by looking at the J-test that is provided by the \textit{summary} method: <<>>= summary(res) @ The null hypothesis that the moment conditions are satisfied is rejected. For nonlinear models, a significant J-test may indicate that we have not reached the global minimum. Furthermore, the standard deviation of the coefficient of $\delta$ indicates that the covariance matrix is nearly singular. Notice also that the convergence code is equal to 1, which indicates that the algorithm did not converge. We could try different starting values, increase the number of iterations in the control option of \textit{optim} or use \textit{nlminb} which allows to put restrictions on the parameter space. The former would work but the latter will allow us to see how to select another optimizer. The option \textit{optfct} can be modified to use this algorithm instead of \textit{optim}. In that case, we can specify the upper and lower bounds of $\theta$. <<>>= res2 <- gmm(g2,x2,t0,optfct="nlminb",lower=c(0,-1,0,-Inf),upper=c(2,1,Inf,Inf)) summary(res2) @ Although the above modification solved the convergence problem, there is another issue that we need to address. The first step estimate used to compute the weighting matrix is almost identical to the starting values. There is therefore a convergence problem in the first step. In fact, choosing the initial $\alpha$ to be on the boundary was not a wise choice. Also, it seems that an initial value of $\beta$ equals to zero makes the objective function harder to minimize. Having a gobal minimum for the first step estimate is important if we care about efficiency. A wrong estimate will cause the weighting matrix not being a consistent estimate of the optimal matrix. The information about convergence is included in the argument 'initialAlgoInfo' of the gmm object. We conclude this example by estimating the parameters for a vector of stock returns from the data set \textit{Finance} that comes with the \pkg{gmm} package. <>= data(Finance) x3 <- Finance[1:1500,"WMK"] t0<-c(alpha = 1.8, beta = 0.1, gamma = sd(x3)/sqrt(2),delta = 0) res3 <- gmm(g2,x3,t0,optfct="nlminb") summary(res3) @ For this sub-sample, the hypothesis that the return follows a stable distribution is rejected. The normality assumption can be analyzed by testing $H_0:\alpha=2,\beta=0$ using \textit{linearHypothesis} from the \pkg{car} package: <<>>= library(car) linearHypothesis(res3,cbind(diag(2),c(0,0),c(0,0)),c(2,0)) @ It is clearly rejected. The result is even stronger if the whole sample is used. \subsection{A linear model with iid moment conditions} We want to estimate a linear model with an endogeneity problem. It is the model used by \cite{carrasco07} to compare several methods which deal with the many instruments problem. We want to estimate $\delta$ from: \[ y_i = \delta W_i + \epsilon_i \] with $\delta=0.1$ and \[ W_i = e^{-x_i^2} + u_i , \] where $(\epsilon_i,u_i) \sim ~iidN(0,\Sigma)$ with \[ \Sigma = \left( \begin{array}{cc} 1&0.5\\ 0.5&1 \end{array} \right) \] Any function of $x_i$ can be used as an instrument because it is orthogonal to $\epsilon_i$ and correlated with $W_i$. There is therefore an infinite number of possible instruments. For this example, $(x_i,x_i^2,x_i^3)$ will be the selected instruments and the sample size is set to $n=400$: <<>>= library(mvtnorm) set.seed(112233) sig <- matrix(c(1,.5,.5,1),2,2) n <- 400 e <- rmvnorm(n,sigma=sig) x4 <- rnorm(n) w <- exp(-x4^2) + e[,1] y <- 0.1*w + e[,2] @ where \textit{rmvnorm} is a multivariate normal distribution random generator which is included in the package \pkg{mvtnorm} (\cite{mvtnorm}). For a linear model, the $g$ argument is a formula that specifies the right- and left-hand sides as for \textit{lm} and $x$ is the matrix of instruments: <<>>= h <- cbind(x4, x4^2, x4^3) g3 <- y~w @ By default, an intercept is added to the formula and a vector of ones to the matrix of instruments. It implies the following moment conditions: \[ E\left(\begin{array}{c} (y_i-\alpha-\delta W_i)\\ (y_i-\alpha-\delta W_i)x_i\\ (y_i-\alpha-\delta W_i)x_i^2\\ (y_i-\alpha-\delta W_i)x_i^3 \end{array} \right)=0 \] In order the remove the intercept, -1 has to be added to the formula. In that case there is no column of ones added to the matrix of instruments. To keep the condition that the expected value of the error terms is zero, the column of ones needs to be included manually. We know that the moment conditions of this example are iid. Therefore, we can add the option \textit{vcov="iid"}. This option tells \textit{gmm} to estimate the covariance matrix of $\sqrt{n}\bar{g}(\theta^*)$ as follows: \[ \hat{\Omega}(\theta^*) = \frac{1}{n}\sum_{i=1}^n g(\theta^*,x_i)g(\theta^*,x_i)' \] However, it is recommended not to set this option to ``iid" in practice with real data because one of the reasons we want to use GMM is to avoid such restrictions. Finally, it is not necessary to provide the gradient when