sandwich/0000755000175400001440000000000012600751712012246 5ustar zeileisuserssandwich/inst/0000755000175400001440000000000012600735100013214 5ustar zeileisuserssandwich/inst/CITATION0000644000175400001440000000264512220001272014352 0ustar zeileisuserscitHeader("To cite sandwich in publications use:") citEntry(entry="Article", title = "Econometric Computing with HC and HAC Covariance Matrix Estimators", author = personList(as.person("Achim Zeileis")), journal = "Journal of Statistical Software", year = "2004", volume = "11", number = "10", pages = "1--17", url = "http://www.jstatsoft.org/v11/i10/", textVersion = paste("Achim Zeileis (2004).", "Econometric Computing with HC and HAC Covariance Matrix Estimators.", "Journal of Statistical Software 11(10), 1-17.", "URL http://www.jstatsoft.org/v11/i10/.") ) citEntry(entry = "Article", title = "Object-Oriented Computation of Sandwich Estimators", author = "Achim Zeileis", journal = "Journal of Statistical Software", year = "2006", volume = "16", number = "9", pages = "1--16", url = "http://www.jstatsoft.org/v16/i09/.", textVersion = paste("Achim Zeileis (2006).", "Object-Oriented Computation of Sandwich Estimators.", "Journal of Statistical Software 16(9), 1-16.", "URL http://www.jstatsoft.org/v16/i09/."), header = "If sandwich is applied to models other than lm() also cite:" ) sandwich/inst/doc/0000755000175400001440000000000012600735100013761 5ustar zeileisuserssandwich/inst/doc/sandwich-OOP.pdf0000644000175400001440000047511412600735102016725 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4617 /Filter /FlateDecode /N 87 /First 727 >> stream x\YsF~oT657Uز[r+ IH(R!)ۙ; (Eak>}s*+t!*L-pաӺ_!vxMk<8_SHa!MW(/< -V!Rr 0xIBx8p t.C 8uDpB <CTrE׾Ny(B5'@!7=DLDaI@S RHAd)ĈqAt'o Qځ-⠀dt t, +2D*@V), @Kry-qQq4ddmA"c A ร%0drd#",P,< [lYƞCT-8$FLStS@8q6?Yj5Vq^^ZB.txeUh??A_TU==`7Zs7v'/dGe|_-擫qz^<>/W\ ^׫) ?^;kWi{VOAwi >O# 2J4~4:JPU!CmȖ(;W/e"|t1a+9G@?ItTMѽ'&(_ ^hz\-W1p#~Z=>Y7$OaAABb1T+fѭjL-r&7K(61Ζ O~[E4׳?"!{TuiD.q%ҹĵ>'{vʹI.ͭh~S4_\𘯎@kYZ\UMM+e{d'{r6OK0:;xOa664A|u.Rwx2r3U#ОRȭ7}w).sZ4g.6Ԑ^Z~۠VF:ů*BׄH#s"AlX yTDy9&r哪>;o.]v=`Sv^W옝6b1ϧӚp eg윝uy^X~g)`36g%|lɖ'bEU9bGNTGlQ%9>2_GBr`F)^.>EWc1zFg'-Sů%==x1XK \W B_ ZD7P ȡ+ǐX:`xsFpt/݋׀EGj9+35ÇuG\p1>=ST$ n@ߡhS/]`Dpeghsܜы)iaҵLh")sPSAiEHB|Cٞ™m3U{̞@~ds (ˋl5|*ao;sTj9M-F?*Ď|A֋x<A'uqcdQgH6zO,FoؠHm~GuQQ,rJ0*//j1߲.ĀE OuVE 5.O~*Yu!ݝ-͍A34kh(dh ﶁQMvzZY@-K-K>-IƓ%7I[#cmJX+Rm HJTCEJkDϣ'"O? /Nȗ*^6fR9YX疅-F? ˬ1w9DZ%lc˱ r,6{=xOE,R=QokAeMŖz!g7θq܍#D7  W£i%EzJcOcv(uDdCumtO/CgnGMaɾ39SXy▊e׶VnM%jNej")e$.&bbネoovz%-朰hFtE9GHH!4X)*l,hH~ocbmN3z;v$kmei;+3Mq|IRERg5B˾F;ygQF?]EI;C+}x~;xƬo!ow?4mC֩_enu[ SLsͬb]經^_W4.3+2/as Zl?^xM Dp{3kHe[V?N of5Vm6;O V/PXؿ7hZ:\haܳ"SO-81ld8l҅!lÑsŀew`;q8VwtR#eMU2uLw1{$$Ǐ|Jekݪw9i31]9%ޒtb1Ћzj/e$բhRGӶ{ѳ㪾)v|[*\?W4w:4VcO.rv#t5]$)p _2ZjuaY>Eѡ U{ {7D9 fBwд&if݆=_#}eKntiOGU=Vl..E3""+n`\=}̒r5%-("Nd' '|\ĉJRLnRX{vΕTVh4%x%GJqU" %] u[+d?SGL"413w%7_N֚֕w֝k [+e)] D)h)Liu%TH5MQ ?pJV'!8՛8A]8J҃GMd):RQ!ySBmޑs۫z]JeiEI,9ma`% N⤔8m9t=u(M܆q֗~:5` %:كoT n|tSV?8c廦BtΎ[A)̧"Q.jn:TE{RavZ5E^W5:7qX>7l8K|iW:UBeSuLFs@ѵd P%ؗ꯬+F3^X5P&J.);U*mkt))B}~ԃ6 'hwGIL5yd]|EaR%0^ 0X88߈ƕ"ξ)fG ׃*ښp Rrpq1HݍO'{st nJ;_.//{9_Yw߉sQPihI4M0EǏe5ʫj\V+ZjL;˦@j5/'C iKCCM5A644 Rv+y?f&1~1){hk6K8i~S:7y puK?{4Zy⟩h>eИS"Jҵ.xP Ff#եB1 >x\Vi|/1Q2NRI'nCdhӽ;pp ejBYdv B5j-Ϫ˽cp֙xY*DQ _w,3) y'L2/%ep78if0H-EgZX"lnϓP` Eq⇶\o0!Mև Z|Q&ЪVYi m.j B` f ǁ m+n?9<&* GSҭ}q{[{>Z" uMWiiPdq;?\endstream endobj 89 0 obj << /Subtype /XML /Type /Metadata /Length 1338 >> stream 2015-09-24T10:54:24+02:00 2015-09-24T10:54:24+02:00 David M. 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A follow-up paper on object object-oriented computation of sandwich estimators is available in \citep{hac:Zeileis:2006}. Data described by econometric models typically contains autocorrelation and/or heteroskedasticity of unknown form and for inference in such models it is essential to use covariance matrix estimators that can consistently estimate the covariance of the model parameters. Hence, suitable heteroskedasticity-consistent (HC) and heteroskedasticity and autocorrelation consistent (HAC) estimators have been receiving attention in the econometric literature over the last 20 years. To apply these estimators in practice, an implementation is needed that preferably translates the conceptual properties of the underlying theoretical frameworks into computational tools. In this paper, such an implementation in the package \pkg{sandwich} in the \proglang{R} system for statistical computing is described and it is shown how the suggested functions provide reusable components that build on readily existing functionality and how they can be integrated easily into new inferential procedures or applications. The toolbox contained in \pkg{sandwich} is extremely flexible and comprehensive, including specific functions for the most important HC and HAC estimators from the econometric literature. Several real-world data sets are used to illustrate how the functionality can be integrated into applications. } \Address{ Achim Zeileis\\ Department of Statistics\\ Faculty of Economics and Statistics\\ Universit\"at Innsbruck\\ Universit\"atsstr.