Package ‘gmm’
October 13, 2022
Version1.7
Date2022-07-30
TitleGeneralized Method of Moments and Generalized Empirical
Likelihood
AuthorPierre Chausse
MaintainerPierre Chausse
Description
It is a complete suite to estimate models based on moment conditions. It includes the two step Gen-
eralized method of moments (Hansen 1982;
tinuous updated estimator (Hansen, Eaton and Yaron 1996;
eral methods that belong to the Generalized Empirical Likelihood family of estima-
tors (Smith 1997;
mura 1997; 0262.2004.00482.x>, and Anatolyev 2005 DependsR (>= 2.10.0), sandwich NeedsCompilationyes Suggestsknitr, mvtnorm, car, stabledist, MASS, timeDate, timeSeries Importsstats, methods, grDevices, graphics LicenseGPL (>= 2) VignetteBuilderknitr RepositoryCRAN Repository/R-Forge/Projectgmm Repository/R-Forge/Revision203 Repository/R-Forge/DateTimeStamp2022-08-12 02:50:41 Date/Publication2022-08-12 12:40:02 UTC Rtopics documented: ATEgel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 bread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 1 2ATEgel bwWilhelm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 charStable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 coef . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 confint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 estfun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 FinRes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 fitted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 getDat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 getImpProb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 getLamb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 getModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 gmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 KTest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 marginal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 momentEstim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 nsw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 smoothG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 specTest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 sysGmm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 tsls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vcov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 wage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Index56 ATEgelATE 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, ATEgel3 Lambdacontrol = list(), model = TRUE, X = FALSE, Y = FALSE, ...) checkConv(obj, tolConv=1e-4, verbose=TRUE, ...) Arguments gA formula asy~z, where codey is the response andzthe treatment indicator. If there is more than one treatment, more indicators can be added orzcan be set as a factor. It can also be of the formg(theta, y, z)for non-linear models. It is however, not implemented yet. objObject of class"ategel"produced yATEgel balmA formula for the moments to be balanced between the treated and control groups (see details) yThe response variable whengis a function. Not implemented yet treatThe treatment indicator whengis a function. Not implemented yet wA formula to add covariates to the main regression. WhenNULL, the default value, the main regression only include treatment indicators. tet0A3×1vector of starting values. If not provided, they are obtained using an OLS regression momTypeHow 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) popMomA 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. familyBy 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. type"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continu- ous 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 prob- abilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution. tol_lamTolerance forλbetween two iterations. The algorithm stops when‖λ i −λ i−1 ‖ reachestol_lamb(seegetLamb) maxiterlamThe algorithm to computeλstops if there is no convergence after "maxiterlam" iterations (seegetLamb). tol_objTolerance for the gradiant of the objective function to computeλ(seegetLamb). optfctAlgorithm used for the parameter estimates tol_momIt is the tolerance for the moment condition ∑ n t=1 p t g(θ(x t ) = 0, wherep t = 1 n Dρ(< g t ,λ >)is the implied probability. It adds a penalty if the solution diverges from its goal. 4ATEgel optlamAlgorithm used to solve for the lagrange multiplier ingetLamb. The algorithm Wu is only fortype="EL". The value ofoptlamis ignored for "CUE" because in that case, the analytical solution exists. dataA data.frame or a matrix with column names (Optional). LambdacontrolControls for the optimization of the vector of Lagrange multipliers used by either optim,nlminborconstrOptim model, X, Ylogicals. IfTRUEthe corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula. verboseIf TRUE, a summary of the convergence is printed tolConvThe tolerance for comparing moments between groups ...More options to give tooptimornlminb. IncheckConv, they are options passed togetImpProb. Details We want to estimate the modelY t =θ 1 +θ 2 treat+ t , whereθ 2 is the treatment effect. GEL is used to balance the sample based on the argumentxabove. For example, if we want the sample mean ofx1andx2to be balanced between the treated and control, we setxto~x1+x2. If we want the sample mean ofx1,x2,x1*x2,x1^2andx2^2, we set xto~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 classgel. References Lee, Seojeong (2016), Asymptotic refinements of misspecified-robust bootstrap for GEL estimators, Journal of Econometrics,192, 86–104. Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood. Econometrica,35, 634-672. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling.Survey Methodology,31(2), page 239. Chausse, P. (2010), Computing Generalized Method of Moments and Generalized Empirical Like- lihood with R.Journal of Statistical Software,34(11), 1–35. URL doi:10.18637/jss.v034.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 bread5 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) breadBread for sandwiches Description Computes the bread of the sandwich covariance matrix Usage ## S3 method for class'gmm' bread(x, ...) ## S3 method for class'gel' bread(x, ...) ## S3 method for class'tsls' bread(x, ...) Arguments xA fitted model of classgmmorgel. ...Other arguments whenbreadis applied to another class object Details When the weighting matrix is not the optimal one, the covariance matrix of the estimated coeffi- cients is:(G ′ WG) −1 G ′ WV WG(G ′ WG) −1 , whereG=d ̄g/dθ,Wis the matrix of weights, and Vis the covariance matrix of the moment function. Therefore, the bread is(G ′ WG) −1 , which is the second derivative of the objective function. The method if not yet available forgelobjects. Value Ak×kmatrix (see details). References Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.i09. 