the model is linear since it is already included in \textit{gmm}. The first results are: <<>>= summary(res <- gmm(g3,x=h)) @ By default, the 2SGMM is computed. Other methods can be chosen by modifying the option ``type". The second possibility is ITGMM: <<>>= res2 <- gmm(g3,x=h,type='iterative',crit=1e-8,itermax=200) coef(res2) @ The procedure iterates until the difference between the estimates of two successive iterations reaches a certain tolerance level, defined by the option \textit{crit} (default is $10^{-7}$), or if the number of iterations reaches \textit{itermax} (default is 100). In the latter case, a message is printed to indicate that the procedure did not converge. The third method is CUE. As you can see, the estimates from ITGMM is used as starting values. However, the starting values are required only when \textit{g} is a function. When \textit{g} is a formula, the default starting values are the ones obtained by setting the matrix of weights equal to the identity matrix. <<>>= res3 <- gmm(g3,x=h,res2$coef,type='cue') coef(res3) @ It is possible to produce confidence intervals by using the method \textit{confint}: <<>>= confint(res3,level=.90) @ Whether \textit{optim} or \textit{nlminb} is used to compute the solution, it is possible to modify their default options by adding \textit{control=list()}. For example, you can keep track of the convergence with \textit{control=list(trace=TRUE)} or increase the number of iterations with \textit{control=list(maxit=1000)}. You can also choose the \textit{BFGS} algorithm with \textit{method="BFGS"} (see \textit{help(optim)} for more details). The methods \textit{fitted} and \textit{residuals} are also available for linear models. We can compare the fitted values of \textit{lm} with the ones from \textit{gmm} to see why this model cannot be estimated by LS. \begin{center} <<>>= plot(w,y,main="LS vs GMM estimation") lines(w,fitted(res),col=2) lines(w,fitted(lm(y~w)),col=3,lty=2) lines(w,.1*w,col=4,lty=3) legend("topleft",c("Data","Fitted GMM","Fitted LS","True line"),pch=c(1,NA,NA,NA),col=1:3,lty=c(NA,1,2,3)) @ \end{center} The LS seems to fit the model better. But the graphics hides the endogeneity problem. LS overestimates the relationship between $y$ and $w$ because it does not take into account the fact that some of the correlation is caused by the fact that $y_i$ and $w_i$ are positively correlated with the error term $\epsilon_i$. Finally, the \textit{plot} method produces some graphics to analyze the properties of the residuals. It can only be applied to \textit{gmm} objects when \textit{g} is a formula because when \textit{g} is a function, residuals are not defined. \subsection{Estimating the AR coefficients of an ARMA process} \label{ar} The estimation of auto-regressive coefficients of ARMA(p,q) processes is better performed by ML or nonlinear LS. But in Monte Carlo experiments, it is often estimated by GMM to study its properties. It gives a good example of linear models with endogeneity problems in which the moment conditions are serially correlated and possibly conditionally heteroskedastic. As opposed to the previous example, the choice of the HAC matrix becomes an important issue. We want to estimate the AR coefficients of the following process: \[ X_t = 1.4 X_{t-1} - 0.6X_{t-2} + u_t \] where $u_t = 0.6\epsilon_{t-1} -0.3 \epsilon_{t-2} + \epsilon_t$ and $\epsilon_t\sim iidN(0,1)$. This model can be estimated by GMM using any $X_{t-s}$ for $s>2$, because they are uncorrelated with $u_t$ and correlated with $X_{t-1}$ and $X_{t-2}$. However, as $s$ increases the quality of the instruments decreases since the stationarity of the process implies that the auto-correlation goes to zero. For this example, the selected instruments are $(X_{t-3},X_{t-4},X_{t-5},X_{t-6})$ and the sample size equals 400. The ARMA(2,2) process is generated by the function \textit{arima.sim}: <<>>= t <- 400 set.seed(345) x5 <- arima.sim(n=t,list(ar=c(1.4,-0.6),ma=c(0.6,-0.3))) x5t<-cbind(x5) for (i in 1:6) x5t<-cbind(x5t,lag(x5,-i)) x5t<-na.omit(x5t) g4<-x5t[,1]~x5t[,2]+x5t[,3] res<-gmm(g4,x5t[,4:7]) summary(res) @ The optimal matrix, when moment conditions are based on time series, is an HAC matrix which is defined by equation (\ref{optw_hat}). Several estimators of this matrix have been proposed in the literature. Given some regularity conditions, they are asymptotically equivalent. However, their impacts on the finite sample properties of GMM estimators may differ. The \pkg{gmm} package uses the \pkg{sandwich} package to compute these estimators which are well explained by \cite{zeileis06} and \cite{zeileis04}. We will therefore briefly summarize the available options. The option \textit{kernel} allows to choose between five kernels: Truncated, Bartlett, Parzen, Tukey-Hanning and Quadratic spectral\footnote{The first three have been proposed by \cite{white84}, \cite{newey-west87a} and \cite{gallant87} respectively and the last two, applied to HAC estimation, by \cite{andrews91}. But the latter gives a good review of all five.