~15\\ 6020 Innsbruck, Austria\\ E-mail: \email{Achim.Zeileis@R-project.org}\\ URL: \url{http://eeecon.uibk.ac.at/~zeileis/} } \begin{document} \SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Econometric Computing with HC and HAC Covariance Matrix Estimators} %\VignetteDepends{sandwich,zoo,lmtest,strucchange,scatterplot3d} %\VignetteKeywords{covariance matrix estimator, heteroskedasticity, autocorrelation, estimating functions, econometric computing, R} %\VignettePackage{sandwich} <>= library("zoo") library("sandwich") library("strucchange") library("lmtest") options(prompt = "R> ", continue = "+ ") @ \section{Introduction} \label{sec:intro} This paper combines two topics that play an important role in applied econometrics: computational tools and robust covariance estimation. Without the aid of statistical and econometric software modern data analysis would not be possible: hence, both practitioners and (applied) researchers rely on computational tools that should preferably implement state-of-the-art methodology and be numerically reliable, easy to use, flexible and extensible. In many situations, economic data arises from time-series or cross-sectional studies which typically exhibit some form of autocorrelation and/or heteroskedasticity. If the covariance structure were known, it could be taken into account in a (parametric) model, but more often than not the form of autocorrelation and heteroskedasticity is unknown. In such cases, model parameters can typically still be estimated consistently using the usual estimating functions, but for valid inference in such models a consistent covariance matrix estimate is essential. Over the last 20 years several procedures for heteroskedasticity consistent (HC) and for heteroskedasticity and autocorrelation consistent (HAC) covariance estimation have been suggested in the econometrics literature \citep[among others]{hac:White:1980,hac:MacKinnon+White:1985,hac:Newey+West:1987,hac:Newey+West:1994,hac:Andrews:1991} and are now routinely used in econometric analyses. Many statistical and econometric software packages implement various HC and HAC estimators for certain inference procedures, so why is there a need for a paper about econometric computing with HC and HAC estimators? Typically, only certain special cases of such estimators---and not the general framework they are taken from---are implemented in statistical and econometric software packages and sometimes they are only available as options to certain inference functions. It is desirable to improve on this for two reasons: First, the literature suggested conceptual frameworks for HC and HAC estimation and it would only be natural to translate these conceptual properties into computational tools that reflect the flexibility of the general framework. Second, it is important, particularly for applied research, to have covariance matrices not only as options to certain tests but as stand-alone functions which can be used as modular building blocks and plugged into various inference procedures. This is becoming more and more relevant, because today, as \cite{hac:Cribari-Neto+Zarkos:2003} point out, applied researchers typically cannot wait until a certain procedure becomes available in the software package of their choice but are often forced to program new techniques themselves. Thus, just as suitable covariance estimators are routinely plugged into formulas in theoretical work, programmers should be enabled to plug in implementations of such estimators in computational work. Hence, the aim of this paper is to present an econometric computing approach to HC and HAC estimation that provides reusable components that can be used as modular building blocks in implementing new inferential techniques and in applications. All functions described are available in the package \pkg{sandwich} implemented in the \proglang{R} system for statistical computing \citep{hac:R:2008} which is currently not the most popular environment for econometric computing but which is finding increasing attention among econometricians \citep{hac:Cribari-Neto+Zarkos:1999,hac:Racine+Hyndman:2002}. Both \proglang{R} itself and the \pkg{sandwich} package (as well as all other packages used in this paper) are available at no cost under the terms of the general public licence (GPL) from the comprehensive \proglang{R} archive network (CRAN, \url{http://CRAN.R-project.org/}). \proglang{R} has no built-in support for HC and HAC estimation and at the time we started writing \pkg{sandwich} there was only one package that implements HC (but not HAC) estimators \citep[the \pkg{car} package][]{hac:Fox:2002} but which does not allow for as much flexibility as the tools presented here. \pkg{sandwich} provides the functions \code{vcovHC} and \code{vcovHAC} implementing general classes of HC and HAC estimators. The names of the functions are chosen to correspond to \code{vcov}, \proglang{R}'s generic function for extracting covariance matrices from fitted model objects. Below, we focus on the general linear regression model estimated by ordinary least squares (OLS), which is typically fitted in \proglang{R} using the function \code{lm} from which the standard covariance matrix (assuming spherical errors) can be extracted by \code{vcov}. Using the tools from \pkg{sandwich}, HC and HAC covariances matrices can now be extracted from the same fitted models using \code{vcovHC} and \code{vcovHAC}. Due to the object orientation of \proglang{R}, these functions are not only limited to the linear regression model but can be easily extended to other models. The HAC estimators are already available for generalized linear models (fitted by \code{glm}) and robust regression (fitted by \code{rlm} in package \pkg{MASS}). Another important feature of \proglang{R} that is used repeatedly below is that functions are first-level objects---i.e., functions can take functions as arguments and return functions---which is particularly useful for defining certain procedures for data-driven computations such as the definition of the structure of covariance matrices in HC estimation and weighting schemes for HAC estimation. The remainder of this paper is structured as follows: To fix notations, Section~\ref{sec:model} describes the linear regression model used and motivates the following sections. Section~\ref{sec:estimating} gives brief literature reviews and describes the conceptual frameworks for HC and HAC estimation respectively and then shows how the conceptual properties are turned into computational tools in \pkg{sandwich}. Section~\ref{sec:applications} provides some illustrations and applications of these tools before a summary is given in Section~\ref{sec:summary}. More details about the \proglang{R} code used are provided in an appendix. \section{The linear regression model} \label{sec:model} To