6bwWilhelm 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)) bwWilhelmWilhelm (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 xAn object of classgmm. order.byEither 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). bwWilhelm7 kerneltype of kernel used to compute the covariance matrix of the vector of sample moment conditions (seekernHACfor more details) approxA character specifying the approximation method if the bandwidth has to be chosen bybwAndrews. weightsnumeric. 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) prewhitelogical or integer. Should the estimating functions be prewhitened? IfTRUEor greater than 0 a VAR model of orderas.integer(prewhite)is fitted viaar with method"ols"anddemean = FALSE. ar.methodcharacter. Themethodargument passed toarfor prewhitening. dataan optional data frame containing the variables in the ’order.by’ model. Value The function ’bwWilhelm’ returns the optimal bandwidth. Note The function was written by Daniel Wilhelm and is based on bwAndrews. References Wilhelm, D. (2015), Optimal Bandwidth Selection for Robust Generalized Method of Moments Estimation.Econometric Theory,31, 1054–1077 Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.i09. 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 8charStable ################################################################### res3 <- gmm(y ~ f1, ~ f1 + f2 + f3, bw=bwWilhelm) summary(res3) charStableThe characteristic function of a stable distribution Description It computes the theoretical characteristic function of a stable distribution for two different parametriza- tions. It is used in the vignette to illustrate the estimation of the parameters using GMM. Usage charStable(theta, tau, pm = 0) Arguments thetaVector of parameters of the stable distribution. See details. tauA vector of numbers at which the function is evaluated. pmThe type of parametization. It takes the values 0 or 1. Details The function returns the vectorΨ(θ,τ,pm)defined asE(e ixτ , whereτis a vector of real numbers,i is the imaginary number,xis a stable random variable with parametersθ=(α,β,γ,δ)andpmis 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:α∈(0,2], β∈[−1,1]andγ >0. For mode details see Nolan(2009). Value It returns a vector of complex numbers with the dimension equals tolength(tau). References Nolan J. P. (2020), Univariate Stable Distributions - Models for Heavy Tailed Data.Springer Se- ries in Operations Research and Financial Engineering. URLhttps://edspace.american.edu/ jpnolan/stable/. coef9 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) coefCoefficients of GEL or GMM Description It extracts the coefficients fromgelorgmmobjects. Usage ## S3 method for class'gmm' coef(object, ...) ## S3 method for class'gel' coef(object, lambda = FALSE, ...) Arguments objectAn object of classgelorgmmreturned by the functiongelorgmm lambdaIf set to TRUE, the lagrange multipliers are extracted instead of the vector of coefficients ...Other arguments whencoefis 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)] 10confint 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) confintConfidence intervals for GMM or GEL Description It produces confidence intervals for the coefficients fromgelorgmmestimation. Usage ## S3 method for class'gel' confint(object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, ...) ## S3 method for class'gmm' confint(object, parm, level = 0.95, ...) ## S3 method for class'ategel' confint(object, parm, level = 0.95, lambda = FALSE, type = c("Wald", "invLR", "invLM", "invJ"), fact = 3, corr = NULL, robToMiss=TRUE, ...) ## S3 method for class'confint' print(x, digits = 5, ...) Arguments objectAn object of classgelorgmmreturned by the functiongelorgmm parmA 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 consid- ered. levelThe confidence level lambdaIf set to TRUE, the confidence intervals for the Lagrange multipliers are pro- duced. type’Wald’ is the usual symetric confidence interval. The thee others are based on the inversion of the LR, LM, and J tests. confint11 factThis 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. corrThis 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) xAn object of classconfintproduced byconfint.gelandconfint.gmm digitsThe number of digits to be printed robToMissIfTRUE, the confidence interval is based on the standard errors that are robust to misspecification ...Other arguments whenconfintis 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. Econometrica,50, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators.Journal of Business and Economic Statistics, 14262-280. Owen, A.B. (2001), Empirical Likelihood.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) 12estfun 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") estfunExtracts the empirical moment function Description It extracts the matrix of empirical moments so that it can be used by thekernHACfunction. Usage ## S3 method for class'gmmFct' estfun(x, y = NULL, theta = NULL, ...) ## S3 method for class'gmm' estfun(x, ...) ## S3 method for class'gel' estfun(x, ...) ## S3 method for class'tsls' estfun(x, ...) ## S3 method for class'tsls' model.matrix(object, ...) Arguments xA function of the formg(θ,y)or an×qmatrix with typical elementg i (θ,y t )for i= 1,...qandt= 1,...,nor an object of classgmm. Seegmmfor more details. Fortsls, it is an object of classtsls. estfun13 objectAn object of classtsls. yThe matrix or vector of data from which the functiong(θ,y)is computed ifgis a function. thetaVector of parameters ifgis a function. ...Other arguments whenestfunis applied to another class object Details Forestfun.gmmFct, it returns an×qmatrix with typical elementg i (θ,y t )fori= 1,...qand t= 1,...,n. It is only used bygmmto obtain the estimates. Forestfun.gmm, it returns the matrix of first order conditions ofmin θ ̄g ′ W ̄g/2, which is an×k matrix with thet th row beingg(θ,y t )WG, whereGisd ̄g/dθ. It allows to compute the sandwich covariance matrix usingkernHACorvcovHACwhenWis not the optimal matrix. The method if not yet available forgelobjects. For tsls, model.matrix and estfun are used byvcov()to compute different covariance matrices using thesandwichpackage. Seevcov.tsls.model.matrixreturns the fitted values frin the first stage regression andesfunthe residuals. Value An×qmatrix (see details). References Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.