}. By default, the Quadratic Spectral kernel is used as it was shown to be optimal by \cite{andrews91} with respect to some mean squared error criterion. In most statistical packages, the Bartlett kernel is used for its simplicity. It makes the estimation of large models less computationally intensive. It may also make the \textit{gmm} algorithm more stable numerically when dealing with highly nonlinear models, especially with CUE. We can compare the results with different choices of kernel: <<>>= res2 <- gmm(g4,x=x5t[,4:7],kernel="Truncated") coef(res2) res3 <- gmm(g4,x=x5t[,4:7],kernel="Bartlett") coef(res3) res4 <- gmm(g4,x=x5t[,4:7],kernel="Parzen") coef(res4) res5<- gmm(g4,x=x5t[,4:7],kernel="Tukey-Hanning") coef(res5) @ The similarity of the results is not surprising since the matrix of weights should only affect the efficiency of the estimator. We can compare the estimated standard deviations using the method \textit{vcov}: <<>>= diag(vcov(res2))^.5 diag(vcov(res3))^.5 diag(vcov(res4))^.5 diag(vcov(res5))^.5 @ which shows, for this example, that the Bartlett kernel generates the estimates with the smallest variances. However, it does not mean it is better. We have to run simulations and compute the true variance if we want to compare them. In fact, we do not know which one produces the most accurate estimate of the variance. The second options is for the bandwidth selection. By default it is the automatic selection proposed by \cite{andrews91}. It is also possible to choose the automatic selection of \cite{newey-west94} by adding \textit{bw=bwNeweyWest} (without quotes because \textit{bwNeweyWest} is a function). A prewhitened kernel estimator can also be computed using the option \textit{prewhite=p}, where $p$ is the order of the vector auto-regressive (VAR) used to compute it. By default, it is set to \textit{FALSE}. \cite{andrews-monahan92} show that a prewhitened kernel estimator improves the properties of hypothesis tests on parameters. Finally, the \textit{plot} method can be applied to \textit{gmm} objects to do a Q-Q plot of the residuals: \begin{center} <<>>= plot(res,which=2) @ \end{center} or to plot the observations with the fitted values: \begin{center} <<>>= plot(res,which=3) @ \end{center} \subsection{Estimating a system of equations: CAPM} We want to test one of the implications of the capital asset pricing model (CAPM). This example comes from \cite{campbell-lo-mackinlay96}. It shows how to apply the \pkg{gmm} package to estimate a system of equations. The theory of CAPM implies that $\mu_i-R_f = \beta_i (\mu_m-R_f)$ $\forall i$, where $\mu_i$ is the expected value of stock i's return, $R_f$ is the risk free rate and $\mu_m$ is the expected value of the market porfolio's return. The theory can be tested by running the following regression: \[ (R_t-R_f) = \alpha + \beta (R_{mt}-R_f) + \epsilon_t, \] where $R_t$ is a $N\times 1$ vector of observed returns on stocks, $R_{mt}$ if the observed return of a proxy for the market portfolio, $R_f$ is the interest rate on short term government bonds and $\epsilon_t$ is a vector of error terms with covariance matrix $\Sigma_t$. When estimated by ML or LS, $\Sigma$ is assumed to be fixed. However, GMM allows $\epsilon_t$ to be heteroskedastic and serially correlated. One implication of the CAPM is that the vector $\alpha$ should be zero. It can be tested by estimating the model with $(R_{mt}-R_f)$ as instruments, and by testing the null hypothesis $H_0:~\alpha=0$. The data, which are included in the package, are the daily returns of twenty selected stocks from January 1993 to February 2009, the risk-free rate and the three factors of Fama and French\footnote{The symbols of the stocks taken from \url{http://ca.finance.yahoo.com/} are ("WMK", "UIS", "ORB", "MAT", "ABAX","T", "EMR", "JCS", "VOXX", "ZOOM", "ROG", "GGG", "PC", "GCO", "EBF", "F", "FNM", "NHP", "AA", "TDW"). The four other series can be found on K. R. French's web site: \url{http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data\_library.html}}. The following test is performed using the returns of 5 stocks and a sample size of 500\footnote{The choice of sample size is arbitrary. The purpose is to show how to estimate a system of equations not to test the CAPM. Besides, the $\beta$'s seem to vary over time. It is therefore a good practice to estimate the model using short periods.