fix notations, we consider the linear regression model \begin{equation} \label{eq:lm} y_i \quad = \quad x_i^\top \beta \; + \; u_i \qquad (i = 1, \dots, n), \end{equation} with dependent variable $y_i$, $k$-dimensional regressor $x_i$ with coefficient vector $\beta$ and error term $u_i$. In the usual matrix notation comprising all $n$ observations this can be formulated as $y = X \beta + u$. In the general linear model, it is typically assumed that the errors have zero mean and variance $\VAR[u] = \Omega$. Under suitable regularity conditions \citep[see e.g.,][]{hac:Greene:1993,hac:White:2000}, the coefficients $\beta$ can be consistently estimated by OLS giving the well-known OLS estimator $\hat \beta$ with corresponding OLS residuals $\hat u_i$: \begin{eqnarray} \hat \beta & = & \left( X^\top X \right)^{-1} X^\top y \\ \hat u & = & (I_n - H) \, y \; = \; (I_n - X \left( X^\top X \right)^{-1} X^\top) \, y \end{eqnarray} where $I_n$ is the $n$-dimensional identity matrix and $H$ is usually called hat matrix. The estimates $\hat \beta$ are unbiased and asymptotically normal \citep{hac:White:2000}. Their covariance matrix $\Psi$ is usually denoted in one of the two following ways: \begin{eqnarray} \Psi \; = \; \VAR[\hat \beta] & = & \left( X^\top X \right)^{-1} X^\top \Omega X \left( X^\top X \right)^{-1} \label{eq:PsiHC} \\ & = & \left( \frac{1}{n} X^\top X \right)^{-1} \frac{1}{n} \Phi \left( \frac{1}{n} X^\top X \right)^{-1} \label{eq:PsiHAC} \end{eqnarray} where $\Phi = n^{-1} X^\top \Omega X$ is essentially the covariance matrix of the scores or estimating functions $V_i(\beta) = x_i (y_i - x_i^\top \beta)$. The estimating functions evaluated at the parameter estimates $\hat V_i = V_i(\hat \beta)$ have then sum zero. For inference in the linear regression model, it is essential to have a consistent estimator for $\Psi$. What kind of estimator should be used for $\Psi$ depends on the assumptions about $\Omega$: In the classical linear model independent and homoskedastic errors with variance $\sigma^2$ are assumed yielding $\Omega = \sigma^2 I_n$ and $\Psi = \sigma^2 (X^\top X)^{-1}$ which can be consistently estimated by plugging in the usual OLS estimator ${\hat \sigma}^2 = (n-k)^{-1} \sum_{i = 1}^n {\hat u_i}^2$. But if the independence and/or homoskedasticity assumption is violated, inference based on this estimator $\hat \Psi_{\mathrm{const}} = \hat \sigma (X^\top X)^{-1}$ will be biased. HC and HAC estimators tackle this problem by plugging an estimate $\hat \Omega$ or $\hat \Phi$ into (\ref{eq:PsiHC}) or (\ref{eq:PsiHAC}) respectively which are consistent in the presence of heteroskedasticity and autocorrelation respectively. Such estimators and their implementation are described in the following section. \section[Estimating the covariance matrix]{Estimating the covariance matrix $\Psi$} \label{sec:estimating} \subsection{Dealing with heteroskedasticity} If it is assumed that the errors $u_i$ are independent but potentially heteroskedastic---a situation which typically arises with cross-sectional data---their covariance matrix $\Omega$ is diagonal but has nonconstant diagonal elements. Therefore, various HC estimators $\hat \Psi_{\mathrm{HC}}$ have been suggested which are constructed by plugging an estimate of type $\hat \Omega = \mathrm{diag}(\omega_1, \dots, \omega_n)$ into Equation~(\ref{eq:PsiHC}). These estimators differ in their choice of the $\omega_i$, an overview of the most important cases is given in the following: \begin{eqnarray*} \mathrm{const:} \quad \omega_i & = & \hat \sigma^2 \\ \mathrm{HC0:} \quad \omega_i & = & {\hat u_i}^2 \\ \mathrm{HC1:} \quad \omega_i & = & \frac{n}{n-k} \, {\hat u_i}^2 \\ \mathrm{HC2:} \quad \omega_i & = & \frac{{\hat u_i}^2}{1 - h_i} \\ \mathrm{HC3:} \quad \omega_i & = & \frac{{\hat u_i}^2}{(1 - h_i)^2} \\ \mathrm{HC4:} \quad \omega_i & = & \frac{{\hat u_i}^2}{(1 - h_i)^{\delta_i}} \end{eqnarray*} where $h_i = H_{ii}$ are the diagonal elements of the hat matrix, $\bar h$ is their mean and $\delta_i = \min\{4, h_i/\bar h\}$. The first equation above yields the standard estimator $\hat \Psi_{\mathrm{const}}$ for homoskedastic errors. All others produce different kinds of HC estimators. The estimator HC0 was suggested in the econometrics literature by \cite{hac:White:1980} and is justified by asymptotic arguments. %% check White, maybe explain ideas The estimators HC1, HC2 and HC3 were suggested by \cite{hac:MacKinnon+White:1985} to improve the performance in small samples. A more extensive study of small sample behaviour was carried out by \cite{hac:Long+Ervin:2000} which arrive at the conclusion that HC3 provides the best performance in small samples as it gives less weight to influential observations. Recently, \cite{hac:Cribari-Neto:2004} suggested the estimator HC4 to further improve small sample performance, especially in the presence of influential observations. All of these HC estimators $\hat \Psi_{\mathrm{HC}}$ have in common that they are determined by $\omega = (\omega_1, \dots, \omega_n)^\top$ which in turn can be computed based on the residuals $\hat u$, the diagonal of the hat matrix $h$ and the degrees of freedom $n-k$. To translate these conceptual properties of this class of HC estimators into a computational tool, a function is required which takes a fitted regression model and the diagonal elements $\omega$ as inputs and returns the corresponding $\hat \Psi_{\mathrm{HC}}$. In \pkg{sandwich}, this is implemented in the function \code{vcovHC} which takes the following arguments: \begin{verbatim} vcovHC(lmobj, omega = NULL, type = "HC3", ...) \end{verbatim} The first argument \code{lmobj} is an object as returned by \code{lm}, \proglang{R}'s standard function for fitting linear regression models. The argument \code{omega} can either be the vector $\omega$ or a function for data-driven computation of $\omega$ based on the residuals $\hat u$, the diagonal of the hat matrix $h$ and the residual degrees of freedom $n-k$. Thus, it has to be of the form \code{omega(residuals, diaghat, df)}: e.g., for computing HC3 \code{omega} is set to \verb+function(residuals, diaghat, df)+ \linebreak \verb+residuals^2/(1 - diaghat)^2+. As a convenience option, a \code{type} argument can be set to \code{"const"}, \code{"HC0"} (or equivalently \code{"HC"}), \code{"HC1"}, \code{"HC2"}, \code{"HC3"} (the default) or \code{"HC4"} and then \code{vcovHC} uses the corresponding \code{omega} function. As soon as \code{omega} is specified by the user, \code{type} is ignored. In summary, by specfying $\omega$---either as a vector or as a function---\code{vcovHC} can compute arbitrary HC covariance matrix estimates from the class of estimators outlined above. In Section~\ref{sec:applications}, it will be illustrated how this function can be used as a building block when doing inference in linear regression models. \subsection{Dealing with autocorrelation} If the error terms $u_i$ are not independent, $\Omega$ is not diagonal and without further specification of a parametic model for the type of dependence it is typically burdensome to estimate $\Omega$ directly. However, if the form of heteroskedasticity and autocorrelation is unknown, a solution to this problem is to estimate $\Phi$ instead which is essentially the covariance matrix of the estimating functions\footnote{Due to the use of estimating functions, this approach is not only feasible in linear models estimated by OLS, but also in nonlinear models using other estimating functions such as maximum likelihood (ML), generalized methods of moments (GMM) or Quasi-ML.}. This is what HAC estimators do: $\hat \Psi_{\mathrm{HAC}}$ is computed by plugging an estimate $\hat \Phi$ into Equation~(\ref{eq:PsiHAC}) with \begin{equation} \label{eq:HAC} \hat \Phi \quad = \quad \frac{1}{n} \sum_{i, j = 1}^n w_{|i-j|} \, {\hat V}_i {{\hat V}_j}^\top \end{equation} where $w = (w_0, \dots, w_{n-1})^\top$ is a vector of weights. An additional finite sample adjustment can be applied by multiplication with $n/(n-k)$. For many data structures, it is a reasonable assumption that the autocorrelations should decrease with increasing lag $\ell = |i-j|$---otherwise $\beta$ can typically not be estimated consistently by OLS---so that it is rather intuitive that the weights $w_\ell$ should also decrease. Starting from \cite{hac:White+Domowitz:1984} and \cite{hac:Newey+West:1987}, different choices for the vector of weights $w$ have been suggested in the econometrics literature which have been placed by \cite{hac:Andrews:1991} in a more general framework of choosing the weights by kernel functions with automatic bandwidth selection. \cite{hac:Andrews+Monahan:1992} show that the bias of the estimators can be reduced by prewhitening the estimating functions $\hat V_i$ using a vector autoregression (VAR) of order $p$ and applying the estimator in Equation~(\ref{eq:HAC}) to the VAR($p$) residuals subsequently. \cite{hac:Lumley+Heagerty:1999} suggest an adaptive weighting scheme where the weights are chosen based on the estimated autocorrelations of the residuals $\hat u$. All the estimators mentioned above are of the form (\ref{eq:HAC}), i.e., a weighted sum of lagged products of the estimating functions corresponding to a fitted regression model. Therefore, a natural implementation for this class of HAC estimators is the following: \begin{verbatim} vcovHAC(lmobj, weights, prewhite = FALSE, adjust = TRUE, sandwich = TRUE, order.by, ar.method, data) \end{verbatim} The most important arguments are again the fitted linear model\footnote{Note, that not only HAC estimators for fitted \emph{linear} models can be computed with \code{vcovHAC}. See \cite{hac:Zeileis:2006} for details.} \code{lmobj}---from which the estimating functions $\hat V_i$ can easily be extracted using the generic function \code{estfun(lmobj)}---and the argument \code{weights} which specifys $w$. The latter can be either the vector $w$ directly or a function to compute it from \code{lmobj}.\footnote{If \code{weights} is a vector with less than $n$ elements, the remaining weights are assumed to be zero.} The argument \code{prewhite} specifies wether prewhitening should be used or not\footnote{The order $p$ is set to \code{as.integer(prewhite)}, hence both \code{prewhite = 1} and \code{prewhite = TRUE} lead to a VAR(1) model, but also \code{prewhite = 2} is possible.} and \code{adjust} determines wether a finite sample correction by multiplication with $n/(n-k)$ should be made or not. By setting \code{sandwich} it can be controlled wether the full sandwich estimator $\hat \Psi_{\mathrm{HAC}}$ or only the ``meat'' $\hat \Phi/n$ of the sandwich should be returned. The remaining arguments are a bit more technical: \code{order.by} specifies by which variable the data should be ordered (the default is that they are already ordered, as is natural with time series data), which \code{ar.method} should be used for fitting the VAR($p$) model (the default is OLS) and \code{data} provides a data frame from which \code{order.by} can be taken (the default is the environment from which \code{vcovHAC} is called).\footnote{More detailed technical documentation of these and other arguments of the functions described are available in the reference manual included in \pkg{sandwich}.