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 foc2 <- estfun(res) 14Finance foc[1:5,] foc2[1:5,] FinanceReturns 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 descrip- tion, company symboles are used. WMKReturns of WEIS MARKETS INC UISReturns of UNISYS CP NEW ORBReturns of ORBITAL SCIENCES CP MATReturns of Mattel, Inc. ABAXReturns of ABAXIS, Inc. TReturns of AT&T INC. EMRReturns of EMERSON ELEC CO JCSReturns of Communications Systems Inc. VOXXReturns of Audiovox Corp. ZOOMReturns of ZOOM Technologies Inc. TDWReturns of TIDEWATER INC ROGReturns of Rogers Corporation GGGReturns of Graco Inc. PCReturns of Panasonic Corporation GCOReturns of Genesco Inc. EBFReturns of ENNIS, INC FReturns of FORD MOTOR CO FNMReturns of FANNIE MAE NHPReturns of NATIONWIDE HLTH PROP AAReturns of ALCOA INC rfRisk-free rate of Fama-French rmReturn of the market portfolio of Fama-French hmlFactor High-Minus-Low of Fama-French smbFactor Small-Minus-Big of Fama-French FinRes15 Source Yahoo Finance andhttps://mba.tuck.dartmouth.edu/pages/faculty/ken.french/ FinResMethod to finalize the result of the momentEstim method Description It computes the final results that will be needed to create the object of classgmm.). Usage ## S3 method for class'baseGmm.res' FinRes(z, object, ...) Arguments zAn object of class determined by the methodmomentEstim. objectAn object produced mygetModel ...Other argument to be passed to otherFinResmethods. Value It returns an object of classgmm. Seegmmfor more details. References Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. Econometrica,50, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators.Journal of Business and Economic Statistics,14262-280. fittedFitted values of GEL and GMM Description Method to extract the fitted values of the model estimated bygelorgmm. Usage ## S3 method for class'gel' fitted(object, ...) ## S3 method for class'gmm' fitted(object, ...) 16fitted Arguments objectAn object of classgelorgelreturned by the functiongelorgmm ...Other arguments whenfittedis applied to an other class object Value It returns a matrix of the estimated meanˆying=y~xas it is done byfitted.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)) plot(y, main = "Fitted ARMA with GEL") lines(fitted(res), col = 2) # 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, y, main = "Fitted model with GMM") lines(x, fitted(res), col = 2) legend("topright", c("Y","Yhat"), col = 1:2, lty = c(1,1)) formula17 formulaFormula method for gel and gmm objects Description Method to extract the formula fromgelorgmmobjects. Usage ## S3 method for class'gel' formula(x, ...) ## S3 method for class'gmm' formula(x, ...) Arguments xAn object of classgelorgmmreturned by the functiongelorgmm ...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 18gel res <- gmm(y ~ x, x) formula(res) gelGeneralized 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 gA function of the formg(θ,x)and which returns an×qmatrix with typical elementg i (θ,x t )fori= 1,...qandt= 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). tet0Ak×1vector of starting values. If the dimension ofθis 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 xThe matrix or vector of data from which the functiong(θ,x)is computed. If "g" is a formula, it is ann×Nhmatrix of instruments (see details below). gel19 gradvA function of the formG(θ,x)which returns aq×kmatrix of derivatives of ̄g(θ) with respect toθ. By default, the numerical algorithmnumericDerivis 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 ˆ θ. If "g" is a formula, the gradiant is not required (see the details below). smoothIf set to TRUE, the moment function is smoothed as proposed by Kitamura(1997) type"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continu- ous 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 prob- abilities are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution. kerneltype of kernel used to compute the covariance matrix of the vector of sample moment conditions (seekernHACfor 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 im- plies a Parzen kernel (see Smith 2004). bwThe method to compute the bandwidth parameter. By default it isbwAndrews which is proposed by Andrews (1991). The alternative isbwNeweyWestof Newey-West(1994). prewhitelogical or integer. Should the estimating functions be prewhitened? IfTRUEor greater than 0 a VAR model of orderas.integer(prewhite)is fitted viaar with method"ols"anddemean = FALSE. ar.methodcharacter. Themethodargument passed toarfor prewhitening. approxa character specifying the approximation method if the bandwidth has to be chosen bybwAndrews. tol_weightsnumeric. Weights that exceedtolare used for computing the covariance matrix, all other weights are treated as 0. tol_lamTolerance forλbetween two iterations. The algorithm stops when‖λ i −λ i−1 ‖ reachestol_lamb(seegetLamb) maxiterlamThe algorithm to computeλstops if there is no convergence after "maxiterlam" iterations (seegetLamb). tol_objTolerance for the gradiant of the objective function to computeλ(seegetLamb). optfctOnly when the dimension ofθis 1, you can choose between the algorithmoptim oroptimize. In that case, the former is unreliable. Ifoptimizeis chosen, "t0" must be1×2which represents the interval in which the algorithm seeks the solution.It is also possible to choose thenlminbalgorithm. In that case, borns for the coefficients can be set by the optionsupper=andlower=. constraintIf set to TRUE, the constraint optimization algorithm is used. SeeconstrOptim 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 constraintuiθ−ci≥0. tol_momIt is the tolerance for the moment condition ∑ n t=1 p t g(θ(x t ) = 0, wherep t = 1 n Dρ(< g t ,λ >)is the implied probability. It adds a penalty if the solution diverges from its goal. 20gel optlamAlgorithm used to solve for the lagrange multiplier ingetLamb. The algorithm Wu is only fortype="EL". The value ofoptlamis ignored for "CUE" because in that case, the analytical solution exists. dataA data.frame or a matrix with column names (Optional). LambdacontrolControls for the optimization of the vector of Lagrange multipliers used by either optim,nlminborconstrOptim model, X, Ylogicals. IfTRUEthe corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula. TypeGelThe name of the class object created by the methodgetModel. It allows devel- opers to extand the package and create other GEL methods. alphaRegularization coefficient for discrete CGEL estimation (experimental). By set- ting alpha to any value, the model is estimated by CGEL of type specified by the optiontype. See Chausse (2011) eqConstEither a named vector (if "g" is a function), a simple vector for the nonlinear case indicating which of theθ 0 is restricted, or a qx2 vector defining equality constraints of the formθ i =c i . Seegmmfor an example. eqConstFullVcov If FALSE, the constrained coefficients are assumed to be fixed and only the co- variance of the unconstrained coefficients is computed. If TRUE, the covariance matrix of the full set of coefficients is computed. onlyCoefficients IfTRUE, only the vector of coefficients and Lagrange multipliers are returned ...More options to give tooptim,optimizeorconstrOptim. Details If we want to estimate a model likeY t =θ 1 +X 2t θ 2 +...+X k θ k + t using the moment conditions Cov( t H t ) = 0, whereH t is a vector ofNhinstruments, than we can define "g" like we do forlm. We would haveg = y~x2+x3+...+xkand the argument "x" above would become the matrix H of instruments. As forlm,Y t can be aNy×1vector which would imply thatk=Nh×Ny. The intercept is included by default so you do not have to add a column of ones to the matrixH. You do not need to provide the gradiant in that case since in that case it is embedded ingel. 