}. <<>>= data(Finance) r <- Finance[1:500,1:5] rm <- Finance[1:500,"rm"] rf <- Finance[1:500,"rf"] z <- as.matrix(r-rf) zm <- as.matrix(rm-rf) res <- gmm(z~zm,x=zm) coef(res) R <- cbind(diag(5),matrix(0,5,5)) c <- rep(0,5) linearHypothesis(res,R,c,test = "Chisq") @ where the asymptotic chi-square is used since the default distribution requires a normality assumption. The same test could have been performed using the names of the coefficients: <>= test <- paste(names(coef(res)[1:5])," = 0",sep="") linearHypothesis(res,test) @ Another way to test the CAPM is to estimate the restricted model ($\alpha=0$), which is over-identified, and to perform a J-test. Adding $-1$ to the formula removes the intercept. In that case, a column of ones has to be added to the matrix of instruments: <<>>= res2<-gmm(z~zm-1,cbind(1,zm)) specTest(res2) @ which confirms the non-rejection of the theory. \subsection{Testing the CAPM using the stochastic discount factor representation} In some cases the theory is directly based on moment conditions. When it is the case, testing the validity of these conditions becomes a way of testing the theory. \cite{jagannathan-skoulakis02} present several GMM applications in finance and one of them is the stochastic discount factor (SDF) representation of the CAPM. The general theory implies that $E(m_tR_{it})=1$ for all $i$, where $m_t$ is the SDF and $R_{it}$ the gross return ($1 + r_{it}$). It can be shown that if the CAPM holds, $m_t=\theta_0 + \theta_0 R_{mt}$ which implies the following moment conditions: \[ E\Big[R_{it}(\theta_0-\theta_1R_{mt})-1 \Big] = 0~~\mbox{for}~~ i=1,...,N \] which can be tested as follows: <<>>= g5 <- function(tet, x) { gmat <- (tet[1] + tet[2] * (1 + c(x[, 1]))) * (1 + x[, 2:6]) - 1 return(gmat) } res_sdf <- gmm(g5, x = as.matrix(cbind(rm, r)), c(0, 0)) specTest(res_sdf) @ which is consistent with the two previous tests. \subsection{Estimating continuous time processes by discrete time approximation} This last example also comes from \cite{jagannathan-skoulakis02}. We want to estimate the coefficients of the following continuous time process which is often used in finance for interest rates: \[ dr_t = (\alpha + \beta r_t)dt + \sigma r_t^\gamma dW_t, \] where $W_t$ is a standard Brownian motion. Special cases of this process are the Brownian motion with drift ($\beta=0$ and $\gamma = 0$), the Ornstein-Uhlenbeck process ($\gamma=0$) and the Cox-Ingersoll-Ross or square root process ( $\gamma = 1/2$). It can be estimated using the following discrete time approximation: \[ r_{t+1}-r_t = \alpha + \beta r_t + \epsilon_{t+1} \] with \[ E_t \epsilon_{t+1}=0,~~\mbox{and}~~E_t(\epsilon_{t+1}^2) = \sigma^2r_t^{2\gamma} \] Notice that ML cannot be used to estimate this model because the distribution depends on $\gamma$. In particular, it is normal for $\gamma=0$ and gamma for $\gamma=1/2$. It can be estimated by GMM using the following moment conditions: \[ E[g(\theta,x_t)] \equiv E\left(\begin{array}{c} \epsilon_{t+1}\\ \epsilon_{t+1}r_t\\ \epsilon_{t+1}^2-\sigma^2r_t^{2\gamma}\\ (\epsilon_{t+1}^2-\sigma^2r_t^{2\gamma})r_t \end{array} \right) = 0 \] The related \textit{g} function, with $\theta=\{\alpha, \beta, \sigma^2, \gamma\}$ is: <<>>= g6 <- function(theta, x) { t <- length(x) et1 <- diff(x) - theta[1] - theta[2] * x[-t] ht <- et1^2 - theta[3] * x[-t]^(2 * theta[4]) g <- cbind(et1, et1 * x[-t], ht, ht * x[-t]) return(g) } @ In order to estimate the model, the vector of interest rates needs to be properly scaled to avoid numerical problems. The transformed series is the annualized interest rates expressed in percentage. Also, the starting values are obtained using LS and some options for \textit{optim} need to be modified. <<>>= rf <- Finance[,"rf"] rf <- ((1 + rf/100)^(365) - 1) * 100 dr <- diff(rf) res_0 <- lm(dr ~ rf[-length(rf)]) tet0 <- c(res_0$coef, var(residuals(res_0)), 0) names(tet0) <- c("alpha", "beta", "sigma^2", "gamma") res_rf <- gmm(g6, rf, tet0, control = list(maxit = 1000, reltol = 1e-10)) coef(res_rf) @ \subsection{Comments on models with panel data} The \pkg{gmm} package is not directly built to easily deal with panel data. However, it is flexible enough to make it possible in most cases. To see that, let us consider the following model (see \cite{wooldridge02} for more details): \[ y_{it} = x_{it} \beta + a_i + \epsilon_{it}\mbox{ for } i=1,...,N\mbox{ and }t=1,...,T, \] where $x_{it}$ is $1\times k$, $\beta$ is $k\times 1$, $\epsilon_{it}$ is an error term and $a_i$ is an unobserved component which is specific to individual $i$. If $a_i$ is correlated with $x_{it}$, it can be removed by subtracting the average of the equation over time, which gives: \[ (y_{it}-\bar{y}_i) = (x_{it}-\bar{x}_i) \beta + (\epsilon_{it}-\bar{\epsilon}_i) \mbox{ for } i=1,...,N\mbox{ and } t=1,...,T, \] which can be estimated by \textit{gmm}. For example, if there are 3 individuals the following corresponds to the GMM fixed effects estimation: <>= y <- rbind(y1-mean(y1),y2-mean(y2),y3-mean(y3)) x <- rbind(x1-mean(x1),x2-mean(x2),x3-mean(x3)) res <- gmm(y~x,h) @ However, if $a_i$ is not correlated with $x_{it}$, the equation represents a random effects model. In that case, it is more efficient not to remove $a_i$ from the equation because of the information it carries about the individuals. The error