} As already pointed out above, all that is required for specifying an estimator $\hat \Psi_{\mathrm{HAC}}$ is the appropriate vector of weights (or a function for data-driven computation of the weights). For the most important estimators from the literature mentioned above there are functions for computing the corresponding weights readily available in \pkg{sandwich}. They are all of the form \code{weights(lmobj, order.by, prewhite, ar.method, data)}, i.e., functions that compute the weights depending on the fitted model object \code{lmobj} and the arguments \code{order.by}, \code{prewhite}, \code{data} which are only needed for ordering and prewhitening. The function \code{weightsAndrews} implements the class of weights of \cite{hac:Andrews:1991} and \code{weightsLumley} implements the class of weights of \cite{hac:Lumley+Heagerty:1999}. Both functions have convenience interfaces: \code{kernHAC} calls \code{vcovHAC} with \code{weightsAndrews} (and different defaults for some parameters) and \code{weave} calls \code{vcovHAC} with \code{weightsLumley}. Finally, a third convenience interface to \code{vcovHAC} is available for computing the estimator(s) of \cite{hac:Newey+West:1987,hac:Newey+West:1994}. \begin{itemize} \item \cite{hac:Newey+West:1987} suggested to use linearly decaying weights \begin{equation} \label{eq:NeweyWest} w_\ell \quad = \quad 1 - \frac{\ell}{L + 1} \end{equation} where $L$ is the maximum lag, all other weights are zero. This is implemented in the function \code{NeweyWest(lmobj, lag = NULL, \dots)} where \code{lag} specifies $L$ and \code{\dots} are (here, and in the following) further arguments passed to other functions, detailed information is always available in the reference manual. If \code{lag} is set to \code{NULL} (the default) the non-parametric bandwidth selection procedure of \cite{hac:Newey+West:1994} is used. This is also available in a stand-alone function \code{bwNeweyWest}, see also below. \setkeys{Gin}{width=.7\textwidth} \begin{figure}[tbh] \begin{center} <>= curve(kweights(x, kernel = "Quadratic", normalize = TRUE), from = 0, to = 3.2, xlab = "x", ylab = "K(x)") curve(kweights(x, kernel = "Bartlett", normalize = TRUE), from = 0, to = 3.2, col = 2, add = TRUE) curve(kweights(x, kernel = "Parzen", normalize = TRUE), from = 0, to = 3.2, col = 3, add = TRUE) curve(kweights(x, kernel = "Tukey", normalize = TRUE), from = 0, to = 3.2, col = 4, add = TRUE) lines(c(0, 0.5), c(1, 1), col = 6) lines(c(0.5, 0.5), c(1, 0), lty = 3, col = 6) lines(c(0.5, 3.2), c(0, 0), col = 6) curve(kweights(x, kernel = "Quadratic", normalize = TRUE), from = 0, to = 3.2, col = 1, add = TRUE) text(0.5, 0.98, "Truncated", pos = 4) text(0.8, kweights(0.8, "Bartlett", normalize = TRUE), "Bartlett", pos = 4) text(1.35, kweights(1.4, "Quadratic", normalize = TRUE), "Quadratic Spectral", pos = 2) text(1.15, 0.29, "Parzen", pos = 4) arrows(1.17, 0.29, 1, kweights(1, "Parzen", normalize = TRUE), length = 0.1) text(1.3, 0.2, "Tukey-Hanning", pos = 4) arrows(1.32, 0.2, 1.1, kweights(1.1, "Tukey", normalize = TRUE), length = 0.1) @ \caption{\label{fig:kweights} Kernel functions for kernel-based HAC estimation.} \end{center} \end{figure} \item \cite{hac:Andrews:1991} placed this and other estimators in a more general class of kernel-based HAC estimators with weights of the form $w_\ell = K(\ell/B)$ where $K(\cdot)$ is a kernel function and $B$ the bandwidth parameter used. The kernel functions considered are the truncated, Bartlett, Parzen, Tukey-Hanning and quadratic spectral kernel which are depicted in Figure~\ref{fig:kweights}. The Bartlett kernel leads to the weights used by \cite{hac:Newey+West:1987} in Equation~(\ref{eq:NeweyWest}) when the bandwidth $B$ is set to $L + 1$. The kernel recommended by \cite{hac:Andrews:1991} and probably most used in the literature is the quadratic spectral kernel which leads to the following weights: \begin{equation} w_\ell \quad = \quad \frac{3}{z^2} \, \left(\frac{\sin(z)}{z} - \cos (z) \right), \end{equation} where $z = 6 \pi/5 \cdot \ell/B$. The definitions for the remaining kernels can be found in \cite{hac:Andrews:1991}. All kernel weights mentioned above are available in \code{weightsAndrews(lmobj, kernel, bw, ...)} where \code{kernel} specifies one of the kernels via a character string (\code{"Truncated"}, \code{"Bartlett"}, \code{"Parzen"}, \code{"Tukey-Hanning"} or \code{"Quadratic Spectral"}) and \code{bw} the bandwidth either as a scalar or as a function. The automatic bandwidth selection described in \cite{hac:Andrews:1991} via AR(1) or ARMA(1,1) approximations is implemented in a function \code{bwAndrews} which is set as the default in \code{weightsAndrews}. For the Bartlett, Parzen and quadratic spectral kernels, \cite{hac:Newey+West:1994} suggested a different nonparametric bandwidth selection procedure, which is implemented in \code{bwNeweyWest} and which can also be passed to \code{weightsAndrews}. As the flexibility of this conceptual framework of estimators leads to a lot of knobs and switches in the computational tools, a convenience function \code{kernHAC} for kernel-based HAC estimation has been added to \pkg{sandwich} that calls \code{vcovHAC} based on \code{weightsAndrews} and \code{bwAndrews} with defaults as motivated by \cite{hac:Andrews:1991} and \cite{hac:Andrews+Monahan:1992}: by default, it computes a quadratic spectral kernel HAC estimator with VAR(1) prewhitening and automatic bandwidth selection based on an AR(1) approximation. But of course, all the options described above can also be changed by the user when calling \code{kernHAC}. \item \cite{hac:Lumley+Heagerty:1999} suggested a different approach for specifying the weights in (\ref{eq:HAC}) based on some estimate $\hat \varrho_\ell$ of the autocorrelation of the residuals $\hat u_i$ at lag $0 = 1, \dots, n-1$. They suggest either to use truncated weights $w_\ell = I\{n \, \hat \varrho^2_\ell > C\}$ (where $I(\cdot)$ is the indicator function) or smoothed weights $w_\ell = \min\{1, C \, n \, \hat \varrho^2_\ell\}$, where for both a suitable constant $C$ has to be specified. \cite{hac:Lumley+Heagerty:1999} suggest using a default of $C = 4$ and $C = 1$ for the truncated and smoothed weights respectively. Note, that the truncated weights are equivalent to the truncated kernel from the framework of \cite{hac:Andrews:1991} but using a different method for computing the truncation lag. To ensure that the weights $|w_\ell|$ are decreasing, the autocorrelations have to be decreasing for increasing lag $\ell$ which can be achieved by using isotonic regression methods. In \pkg{sandwich}, these two weighting schemes are implemented in a function \code{weightsLumley} with a convenience interface \code{weave} (which stands for \underline{w}eighted \underline{e}mpirical \underline{a}daptive \underline{v}ariance \underline{e}stimators) which again sets up the weights and then calls \code{vcovHAC}. Its most important arguments are \code{weave(lmobj, method, C, ...)