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 ∑ n t=1 g(θ,x t )is replaced by: ∑ n t=1 g k (θ,x t ), whereg k (θ,x t ) = ∑ r i=−r k(i)g(θ,x t+i ), whereris a truncated parameter that depends on the bandwidth andk(i)are normalized weights so that they sum to 1. The method solves ˆ θ= arg min [ arg max λ 1 n ∑ n t=1 ρ(< g(θ,x t ),λ >)−ρ(0) ] evalGelgenerates the object of class "gel" for a fixed vector of parameters. There is no estimation forθ, but the optimal vector of Lagrange multipliersλis computed. The objective function is then the profiled likelihood for a givenθ. It can be used to construct a confidence interval by inverting the likelihood ratio test. gel21 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: coefficientsk×1vector of parameters residualsthe residuals, that is response minus fitted values if "g" is a formula. fitted.valuesthe fitted mean values if "g" is a formula. lambdaq×1vector of Lagrange multipliers. vcov_parthe covariance matrix of "coefficients" vcov_lambdathe covariance matrix of "lambda" ptThe implied probabilities objectivethe value of the objective function conv_lambdaConvergence code for "lambda" (seegetLamb) conv_mesConvergence message for "lambda" (seegetLamb) conv_parConvergence code for "coefficients" (seeoptim,optimizeorconstrOptim) termsthetermsobject used when g is a formula. callthe matched call. yif requested, the response used (if "g" is a formula). xif requested, the model matrix used if "g" is a formula or the data if "g" is a function. modelif requested (the default), the model frame used if "g" is a formula. References Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias.Econometrica,73, 983-1002. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti- mation.Econometrica,59, 817–858. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.The Annals of Statistics,25, 2084-2102. Kitamura, Y. and Otsu, T. and Evdokimov, K. (2013), Robustness, Infinitesimal Neighborhoods and Moment Restrictions.Econometrica,81, 1185-1201. Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators.Econometrica,72, 219-255. Smith, R.J. (2004), GEL Criteria for Moment Condition Models.Working paper, CEMMAP. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Auto- correlation Consistent Covariance Matrix.Econometrica,55, 703–708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.Review of Economic Studies,61, 631-653. 22gel Schennach, Susanne, M. (2007), Point Estimation with Exponentially Tilted Empirical Likelihood. Econometrica,35, 634-672. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling.Survey Methodology,31(2), page 239. Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.i09. Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likeli- hood with R.Journal of Statistical Software,34(11), 1–35. URL doi:10.18637/jss.v034.i11. Chausse (2011), Generalized Empirical likelihood for a continumm of moment conditions.Working Paper,Department of Economics,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 getDat23 res <- gel(g, x, c(0, .3, .6)) summary(res) # Using evalGel to create the object without estimation res <- evalGel(g, x, res$coefficients) getDatExtracting 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 functiong(θ,x)that is required bygmmandgel. Usage getDat(formula, h, data, error=TRUE) Arguments formulaA formula that defines the linear model to be estimated (see details). hAn×nhmatrix of intruments(see details). dataA data.frame or a matrix with colnames (Optionnal). errorIf FALSE, the data is generated without giving any error message Details The model to be estimated is based on the moment conditions< h,(y−zθ)>= 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 forsysGmmwith common coefficients because the check is only valid for equation by equation identification. Value x: An×lmatrix, wherel=ncol(y) +ncol(z) +ncol(h) + 2if "intercept" is TRUE and ncol(y) +ncol(z) +xcol(h)if "intercept" is FALSE. nh: dimension of h k: dimension of z ny: dimension of y 24getImpProb 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) getImpProbImplied Probabilities Description It computes the implied probabilities from objects of classgelwith additional options. Usage ## S3 method for class'gel' getImpProb(object, posProb=TRUE, normalize=TRUE, checkConv=FALSE,...) Arguments objectObject of classgel. posProbShould the implied probabilities be transformed into positive probabilities? normalizeShould we normalize the probabilities so that they sum to one? checkConvShould we add the attribute convergence to check the sum of the probabilities and the weighted sum of the moment conditions? ...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.Econometrica,72, 219-255. getLamb25 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) getLambSolving for the Lagrange multipliers of Generalized Empirical Likeli- hood (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 gtAn×qmatrix with typical elementg i (θ,x t ) l0Vector of starting values for lambda type"EL" for empirical likelihood, "ET" for exponential tilting, "CUE" for continu- ous updated estimator, and "HD" for Hellinger Distance. See details for "ETEL" and "ETHD". "RCUE" is a restricted version of "CUE" in which the probabili- ties are bounded below by zero. In that case, an analytical Kuhn-Tucker method is used to find the solution. tol_lamTolerance forλbetween two iterations. The algorithm stops when‖λ i −λ i−1 ‖ reachestol_lam 26getLamb maxiterlamThe algorithm stops if there is no convergence after "maxiterlam" iterations. tol_objTolerance for the gradiant of the objective function. The algorithm returns a non-convergence message ifmax(|gradiant|)does not reachtol_obj. It helps thegelalgorithm to select the right space to look forθ kIt represents the ratio k1/k2, wherek1 = ∫ ∞ −∞ k(s)dsandk2 = ∫ ∞ −∞ k(s) 2 ds. See Smith(2004). methodThe iterative procedure uses a Newton method for solving the FOC. It i however recommended to useoptimornlminb. If type is set to "EL" and method to "optim",constrOptimis called to preventlog(1−gt ′ λ)from producing NA. The gradient and hessian is provided tonlminbwhich 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 ofmethodis ignored for "CUE" because in that case, the analytical solution exists. controlControls to send tooptim,nlminborconstrOptim Details It solves the problemmax λ 1 n ∑ n t=1 ρ(gt ′ λ). For the type "ETEL", it is only used bygel. In that caseλis obtained by maximizing 1 n ∑ n t=1 ρ(gt ′ λ), usingρ(v) =−expv(so ET) andθby minimizing the same equation but withρ(v)−log (1−v). To avoid NA’s,constrOptimis used with the restrictionλ ′ g t <1. The type "ETHD" is experimental and proposed by Antoine-Dovonon (2015). The paper is not yet available. Value lambda: Aq×1vector of Lagrange multipliers which solve the system of equations given above. 