terms are then combined in a single one, $\eta_{it}=(a_i + \epsilon_{it})$ to produce the linear model: \[ y_{it} = x_{it} \beta + \eta_{it} \] This model cannot be efficiently estimated by OLS because the presence of the common factor $a_i$ at each period implies that $\eta_{it}$ is serially correlated. However, GMM is well suited to deal with such specifications. The following will therefore produce a GMM random effects estimation: <>= y <- rbind(y1,y2,y3) x <- rbind(x1,x2,x3) res <- gmm(y~x,h) @ The package \pkg{plm} of \cite{plm} offers several functions to manipulate panel data. It could therefore be combined with \pkg{gmm} when estimating such models. It also offers a way to estimate them with its own GMM algorithm for panel data. \subsection{GMM and the sandwich package} In the \pkg{gmm} package, the estimation of the optimal weighting matrices are obtained using the \pkg{sandwich} package of \cite{zeileis06}. For example, the weighting matrix of the two-step GMM defined as: \[ W = \left[\lim_{n\rightarrow \infty} Var{(\sqrt{n}\bar{g})}\right]^{-1} \] is estimated as follows: <>= gt <- g(t0, x) V <- kernHAC(lm(gt~1),sandwich = FALSE) W <- solve(V) @ where $t0$ is any consistent estimate. As long as the optimal matrix is used, the covariance matrix of the coefficients can be estimated as follows: \[ (\hat{G}'W\hat{G})^{-1}/n \equiv (\hat{G}'\hat{V}^{-1}\hat{G})^{-1}/n, \] where $\hat{G}=d\bar{g}(\hat{\theta})/d\theta$ and $\hat{V}$ is obtained using \textit{kernHAC()}. It is not a sandwich covariance matrix and is computed using the \textit{vcov()} method included in \pkg{gmm}. However, if any other weighting matrix is used, say $W$, the estimated covariance matrix of the coefficients must then be estimated as follows: \[ (\hat{G}'W\hat{G})^{-1}\hat{G}'W\hat{V}W\hat{G}(\hat{G}'W\hat{G})^{-1}/n. \] A \textit{bread()} and \textit{estfun()} methods are available for \textit{gmm} objects which allows to compute the above matrix using the \pkg{sandwich} package. The \textit{bread()} method computes $(\hat{G}'W\hat{G})^{-1}$ while the \textit{estfun()} method returns a $T\times q$ matrix with the $t^{th}$ row equals to $g(\hat{\theta}, x_t) W \hat{G}$. The \textit{meatHAC()} method applied to the latter produces the right meat. Let us consider the example of section (\ref{ar}). Suppose we want to use the identity matrix to eliminate one source of bias, at the cost of lower efficiency. In that case, a consistent estimate of the covariance matrix is \[ (\hat{G}'\hat{G})^{-1}\hat{G}'\hat{V}\hat{G}(\hat{G}'\hat{G})^{-1}/n, \] which can be computed as: <<>>= print(res<-gmm(g4,x5t[,4:7],wmatrix="ident")) diag(vcovHAC(res))^.5 @ which is more robust than using \textit{vcov()}: <<>>= diag(vcov(res))^.5 @ Notice that it is possible to fixe $W$. Therefore, the above results can also be obtained as: <<>>= print(res<-gmm(g4,x5t[,4:7], weightsMatrix = diag(5))) @ In this case, the choice of the type of GMM is irrelevant since the weighting matrix is fixed. \section{Generalized empirical likelihood} The GEL is a new family of estimation methods which, as GMM, is based on moment conditions. It follows \cite{owen01} who developed the idea of empirical likelihood estimation which was meant to improve the confidence regions of estimators. We present here a brief discussion on the method without going into too much details. For a complete review, see \cite{smith97}, \cite{newey-smith04} or \cite{anatolyev05}. The estimation is based on \[ E(g(\theta_0,x_i))=0, \] which can be estimated in general by \[ \tilde{g}(\theta) = \sum_{i=1}^n p_i g(\theta,x_i) =0, \] where $p_i$ is called the implied probability associated with the observation $x_i$. For the GEL method, it is assumed that $q>p$ because otherwise it would correspond to GMM. Therefore, as it is the case for GMM, there is no solution to $\bar{g}(\theta)=0$. However, there is a solution to $\tilde{g}(\theta)=0$ for some choice of the probabilities $p_i$ such that $\sum_i p_i=1$. In fact, there is an infinite number of solutions since there are $(n+q)$ unknowns and only $q+1$ equations. GEL selects among them the one for which the distance between the vector of probabilities $p$ and the empirical density $1/n$ is minimized. The empirical likelihood of \cite{owen01} is a special case in which the distance is the likelihood ratio. The other methods that belong to the GEL family of estimators use different metrics. If the moment conditions hold, the implied probabilities carry a lot of information about the stochastic properties of $x_i$. For GEL, the estimations of the expected value of the Jacobian and the covariance matrix of the moment conditions, which are required to estimate $\theta$, are based on $p_i$ while in GMM they are estimated using $1/n$. \cite{newey-smith04} show that this difference explains partially why the second order properties of GEL are better. Another difference between GEL