} where \code{method} can be either \code{"truncate"} or \code{"smooth"} and \code{C} is by default 4 or 1 respectively. \end{itemize} To sum up, \code{vcovHAC} provides a simple yet flexible interface for general HAC estimation as defined in Equation~(\ref{eq:HAC}). Arbitrary weights can be supplied either as vectors or functions for data-driven computation of the weights. As the latter might easily become rather complex, in particular due to the automatic choice of bandwidth or lag truncation parameters, three strategies suggested in the literature are readily available in \pkg{sandwich}: First, the Bartlett kernel weights suggested by \cite{hac:Newey+West:1987,hac:Newey+West:1994} are used in \code{NeweyWest} which by default uses the bandwidth selection function \code{bwNeweyWest}. Second, the weighting scheme introduced by \cite{hac:Andrews:1991} for kernel-based HAC estimation with automatic bandwidth selection is implemented in \code{weightsAndrews} and \code{bwAndrews} with corresponding convenience interface \code{kernHAC}. Third, the weighted empirical adaptive variance estimation scheme suggested by \cite{hac:Lumley+Heagerty:1999} is available in \code{weightsLumley} with convenience interface \code{weave}. It is illustrated in the following section how these functions can be easily used in applications. \section{Applications and illustrations} \label{sec:applications} In econometric analyses, the practitioner is only seldom interested in the covariance matrix $\hat \Psi$ (or $\hat \Omega$ or $\hat \Phi$) \emph{per se}, but mainly wants to compute them to use them for inferential procedures. Therefore, it is important that the functions \code{vcovHC} and \code{vcovHAC} described in the previous section can be easily supplied to other procedures such that the user does not necessarily have to compute the variances in advance. A typical field of application for HC and HAC covariances are partial $t$ or $z$ tests for assessing whether a parameter $\beta_j$ is significantly different from zero. Exploiting the (asymptotic) normality of the estimates, these tests are based on the $t$ ratio $\hat \beta_j/\sqrt{\hat \Psi_{jj}}$ and either use the asymptotic normal distribution or the $t$ distribution with $n-k$ degrees of freedom for computing $p$ values \citep{hac:White:2000}. This procedure is available in the \proglang{R} package \pkg{lmtest} \citep{hac:Zeileis+Hothorn:2002} in the generic function \code{coeftest} which has a default method applicable to fitted \code{"lm"} objects. \begin{verbatim} coeftest(lmobj, vcov = NULL, df = NULL, ...) \end{verbatim} where \code{vcov} specifies the covariances either as a matrix (corresponding to the covariance matrix estimate) or as a function computing it from \code{lmobj} (corresponding to the covariance matrix estimator). By default, it uses the \code{vcov} method which computes $\hat \Psi_{\mathrm{const}}$ assuming spherical errors. The \code{df} argument determines the degrees of freedom: if \code{df} is finite and positive, a $t$ distribution with \code{df} degrees of freedom is used, otherwise a normal approximation is employed. The default is to set \code{df} to $n-k$. Inference based on HC and HAC estimators is illustrated in the following using three real-world data sets: testing coefficients in two models from \cite{hac:Greene:1993} and a structural change problem from \cite{hac:Bai+Perron:2003}. To make the results exactly reproducible for the reader, the commands for the inferential procedures is given along with their output within the text. A full list of commands, including those which produce the figures in the text, are provided (without output) in the appendix along with the versions of \proglang{R} and the packages used. Before we start with the examples, the \pkg{sandwich} and \pkg{lmtest} package have to be loaded: <>= library("sandwich") library("lmtest") @ \subsection{Testing coefficients in cross-sectional data} A quadratic regression model for per capita expenditures on public schools explained by per capita income in the United States in 1979 has been analyzed by \cite{hac:Greene:1993} and re-analyzed in \cite{hac:Cribari-Neto:2004}. The corresponding cross-sectional data for the 51 US states is given in Table 14.1 in \cite{hac:Greene:1993} and available in \pkg{sandwich} in the data frame \code{PublicSchools} which can be loaded by: <>= data("PublicSchools") ps <- na.omit(PublicSchools) ps$Income <- ps$Income * 0.0001 @ where the second line omits a missing value (\code{NA}) in Wisconsin and assigns the result to a new data frame \code{ps} and the third line transforms the income to be in USD $10,000$. The quadratic regression can now easily be fit using the function \code{lm} which fits linear regression models specified by a symbolic formula via OLS. <>= fm.ps <- lm(Expenditure ~ Income + I(Income^2), data = ps) @ The fitted \code{"lm"} object \code{fm.ps} now contains the regression of the variable \code{Expenditure} on the variable \code{Income} and its sqared value, both variables are taken from the data frame \code{ps}. The question in this data set is whether the quadratic term is really needed, i.e., whether the coefficient of \verb/I(Income^2)/ is significantly different from zero. The partial quasi-$t$~tests (or $z$~tests) for all coefficients can be computed using the function \code{coeftest}. \cite{hac:Greene:1993} assesses the significance using the HC0 estimator of \cite{hac:White:1980}. <>= coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC0")) @ The \code{vcov} argument specifies the covariance matrix as a matrix (as opposed to a function) which is returned by \code{vcovHC(fm.ps, type = "HC0")}. As \code{df} is set to infinity (\code{Inf}) a normal approximation is used for computing the $p$ values which seem to suggest that the quadratic term might be weakly significant. In his analysis, \cite{hac:Cribari-Neto:2004} uses his HC4 estimator (among others) giving the following result: <>= coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC4")) @ The quadratic term is clearly non-significant. The reason for this result is depicted in Figure~\ref{fig:hc} which shows the data along with the fitted linear and quadratic model---the latter being obviously heavily influenced by a single outlier: Alaska. Thus, the improved performance of the HC4 as compared to the HC0 estimator is due to the correction for high leverage points. \setkeys{Gin}{width=.6\textwidth} \begin{figure}[tbh] \begin{center} <>= plot(Expenditure ~ Income, data = ps, xlab = "per capita income", ylab = "per capita spending on public schools") inc <- seq(0.5, 1.2, by = 0.001) lines(inc, predict(fm.ps, data.frame(Income = inc)), col = 4, lty = 2) fm.ps2 <- lm(Expenditure ~ Income, data = ps) abline(fm.ps2, col = 4) text(ps[2,2], ps[2,1], rownames(ps)[2], pos = 2) @ \caption{\label{fig:hc} Expenditure on public schools and income with fitted models.} \end{center} \end{figure} \subsection{Testing coefficients in time-series data} \cite{hac:Greene:1993} also anayzes a time-series regression model based on robust covariance matrix estimates: his Table 15.1 provides data on the nominal gross national product (GNP), nominal gross private domestic investment, a price index and an interest rate which is used to formulate a model that explains real investment by real GNP and real interest. The corresponding transformed variables \code{RealInv}, \code{RealGNP} and \code{RealInt} are stored in the data frame \code{Investment} in \pkg{sandwich} which can be loaded by: <>= data("Investment") @ Subsequently, the fitted linear regression model is computed by: <>= fm.inv <- lm(RealInv ~ RealGNP + RealInt, data = Investment) @ and the significance of the coefficients can again be assessed by partial $z$ tests using \code{coeftest}. \cite{hac:Greene:1993} uses the estimator of \cite{hac:Newey+West:1987} without prewhitening and with lag $L = 4$ for this purpose which is here passed as a matrix (as opposed to a function) to \code{coeftest}. <>= coeftest(fm.inv, df = Inf, vcov = NeweyWest(fm.inv, lag = 4, prewhite = FALSE)) @ If alternatively the automatic bandwidth selection procedure of \cite{hac:Newey+West:1994} with prewhitening should be used, this can be passed as a function to \code{coeftest}. <>= coeftest(fm.inv, df = Inf, vcov = NeweyWest) @ For illustration purposes, we show how a new function implementing a particular HAC estimator can be easily set up using the tools provided by \pkg{sandwich}. This is particularly helpful if the same estimator is to be applied several times in the course of an analysis. Suppose, we want to use a Parzen kernel with VAR(2) prewhitening, no finite sample adjustment and automatic bandwidth selection according to \cite{hac:Newey+West:1994}. First, we set up the function \code{parzenHAC} and then pass this function to \code{coeftest}. <>= parzenHAC <- function(x, ...) kernHAC(x, kernel = "Parzen", prewhite = 2, adjust = FALSE, bw = bwNeweyWest, ...) coeftest(fm.inv, df = Inf, vcov = parzenHAC) @ The three estimators leads to slightly different standard errors, but all tests agree that real GNP has a highly significant influence while the real interest rate has not. The data along with the fitted regression are depicted in Figure~\ref{fig:hac}. \setkeys{Gin}{width=.6\textwidth} \begin{figure}[tbh] \begin{center} <>= library("scatterplot3d") s3d <- scatterplot3d(Investment[,c(5,7,6)], type = "b", angle = 65, scale.y = 1, pch = 16) s3d$plane3d(fm.inv, lty.box = "solid", col = 4) @ \caption{\label{fig:hac} Investment equation data with fitted model.} \end{center} \end{figure} \subsection[Testing and dating structural changes in the presence of heteroskedasticity and autocorrelation]{Testing and dating structural changes in the presence of\\ heteroskedasticity and autocorrelation} To illustrate that the functionality provided by the covariance estimators implemented in \pkg{sandwich} cannot only be used in simple settings, such as partial quasi-$t$~tests, but also for more complicated tasks, we employ the real interest time series analyzed by \cite{hac:Bai+Perron:2003}. This series contains changes in the mean (see Figure~\ref{fig:sc}, right panel) which \cite{hac:Bai+Perron:2003} detect using several structural change tests based on $F$ statistics and date using a dynamic programming algorithm. As the visualization suggests, this series exhibits both heteroskedasticity and autocorrelation, hence \cite{hac:Bai+Perron:2003} use a quadratic spectral kernel HAC estimator in their analysis. Here, we use the same dating procedure but assess the significance using an OLS-based CUSUM test \citep{hac:Ploberger+Kraemer:1992} based on the same HAC estimator. The data are available in the package \pkg{strucchange} as the quarterly time series \code{RealInt} containing the US ex-post real interest rate from 1961(1) to 1986(3) and they are analyzed by a simple regression on the mean. Under the assumptions in the classical linear model with spherical errors, the test statistic of the OLS-based CUSUM test is \begin{equation} \sup_{j = 1, \dots, n} \left| \frac{1}{\sqrt{n \, \hat \sigma^2}} \; \sum_{i = 1}^{j} \hat u_i \right|. \end{equation} If autocorrelation and heteroskedasticity are present in the data, a robust variance estimator should be used: if $x_i$ is equal to unity, this can simply be achieved by replacing $\hat \sigma^2$ with $\hat \Phi$ or $n \hat \Psi$ respectively. Here, we use