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.Econometrica,72, 219-255. Smith, R.J. (2004), GEL Criteria for Moment Condition Models.Working paper, CEMMAP. Wu, C. (2005), Algorithms and R codes for the pseudo empirical likelihood method in survey sampling.Survey Methodology,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 getModel27 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") getModelMethod for setting the properties of a model Description It collects what is needed by the methodmomentEstim(see details). Usage ## S3 method for class'baseGmm' getModel(object, ...) ## S3 method for class'sysGmm' getModel(object, ...) ## S3 method for class'baseGel' getModel(object, ...) ## S3 method for class'constGel' getModel(object, ...) ## S3 method for class'constGel' getModel(object, ...) ## S3 method for class'tsls' getModel(object, ...) ## S3 method for class'ateGel' getModel(object, ...) Arguments objectAn object of classbaseGmm ...Other arguments whengetModelis applied to another class object Value It returns an object of the right class which determines how the methodmomentEstimwill treat it. For example, ifgis a formula andtypeis set to "cue", it creates an object of classbaseGmm.cue.formula. It this case,momentEstim, applied to this object, computes the continuously updated GMM of a lin- ear 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 associatedmomentEstimemethod. 28gmm gmmGeneralized 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 gA function of the formg(θ,x)and which returns an×qmatrix with typical elementg i (θ,x t )fori= 1,...qandt= 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). xThe matrix or vector of data from which the functiong(θ,x)is computed. If "g" is a formula, it is ann×Nhmatrix of instruments or a formula (see details below). t0Ak×1vector of starting values. It is required only when "g" is a function because only then a numerical algorithm is used to minimize the objective func- tion. If the dimension ofθis one, see the argument "optfct". tetwAk×1vector to compute the weighting matrix. gradvA function of the formG(θ,x)which returns aq×kmatrix of derivatives of ̄g(θ) with respect toθ. By default, the numerical algorithmnumericDerivis 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 ˆ θand to obtain the analytical gradient of the objective function if the method is set to "CG" or "BFGS" inoptimand if "type" is not set to "cue". If "g" is a formula, the gradiant is not required (see the details below). gmm29 typeThe 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) wmatrixWhich weighting matrix should be used in the objective function. By default, it is the inverse of the covariance matrix ofg(θ,x). The other choice is the identity matrix which is usually used to obtain a first step estimate ofθ vcovAssumption 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. kerneltype of kernel used to compute the covariance matrix of the vector of sample moment conditions (seekernHACfor more details) critThe stopping rule for the iterative GMM. It can be reduce to increase the preci- sion. bwThe method to compute the bandwidth parameter in the HAC weighting matrix. The default islink{bwAndrews}(as proposed in Andrews (1991)), which mini- mizes the MSE of the weighting matrix. Alternatives arelink{bwWilhelm}(as proposed in Wilhelm (2015)), which minimizes the mean-square error (MSE) of the resulting GMM estimator, andlink{bwNeweyWest}(as proposed in Newey- West(1994)). prewhitelogical or integer. Should the estimating functions be prewhitened? IfTRUEor greater than 0 a VAR model of orderas.integer(prewhite)is fitted viaar with method"ols"anddemean = FALSE. ar.methodcharacter. Themethodargument passed toarfor prewhitening. approxA character specifying the approximation method if the bandwidth has to be chosen bybwAndrews. tolWeights that exceedtolare used for computing the covariance matrix, all other weights are treated as 0. itermaxThe maximum number of iterations for the iterative GMM. It is unlikely that the algorithm does not converge but we keep it as a safety. optfctOnly when the dimension ofθis 1, you can choose between the algorithmoptim oroptimize. In that case, the former is unreliable. Ifoptimizeis chosen, "t0" must be1×2which represents the interval in which the algorithm seeks the solution. It is also possible to choose thenlminbalgorithm. In that case, boundaries for the coefficients can be set by the optionsupper=andlower=. TheconstrOptimis 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. model, X, Ylogical. IfTRUEthe corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula. TypeGmmThe name of the class object created by the methodgetModel. It allows devel- opers to extend the package and create other GMM methods. centeredVcovShould the moment function be centered when computing its covariance matrix. Doing so may improve inference. 30gmm weightsMatrixIt allows users to providegmmwith a fixed weighting matrix. This matrix must beq×q, symmetric and strictly positive definite. When provided, thetype option becomes irrelevant. traceIterTracing information for GMM of type "iter" dataA data.frame or a matrix with column names (Optional). eqConstEither a named vector (if "g" is a function), a simple vector for the nonlinear case indicating which of theθ 0 is restricted, or a qx2 vector defining equality constraints of the formθ i =c i . See below for an example. which, valueThe 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. objObject of class "gmm" eqConstFullVcov If FALSE, the constrained coefficients are assumed to be fixed and only the co- variance of the unconstrained coefficients is computed. If TRUE, the covariance matrix of the full set of coefficients is computed. mustarIf 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. onlyCoefficients If set toTRUE, the function only returns the coefficient estimates. It may be of interest when the standard errors are not needed ...More options to give tooptim. Details If we want to estimate a model likeY t =θ 1 +X 2t θ 2 +···+X k θ k + t using the moment conditions Cov( t H t ) = 0, whereH t is a vector ofNhinstruments, than we can define "g" like we do forlm. We would haveg=y ̃x2 +x3 +···+xkand the argument "x" above would become the matrix H of instruments. As forlm,Y t can be aNy×1vector which would imply thatk=Nh×Ny. The intercept is included by default so you do not have to add a column of ones to the matrixH. You do not need to provide the gradiant in that case since in that case it is embedded ingmm. 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 {1,z 1 ,z 2 ,z 3 }, we can set "x" to ̃z1 +z2 +z3. By default, a column of ones is added. To remove it, set "x" to ̃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(μ,σ)we have the following three moment condi- tions: m 1 =μ−x i m 2 =σ 2 −(x i −μ) 2 m 3 =x 3 i −μ(μ 2 + 3σ 2 ) gmm31 m 1 ,m 2 can be directly obtained by the definition of(μ,σ). The third moment condition comes from the third derivative of the moment generating function (MGF) M X (t) =exp ( μt+ σ 2 t 2 2 ) evaluated at(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): 10 −2μ+ 2x2σ −3μ 2 −3σ 2 −6μσ gmmWithConst()re-estimates an unrestricted model by adding an equality constraint.