and GMM is how they deal with the fact that $g(\theta,x_i)$ can be a conditionally heteroskedastic and weakly dependent process. GEL does not require to compute explicitly the HAC matrix of the moment conditions. However, if it does not take it into account, its estimators may not only be inefficient but may also fail to be consistent. \cite{smith01} proposes to replace $g(\theta,x_i)$ by: \[ g^w(\theta,x_i) = \sum_{s=-m}^m w(s)g(\theta,x_{i-s}) \] where $w(s)$ are kernel based weights that sum to one (see also \cite{kitamura-stutzer97} and \cite{smith97}). The sample moment conditions become: \begin{equation}\label{gel_mcond} \tilde{g}(\theta) = \sum_{i=1}^n p_i g^w(\theta,x_i) =0 \end{equation} The estimator is defined as the solution to the following constrained minimization problem: \begin{eqnarray} \hat{\theta}_n &=& \arg\min_{\theta,p_i} \sum_{i=1}^n h_n(p_i) ,\\ && \mbox{subject to } \\ && \sum_{i=1}^n p_ig^w(\theta,x_i) = 0 ~~\mbox{and} \label{const}\\ && \sum_{i=1}^n p_i=1, \end{eqnarray} where $h_n(p_i)$ has to belong to the following Cressie-Read family of discrepancies: \[ h_n(p_i) = \frac{[\gamma(\gamma+1)]^{-1}[(np_i)^{\gamma+1}-1]}{n} . \] \cite{smith97} showed that the empirical likelihood method (EL) of \cite{owen01} ($\gamma=0$) and the exponential tilting of \cite{kitamura-stutzer97} ($\gamma=-1$) belong to the GEL family of estimators while \cite{newey-smith04} show that it is also the case for the continuous updated estimator of \cite{hansen-heaton-yaron96} ($\gamma=1$). What makes them part of the same GEL family of estimation methods is the existence of a dual problem which is defined as: \begin{equation}\label{gel_obj} \hat{\theta} = \arg\min_{\theta}\left[\max_{\lambda} P_n(\theta,\lambda) = \frac{1}{n}\sum_{i=1}^n\rho\left(\lambda'g^w(\theta,x_i)\right)\right] \end{equation} where $\lambda$ is the Lagrange multiplier associated with the constraint (\ref{const}) and $\rho(v)$ is a strictly concave function normalized so that $\rho'(0)=\rho''(0)=-1$. It can be shown that $\rho(v)=\ln{(1-v)}$ corresponds to EL , $\rho(v)=-\exp{(v)}$ to ET and to CUE if it is quadratic. The equivalence of the primal and dual problems can easily be verified by showing that they both share the same following first order conditions: \begin{equation}\label{lam} \sum_{i=1}^n p_i g^w(\theta,x_i) = 0, \end{equation} \begin{equation} \sum_{i=1}^n p_i \lambda'\left(\frac{\partial g^w(\theta,x_i)}{\partial \theta}\right) = 0 , \end{equation} with \begin{equation} p_i = \frac{1}{n}\rho'\left(\lambda'g^w(\theta,x_i)\right) . \end{equation} Equation (\ref{gel_obj}) represents a saddle point problem. The solution is obtained by solving simultaneously two optimization problems. We can solve for $\theta$ by minimizing $P_n(\theta,\lambda(\theta))$, where $\lambda(\theta)$ is the solution to $\arg\max_\lambda P_n(\theta,\lambda)$ for a given $\theta$. Therefore an optimization algorithm needs to be called inside the $P_n(\theta,\lambda)$ function. It makes the GEL very hard to implement numerically. For example, \cite{guggenberger08}, who analyzes the small sample properties of GEL, uses an iterative procedure based on the Newton method for $\lambda$ and a grid search for $\theta$ in order to confidently reach the absolute minimum. Using such iterative procedures for $\lambda$ makes the problem less computationally demanding and does not seem to affect the properties of the estimator of $\theta_0$. Indeed, \cite{guggenberger-hahn05} show that going beyond two iterations for $\lambda$ does not improve the second order asymptotic properties of the estimator of $\theta_0$. The function \textit{gel} offers two options. By default, $\lambda(\theta)$ is obtained by the following iterative method: \[ \lambda_l = \lambda_{l-1} - \left[\frac{1}{n}\sum_{i=1}^n \rho''(\lambda_{l-1}'g_t)g_tg_t'\right]^{-1}\left[\frac{1}{n} \sum_{i=1}^n \rho'(\lambda_{l-1}'g_i) g_i \right] \] starting with $\lambda=0$, which corresponds to its asymptotic value. The algorithm stops when $\|\lambda_l-\lambda_{l-1}\|$ reaches a certain tolerance level. The second option is to let \textit{optim} solve the problem. Then, as for \textit{gmm}, the minimization problem is solved either by \textit{optim}, \textit{nlminb} or \textit{constrOptim}. In order to test the over-identifying restrictions, \cite{smith04} proposes three tests which are all asymptotically distributed as a $\chi^2_{q-p}$. The first one is the J-test: \[ n \bar{g}^w(\hat{\theta})'[\hat{\Omega}(\hat{\theta})]^{-1}\bar{g}^w(\hat{\theta}), \] the second is a Lagrange multiplier test (LM): \[ LM = n\hat{\lambda}'\hat{\Omega}(\hat{\theta})\hat{\lambda} \] and the last one is a likelihood ratio test (LR): \[ LR = 2\sum_{i=1}^n\left[ \rho\left(\hat{\lambda}'g^w(\hat{\theta},x_i)\right) - \rho(0)\right] \] \section{GEL with R} \subsection{Estimating the parameters of a normal distribution} For this example, we can leave the option \textit{smooth} at its default value, which is \textit{FALSE}, because of the iid properties of $x$. A good starting value is very important for GEL. The best choice is the sample mean and the standard deviation. By default the option \textit{type} is set to \textit{EL}. The same methods that apply to \textit{gmm} objects, can also be applied to \textit{gel} objects. <<>>= tet0 <- c(mu = mean(x1), sig = sd(x1)) res_el <- gel(g1,x1,tet0) summary(res_el) @ Each Lagrange multiplier represents a shadow price of the constraint implied by moment condition. A binding constraint will produce a multiplier different from zero. Therefore, its value informs us on the validity of the moment condition. In the above results, the $\lambda$'s are significantly different from zero which would normally suggest that the moment conditions associated with them are violated. As a result, the LM test also rejects the null hypothesis since it is based on the $\lambda$'s. Notice that \textit{summary} reports two convergence codes, one for $\lambda$ et another for $\theta$. The ET and CUE estimates can be obtained as follows: <<>>= res_et <- gel(g1,x1,tet0,type="ET") coef(res_et) @ <<>>= res_cue <- gel(g1,x1,tet0,type="CUE") coef(res_cue) @ A fourth method is available which is called the exponentially tilted empirical likelihood (ETEL) and was proposed by \cite{schennach07}. However, it does not belong to the family of GEL estimators. It solves the problem of misspecified models. In such models there may not exist any pseudo value to which $\hat{\theta}$ converges as the sample size increases. ETEL uses the $\rho()$ of ET to solve for $\lambda$ and the $\rho()$ of EL to solve for $\theta$. \cite{schennach07} shows that ETEL shares the same asymptotic properties of EL without having to impose restrictions on the domain of $\rho(v)$ when solving for $\lambda$. <<>>= res_etel <- gel(g1,x1,c(mu=1,sig=1),type="ETEL") coef(res_etel) @ The type ETEL is experimental for now. Although it is supposed to be more stable because no restrictions are required to solve for $\lambda$, once we substitute $\lambda(\theta)$ in the EL objective function to estimate $\theta$, we still need to restrict $\lambda'g_t$ to avoid having NA's. The solution used in gel() is to obtain $\lambda(\theta)$ with \textit{constrOptim} with the restriction $\lambda'gt > 1$ even if it is not required by ET ($\rho(v)=-\exp{(v)}$). It is however sensitive to starting values. That's the reason why we used different ones above. \subsection{Estimating the AR coefficients of an ARMA process} Because the moment conditions are weakly dependent, we need to set the option \textit{smooth=TRUE}. Before going to the estimation procedure, we need to understand the relationship between the smoothing kernel and the HAC estimator. The reason why we need to smooth the moment function is that GEL estimates the covariance matrix of $\bar{g}(\theta,x_t)$, as if we had iid observations, using the expression $(1/T)\sum_{t=1}^T(g_tg_t')$. We can show that substituting $g_t$ by $g^w_t$ in this expression results in an HAC estimator. However, the relationship between the smoothing kernel and the kernel that appears in the HAC estimator is not obvious. For example, we can show that if the smoothing kernel is Truncated, then the kernel in the HAC estimator is the Bartlett. Let us consider the truncated kernel with a bandwidth of 2. This implies that $w(s)=1/5$ for $|s|\leq 2$ and 0 otherwise. Then, the expression for the covariance matrix becomes: \begin{eqnarray*} \frac{1}{T}\sum_{t=1}^T g^w_t (g^w_t)' &=& \frac{1}{T}\sum_{t=1}^T \left( \sum_{s=-2}^2 \frac{1}{5} g_{t+s} \right) \left( \sum_{l=-2}^2 \frac{1}{5} g_{t+l}' \right),\\ &=& \frac{1}{25}\sum_{s=-2}^2\sum_{l=-2}^2\left( \frac{1}{T}\sum_{t=1}^T g_{t+s}g_{t+l}' \right),\\ &=& \frac{1}{25}\sum_{s=-2}^2\sum_{l=-2}^2\hat{\Gamma}_{s-l} ,\\ &=& \frac{1}{25}\sum_{s=-4}^4 (5-|s|)\hat{\Gamma}_{s},\\ &=& \sum_{s=-4}^4 \left(\frac{1}{5}-\frac{|s|}{25}\right)\hat{\Gamma}_{s},\\ &=& \sum_{s=-T+1}^{T-1} k_5(s) \hat{\Gamma}_{s}, \end{eqnarray*} where $k_5(s)$ is the Bartlett kernel with a bandwidth of 5 defined as \[ K_5(s) = \left\{ \begin{array}{ccc} 1/5 - |s|/25 & \mbox{if}&|s|\leq 5\\ 0 &\mbox{otherwise}& \end{array}\right. . \] See \cite{smith01} for more details. The model will therefore be estimated using the kernel Truncated. The GMM estimate with the identity matrix is selected as starting value. <>= tet0 <- gmm(g4,x=x5t[,4:7],wmatrix="ident")$coef res <- gel(g4,x=x5t[,4:7],tet0,smooth=TRUE,kernel="Truncated") summary(res) @ The \textit{specTest} method applied to a \textit{gel} object computes the three tests proposed by \cite{smith04}: <<>>= specTest(res) @ The \textit{plot} method produces one more graphics when applied to a \textit{gel} object. It shows the implied probabilities along with the empirical density $(1/T)$. It allows to see which observations have