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: coefficientsk×1vector of coefficients residualsthe residuals, that is response minus fitted values if "g" is a formula. fitted.valuesthe fitted mean values if "g" is a formula. vcovthe covariance matrix of the coefficients objectivethe value of the objective function‖var( ̄g) −1/2 ̄g‖ 2 termsthetermsobject used when g is a formula. callthe matched call. yif requested, the response used (if "g" is a formula). xif requested, the model matrix used if "g" is a formula or the data if "g" is a function. modelif requested (the default), the model frame used if "g" is a formula. algoInfoInformation produced by eitheroptimornlminbrelated to the convergence if "g" is a function. It is printed by thesummary.gmmmethod. 32gmm References Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.i09. Pierre Chausse (2010), Computing Generalized Method of Moments and Generalized Empirical Likelihood with R.Journal of Statistical Software,34(11), 1–35. URL doi:10.18637/jss.v034.i11. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti- mation.Econometrica,59, 817–858. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Auto- correlation Consistent Covariance Matrix.Econometrica,55, 703–708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.Review of Economic Studies,61, 631-653. Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. Econometrica,50, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finite-Sample Properties of Some Alternative GMM Estimators.Journal of Business and Economic Statistics,14262-280. 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)$coefficients gmm33 ## APT test with Fama-French factors and GMM f1 <- zm f2 <- Finance[1:300, "hml"] f3 <- Finance[1:300, "smb"] h <- cbind(f1, f2, f3) res2 <- gmm(z ~ f1 + f2 + f3, x = h) coef(res2) summary(res2)$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) 34gmm 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. Growth35 ###################################### 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) GrowthGrowth 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. Country_IDCountry identification number COM1 if the country is in a communist regime, 0 otherwise OPEC1 if the country is part of the OPEC, 0 otherwise YearYear GDPPer capita GDP (in thousands) in 1985 U.S. dollars. LagGDPGDP of the previous period SavRateSaving rate measured as the ratio of real investment to real GDP LagSavRateSavRate of the previous period CountryCountry names PopPopulation in thousands LagPopPopulation of the previous period 36KTest Source http://fhayashi.fc2web.com/datasets.htm KTestCompute 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) ## S3 method for class'gmmTests' print(x, digits = 5, ...) Arguments objObject of class "gmm" returned bygmm theta0The null hypothesis being tested. See details. alphaK, alphaJThe size of the J and K tests when combining the two. The overall size is alphaK+alphaJ. xAn object of classgmmTestsreturned byKTest digitsThe number of digits to be printed ...Other arguments whenprintis applied to another class object Details The function produces the J-test and K-statistics which are robust to weak identification. The test is eitherH0 :θ=theta 0 , in which case theta0 must be provided, orβ=β 0 , whereθ= (α ′ ,β ′ ) ′ , and αis assumed to be identified. In the latter case, theta0 is NULL and obj is a restricted estimation in whichβis fixed toβ 0 . Seegmmand 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.Econo- metrica,73, 1103-1123, marginal37 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) marginalMarginal effects Summary Description It produces the summary table of marginal effects for GLM estimation with GEL. Only imple- mented for ATEgel. Usage ## S3 method for class'ategel' marginal(object, ...) Arguments objectAn object of classategelreturned by the functionATEgel ...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. 38momentEstim References Owen, A.B. (2001), Empirical Likelihood.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) momentEstimMethod for estimating models based on moment conditions Description It estimates a model which is caracterized by the methodgetModel(see details). Usage ## S3 method for class'baseGmm.twoStep' momentEstim(object, ...) ## S3 method for class'baseGmm.twoStep.formula' momentEstim(object, ...) ## S3 method for class'sysGmm.twoStep.formula' momentEstim(object, ...) ## S3 method for class'tsls.twoStep.formula' nsw39 momentEstim(object, ...) ## S3 method for class'baseGmm.iterative.formula' momentEstim(object, ...) ## S3 method for class'baseGmm.iterative' momentEstim(object, ...) ## S3 method for class'baseGmm.cue.formula' momentEstim(object, ...) ## S3 method for class'baseGmm.cue' momentEstim(object, ...) ## S3 method for class'baseGmm.eval' momentEstim(object, ...) ## S3 method for class'baseGel.mod' momentEstim(object, ...) ## S3 method for class'baseGel.modFormula' momentEstim(object, ...) ## S3 method for class'baseGel.eval' momentEstim(object, ...) Arguments objectAn object created by the methodgetModel ...Other arguments whenmomentEstimis applied to an other class object Value It returns an object of class determined by the argument "TypeGMM" ofgmm. By default, it is of classbaseGmm.res. It estimates the model and organize the results that will be finalized by the methodFinRes. 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. Econometrica,50, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators.Journal of Business and Economic Statistics,14262-280. nswLalonde 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). 40nsw Usage data(nsw) Format A data frame containing 9 variables. treatTreatment assignment ageAge edYears of Education black1 if Black, 0 otherwise hisp1 if Hispanic 0 otherwise married1 if married 0 otherwise nodeg1 if no college degree 0 otherwise re751975 earnings re781978 earnings Details The dataset was obtained from the ATE package (see reference). Source "NSW Data Files" from Rajeev Dehejia’s website. 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: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.https://CRAN.R-project.org/package=ATE plot41 plotPlot Diagnostics for gel and gmm objects Description It is a plot method forgelorgmmobjects. Usage ## S3 method for class'gel' plot(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")) ## S3 method for class'gmm' plot(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 x gelorgmmobject, typically result ofgelorgmm. whichif a subset of the plots is required, specify a subset of the numbers1:4forgel or1:3forgmm. mainVector of titles for each plot. panelpanel function. The useful alternative topoints,panel.smoothcan be chosen byadd.smooth = TRUE. asklogical; ifTRUE, the user isasked before each plot, seepar(ask=.). ...other parameters to be passed through to plotting functions. add.smoothlogical indicating if a smoother should be added to most plots; see alsopanel above. Details It is a beta version of a plot method forgelobjects. It is a modified version ofplot.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 byplot.lm, the third is the dependant variableywith its meanˆy(the fitted values) and the last plots the implied probabilities with the empirical density1/T. 42print 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) printPrinting a gmm or gel object Description It is a printing method forgmmorgelobjects. Usage ## S3 method for class'gmm' print(x, digits = 5, ...) ## S3 method for class'gel' print(x, digits = 5, ...) ## S3 method for class'sysGmm' print(x, digits = 5, ...) Arguments xAn object of classgmmorgelreturned by the functiongmmorgel digitsThe number of digits to be printed ...Other arguments when print is applied to an other class object residuals43 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) residualsResiduals of GEL or GMM Description Method to extract the residuals of the model estimated bygmmorgel. Usage ## S3 method for class'gel' residuals(object, ...) ## S3 method for class'gmm' residuals(object, ...) Arguments objectAn object of classgmmorgelreturned by the functiongmmorgel ...Other arguments whenresidualsis applied to an other classe object 44smoothG Value It returns the matrix of residuals(y−ˆy)ing=y~xas it is done byresiduals.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") smoothGKernel 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 thegelfunction. smoothG45 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 xan×qmatrix of time series, where n is the sample size. bwThe method to compute the bandwidth parameter. By default, it uses the band- width proposed by Andrews(1991). As an alternative, we can choose bw=bwNeweyWest (without "") which is proposed by Newey-West(1996). prewhitelogical or integer. Should the estimating functions be prewhitened? IfTRUEor greater than 0 a VAR model of orderas.integer(prewhite)is fitted viaar with method"ols"anddemean = FALSE. ar.methodcharacter. Themethodargument passed toarfor prewhitening. weightsThe smoothing weights can be computed byweightsAndrewsof it can be pro- vided manually. If provided, it has to be ar×1vector (see details). approxa character specifying the approximation method if the bandwidth has to be chosen bybwAndrews. tolnumeric. Weights that exceedtolare used for computing the covariance matrix, all other weights are treated as 0. kernelThe choice of kernel Details The sample moment conditions ∑ n t=1 g(θ,x t )is replaced by: ∑ n t=1 g k (θ,x t ), whereg k (θ,x t ) = ∑ r i=−r k(i)g(θ,x t+i ), whereris a truncated parameter that depends on the bandwidth andk(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),k(i)will become(.25,.5,1,.5,.25)/(.25 +.5 + 1 +.5 +.25) = (.1,.2,.4,.2,.1) Value smoothx: Aq×qmatrix containing an estimator of the asymptotic variance of √ n ̄x, where ̄xis q×1vector with typical element ̄x i = 1 n ∑ n j=1 x ji . This function is called bygelbut can also be used by itself. kern_weights: Vector of weights used for the smoothing. References Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias.Econometrica,73, 983-1002. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti- mation.Econometrica,59, 817–858. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.The Annals of Statistics,25, 2084-2102. 46specTest Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.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]) specTestCompute tests of specification Description Generic function for testing the specification of estimated models. It computes the J-test fromgmm objects and J-test, LR-test and LM-test fromgelobjects. Usage ## S3 method for class'gmm' specTest(x, ...) ## S3 method for class'gel' specTest(x, ...) ## S3 method for class'specTest' print(x, digits = 5, ...) specTest(x, ...) Arguments xA fitted model object. digitsThe number of digits to be printed. ...Arguments passed to methods. Value Tests and p-values summary47 References Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. Econometrica,50, 1029-1054, Smith, R. J. (2004), GEL Criteria for Moment Condition Models.CeMMAP working papers, Insti- tute 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) summaryMethod for object of class gmm or gel Description It presents the results from thegmmorgelestimation in the same fashion assummarydoes for the lmclass objects for example. It also compute the tests for overidentifying restrictions. Usage ## S3 method for class'gmm' summary(object, ...) ## S3 method for class'sysGmm' summary(object, ...) ## S3 method for class'gel' summary(object, ...) 48summary ## S3 method for class'ategel' summary(object, robToMiss = TRUE, ...) ## S3 method for class'tsls' summary(object, vcov = NULL, ...) ## S3 method for class'summary.gmm' print(x, digits = 5, ...) ## S3 method for class'summary.sysGmm' print(x, digits = 5, ...) ## S3 method for class'summary.gel' print(x, digits = 5, ...) ## S3 method for class'summary.tsls' print(x, digits = 5, ...) Arguments objectAn object of classgmmorgelreturned by the functiongmmorgel xAn object of classsummary.gmmorsummary.gelreturned by the functionsummary.gmm summary.gel digitsThe number of digits to be printed vcovAn alternative covariance matrix computed withvcov.tsls robToMissIfTRUE, it computes the robust to misspecification covariance matrix ...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 thatE(g(θ,X) = 0 References Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. Econometrica,50, 1029-1054, Hansen, L.P. and Heaton, J. and Yaron, A.(1996), Finit-Sample Properties of Some Alternative GMM Estimators.Journal of Business and Economic Statistics,14262-280. Anatolyev, S. (2005), GMM, GEL, Serial Correlation, and Asymptotic Bias.Econometrica,73, 983-1002. Kitamura, Yuichi (1997), Empirical Likelihood Methods With Weakly Dependent Processes.The Annals of Statistics,25, 2084-2102. Newey, W.K. and Smith, R.J. (2004), Higher Order Properties of GMM and Generalized Empirical Likelihood Estimators.Econometrica,72, 219-255. Examples # GMM # set.seed(444) n = 500 sysGmm49 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) sysGmmGeneralized 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) 50sysGmm Arguments gA possibly named list of formulas hA formula if the same instruments are used in each equation or a list of formulas. wmatrixWhich weighting matrix should be used in the objective function. By default, it is the inverse of the covariance matrix ofg(θ,x). The other choice is the identity matrix. vcovAssumption on the properties of the moment vector. By default, it is a martin- gale difference sequence. "HAC" is for weakly dependent processes and "Cond- Hom" implies conditional homoscedasticity. The option "TrueFixed" is used only when the matrix of weights is provided and it is the optimal one. kerneltype of kernel used to compute the covariance matrix of the vector of sample moment conditions (seekernHACfor more details) critThe stopping rule for the iterative GMM. It can be reduce to increase the preci- sion. bwThe method to compute the bandwidth parameter. By default it isbwAndrews which is proposed by Andrews (1991). The alternative isbwNeweyWestof Newey-West(1994). prewhitelogical or