more influence: \begin{center} <<>>= plot(res,which=4) @ \end{center} We can also select \textit{optfct="nlminb"} or \textit{constraint=TRUE} in order to impose restrictions on the coefficients. The former sets lower and upper bounds for the coefficients, while the latter imposes linear constraints using the algorithm \textit{constrOptim}. In this example we want the sum of the AR coefficients to be less than one. \textit{constrOptim} imposes the constraint $ui\theta-ci \geq 0$. Therefore, we need to set: <>= ui=cbind(0,-1,-1) ci <- -1 @ and rerun the estimation as <>= res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE,kernel="Truncated", constraint=TRUE, ui=ui,ci=ci) @ The result, which is not shown, is identical. They are also many option to compute the $\lambda$'s. From version 1.4-0, the default algorithm is \textit{nlminb} because the gradient and Hessian matrix are well defined analytically which speed up convergence. The other choices are \textit{optim} or "iter" which uses a Newton method to solve the first order condition. If the option optlam is set to "optim" and the type is EL, \textit{contrOptim} is selected automatically to restrict $\lambda'g_t$ to be less than 1. It is also possible to change the default values in the control list of the optimizer with the option LambdaControl (see \textit{?nlminb} or \textit{?optim}). Here are some examples: <>= res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim") res <- gel(g4,x=dat5[,4:7],tet0,smooth=TRUE, optlam="optim", LambdaControl=list(trace=TRUE, parscale=rep(.1,5))) @ \subsection{Comments} The GEL method is very unstable numerically. This fact has been reported many times in the recent literature. The method has been included in the \pkg{gmm} package because recent theoretical evidence suggests that it may produce better estimators than GMM. Because \proglang{R} is an open source statistical package, it offers a good platform to experiment with numerical properties of estimators. \section{Conclusion} The \pkg{gmm} package offers complete and flexible algorithms to estimate models by GMM and GEL. Several options are available which allow to choose among several GMM and GEL methods and many different HAC matrix estimators. In order to estimate the vector of parameters, users can select their preferred optimization algorithm depending on whether inequality constraints are required. For the vector of Lagrange multiplier of GEL, it can be computed by an iterative procedure based on the Newton method which increases the speed of convergence and reduce the instability of the estimation procedure. It could then easily be used by those who are interested in studying the numerical properties of both methods. The package also offers an interface which is comparable to the least squares method \textit{lm}. Linear model are estimated using formula and methods such as \textit{summary}, \textit{vcov}, \textit{coef}, \textit{confint}, \textit{plot}. \textit{residuals} or \textit{fitted} are available for the objects of class \textit{gmm} and \textit{gel}. \proglang{R} users will therefore have little difficulty in using the package. \section{Computational Details} The package \pkg{gmm} is written entirely in \proglang{R} and S3-classes with methods are used. It can be found on the comprehensive \proglang{R} archive network (CRAN,\url{http://CRAN.R-project.org/}). It is also hosted on R-Forge (\url{http://r-forge.r-project.org/projects/gmm}). It is shipped with a NAMESPACE. The version used to produce this paper is 1.4-0. It depends on the \pkg{sandwich} package of \cite{zeileis06}, which is used to compute de HAC matrices. The packages \pkg{car} (\cite{car}), \pkg{mvtnorm} (\cite{mvtnorm}), \pkg{stabledist} (\cite{stabledist}), \pkg{MASS} (\cite{MASS}), \pkg{timeDate} (\cite{timeDate}) and \pkg{timeSeries} (\cite{timeSeries}) are suggested in order to reproduce the examples. \section*{Acknowledgments} I am grateful to the three anonymous referees of the Journal of Statistical Software for great comments on the paper and the package. I also want to thank Achim Zeileis for his suggestions regarding the way the \pkg{sandwich} package can be used within \textit{gmm}. \bibliography{empir} \end{document} gmm/inst/CITATION0000644000176200001440000000132714017527033013151 0ustar liggesuserscitHeader("To cite gmm in publications use:") citEntry(entry = "Article", title = "Computing Generalized Method of Moments and Generalized Empirical Likelihood with {R}", author = personList(as.person("Pierre Chauss{\\'e}")), journal = "Journal of Statistical Software", year = "2010", volume = "34", number = "11", pages = "1--35", url = "https://www.jstatsoft.org/v34/i11/", textVersion = paste("Pierre Chausse (2010).", "Computing Generalized Method of Moments and Generalized Empirical Likelihood with R.", "Journal of Statistical Software, 34(11), 1-35.", "URL https://www.jstatsoft.org/v34/i11/.") )