integer. Should the estimating functions be prewhitened? IfTRUEor greater than 0 a VAR model of orderas.integer(prewhite)is fitted viaar with method"ols"anddemean = FALSE. ar.methodcharacter. Themethodargument passed toarfor prewhitening. approxA character specifying the approximation method if the bandwidth has to be chosen bybwAndrews. tolWeights that exceedtolare used for computing the covariance matrix, all other weights are treated as 0. model, X, Ylogical. IfTRUEthe corresponding components of the fit (the model frame, the model matrix, the response) are returned if g is a formula. centeredVcovShould the moment function be centered when computing its covariance matrix. Doing so may improve inference. weightsMatrixIt allows users to providegmmwith a fixed weighting matrix. This matrix must beq×q, symmetric and strictly positive definite. When provided, thetype option becomes irrelevant. dataA data.frame or a matrix with column names (Optional). commonCoefIf true, coefficients accross equations are the same crossEquConstOnly 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 formulasg. Setting it to1:kis equivalent to setting commonCoef to TRUE. Details This set of functions implement the estimation of system of equations as presented in Hayashi (2000) sysGmm51 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: coefficientslist of vectors of coefficients for each equation residualslist of the residuals for each equation. fitted.valueslist of the fitted values for each equation. vcovthe covariance matrix of the stacked coefficients objectivethe value of the objective function‖var( ̄g) −1/2 ̄g‖ 2 termsThe list oftermsobjects for each equation callthe matched call. yIf requested, a list of response variables. xif requested, a list of the model matrices. modelif requested (the default), a list of the model frames. References Zeileis A (2006), Object-oriented Computation of Sandwich Estimators.Journal of Statistical Soft- ware,16(9), 1–16. URL doi:10.18637/jss.v016.i09. Andrews DWK (1991), Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Esti- mation.Econometrica,59, 817–858. Newey WK & West KD (1987), A Simple, Positive Semi-Definite, Heteroskedasticity and Auto- correlation Consistent Covariance Matrix.Econometrica,55, 703–708. Newey WK & West KD (1994), Automatic Lag Selection in Covariance Matrix Estimation.Review of Economic Studies,61, 631-653. Hayashi, F. (2000), Econometrics.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) 52tsls 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) tslsTwo stage least squares estimation Description Function to estimate a linear model by the two stage least squares method. Usage tsls(g,x,data) Arguments gA formula describing the linear regression model (see details below). xThe matrix of instruments (see details below). dataA data.frame or a matrix with column names (Optionnal). Details The function just callsgmmwith the option vcov="iid". It just simplifies the the implementation of 2SLS. The users don’t have to worry about all the options offered ingmm. The model is Y i =X i β+u i In the first step,lmis used to regressX i on the set of instrumentsZ i . The second step also useslm to regressY 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: vcov53 coefficientsk×1vector of coefficients residualsthe residuals, that is response minus fitted values if "g" is a formula. fitted.valuesthe fitted mean values if "g" is a formula. vcovthe covariance matrix of the coefficients objectivethe value of the objective function‖var( ̄g) −1/2 ̄g‖ 2 termsthetermsobject used when g is a formula. callthe matched call. yif requested, the response used (if "g" is a formula). xif requested, the model matrix used if "g" is a formula or the data if "g" is a function. modelif requested (the default), the model frame used if "g" is a formula. algoInfoInformation produced by eitheroptimornlminbrelated to the convergence if "g" is a function. It is printed by thesummary.gmmmethod. References Hansen, L.P. (1982), Large Sample Properties of Generalized Method of Moments Estimators. Econometrica,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 vcovVariance-covariance matrix of GMM or GEL Description It extracts the matrix of variances and covariances fromgmmorgelobjects. 54vcov Usage ## S3 method for class'gmm' vcov(object, ...) ## S3 method for class'gel' vcov(object, lambda = FALSE, ...) ## S3 method for class'tsls' vcov(object, type=c("Classical","HC0","HC1","HAC"), hacProp = list(), ...) ## S3 method for class'ategel' vcov(object, lambda = FALSE, robToMiss = TRUE, ...) Arguments objectAn object of classgmmorgmmreturned by the functiongmmorgel lambdaIf set to TRUE, the covariance matrix of the Lagrange multipliers is produced. typeType of covariance matrix for the meat hacPropA list of arguments to pass tokernHAC robToMissIfTRUE, it computes the robust to misspecification covariance matrix ...Other arguments whenvcovis 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) wage55 ## GEL ## t0 <- c(0,.5,.5) res <- gel(g, x, t0) vcov(res) vcov(res, lambda = TRUE) wageLabor Data Description Data used to measure return to education by Griliches (1976) Usage data(wage) Format A data frame containing 20 cross-sectional vectors. AGE, AGE80Age in 1969 and 1980 respetively EXPR, EXPR80Working experience in 1969 and 1980 respetively IQIQ measure of the individual KWWA test score LW, LW80Log wage in 1969 and 1980 respectively MEDMother education MRT, MRT80 RNS, RNS80 S, S80Schooling in 1969 and 1980 respetively SMSA, SMSA80 TENURE, TENURE80Tenure in 1969 and 1980 respetively YEAR Source http://fhayashi.fc2web.com/datasets.htm Index ∗datasets Finance, 14 Growth, 35 nsw, 39 wage, 55 ar,7,19,29,45,50 ATEgel, 2,37 bread, 5 bwAndrews,7,19,50 bwNeweyWest,19,50 bwWilhelm, 6 charStable, 8 checkConv(ATEgel), 2 coef, 9 confint, 10 constrOptim,4,19–21,26,29 estfun, 12 evalGel,20 evalGel(gel), 18 evalGmm(gmm), 28 Finance, 14 FinRes, 15 fitted, 15 five(sysGmm), 49 formula, 17 gel,4,9, 10,15–17, 18,20,23,26,41–43,45, 48,54 getDat, 23 getImpProb,4, 24 getLamb,3, 4,19–21, 25 getModel, 27 gmm,9, 10,12,15–17,20,23, 28,30,36,39, 41–43,48,52,54 gmmWithConst(gmm), 28 Growth, 35 kernHAC,7,12, 13,19,29,50,54 KTest, 36 lm,20,30,52 marginal, 37 model.matrix.tsls(estfun), 12 momentEstim, 38 nlminb,4,19, 20,26,29,31,53 nsw, 39 optim,4,19–21,26,28–31,53 optimize,19–21,29 panel.smooth,41 par,41 plot, 41 points,41 print, 42 print.confint(confint), 10 print.gmmTests(KTest), 36 print.specTest(specTest), 46 print.summary.gel(summary), 47 print.summary.gmm(summary), 47 print.summary.sysGmm(summary), 47 print.summary.tsls(summary), 47 randEffect(sysGmm), 49 residuals, 43 sandwich,13 smoothG, 44 specTest, 46 summary, 47 summary.ategel(summary), 47 summary.gel,48 summary.gel(summary), 47 summary.gmm,48 summary.gmm(summary), 47 summary.sysGmm(summary), 47 56 INDEX57 summary.tsls(summary), 47 sur(sysGmm), 49 sysGmm,23, 49 terms,21,31,51,53 threeSLS(sysGmm), 49 tsls,12, 52 vcov, 53 vcov.tsls,13 vcovHAC,13 wage, 55 weightsAndrews,45