the quadratic spectral kernel HAC estimator of \cite{hac:Andrews:1991} with VAR(1) prewhitening and automatic bandwidth selection based on an AR(1) approximation as implemented in the function \code{kernHAC}. The $p$ values for the OLS-based CUSUM test can be computed from the distribution of the supremum of a Brownian bridge \citep[see e.g.,][]{hac:Ploberger+Kraemer:1992}. This and other methods for testing, dating and monitoring structural changes are implemented in the \proglang{R} package \pkg{strucchange} \citep{hac:Zeileis+Leisch+Hornik:2002} which contains the function \code{gefp} for fitting and assessing fluctuation processes including OLS-based CUSUM processes \citep[see][for more details]{hac:Zeileis:2004}. After loading the package and the data, <>= library("strucchange") data("RealInt") @ the command <>= ocus <- gefp(RealInt ~ 1, fit = lm, vcov = kernHAC) @ fits the OLS-based CUSUM process for a regression on the mean (\verb/RealInt ~ 1/), using the function \code{lm} and estimating the variance using the function \code{kernHAC}. The fitted OLS-based CUSUM process can then be visualized together with its 5\% critical value (horizontal lines) by \code{plot(scus)} which leads to a similar plot as in the left panel of Figure~\ref{fig:sc} (see the appendix for more details). As the process crosses its boundary, there is a significant change in the mean, while the clear peak in the process conveys that there is at least one strong break in the early 1980s. A formal significance test can also be carried out by \code{sctest(ocus)} which leads to a highly significant $p$ value of \Sexpr{round(sctest(ocus)$p.value, digits = 4)}. Similarly, the same quadratic spectral kernel HAC estimator could also be used for computing and visualizing the sup$F$ test of \cite{hac:Andrews:1993}, the code is provided in the appendix. Finally, the breakpoints in this model along with their confidence intervals can be computed by: <>= bp <- breakpoints(RealInt ~ 1) confint(bp, vcov = kernHAC) @ The dating algorithm \code{breakpoints} implements the procedure described in \cite{hac:Bai+Perron:2003} and estimates the timing of the structural changes by OLS. Therefore, in this step no covariance matrix estimate is required, but for computing the confidence intervals using a consistent covariance matrix estimator is again essential. The \code{confint} method for computing confidence intervals takes again a \code{vcov} argument which has to be a function (and not a matrix) because it has to be applied to several segments of the data. By default, it computes the breakpoints for the minimum BIC partition which gives in this case two breaks.\footnote{By choosing the number of breakpoints with sequential tests and not the BIC, \cite{hac:Bai+Perron:2003} arrive at a model with an additional breakpoint which has rather wide confidence intervals \citep[see also][]{hac:Zeileis+Kleiber:2004}} The fitted three-segment model along with the breakpoints and their confidence intervals is depicted in the right panel of Figure~\ref{fig:sc}. \setkeys{Gin}{width=\textwidth} \begin{figure}[tbh] \begin{center} <>= par(mfrow = c(1, 2)) plot(ocus, aggregate = FALSE, main = "") plot(RealInt, ylab = "Real interest rate") lines(ts(fitted(bp), start = start(RealInt), freq = 4), col = 4) lines(confint(bp, vcov = kernHAC)) @ \caption{\label{fig:sc} OLS-based CUSUM test (left) and fitted model (right) for real interest data.} \end{center} \end{figure} \section{Summary} \label{sec:summary} This paper briefly reviews a class of heteroskedasticity-consistent (HC) and a class of heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators suggested in the econometric literature over the last 20 years and introduces unified computational tools that reflect the flexibility and the conceptual ideas of the underlying theoretical frameworks. Based on these general tools, a number of special cases of HC and HAC estimators is provided including the most popular in applied econometric research. All the functions suggested are implemented in the package \pkg{sandwich} in the \proglang{R} system for statistical computing and designed in such a way that they build on readily available model fitting functions and provide building blocks that can be easily integrated into other programs or applications. To achieve this flexibility, the object orientation mechanism of \proglang{R} and the fact that functions are first-level objects are of prime importance. \section*{Acknowledgments} We are grateful to Thomas Lumley for putting his code in the \pkg{weave} package at disposal and for advice in the design of \pkg{sandwich}, and to Christian Kleiber for helpful suggestions in the development of \pkg{sandwich}. \bibliography{hac} \clearpage \begin{appendix} %% for "plain pretty" printing \DefineVerbatimEnvironment{Sinput}{Verbatim}{} <>= options(prompt = " ") @ \section[R code]{\proglang{R} code} The packages \pkg{sandwich}, \pkg{lmtest} and \pkg{strucchange} are required for the applications in this paper. Furthermore, the packages depend on \pkg{zoo}. For the computations in this paper \proglang{R} \Sexpr{paste(R.Version()[6:7], collapse = ".")} and \pkg{sandwich} \Sexpr{gsub("-", "--", packageDescription("sandwich")$Version)}, \pkg{lmtest} \Sexpr{gsub("-", "--", packageDescription("lmtest")$Version)}, \pkg{strucchange} \Sexpr{gsub("-", "--", packageDescription("strucchange")$Version)} and \pkg{zoo} \Sexpr{gsub("-", "--", packageDescription("zoo")$Version)} have been used. \proglang{R} itself and all packages used are available from CRAN at \url{http://CRAN.R-project.org/}. To make the packages available for the examples the following commands are necessary: <>= <> library("strucchange") @ \subsection{Testing coefficients in cross-sectional data} Load public schools data, omit \code{NA} in Wisconsin and scale income: <>= <> @ Fit quadratic regression model: <>= <> @ Compare standard errors: <>= sqrt(diag(vcov(fm.ps))) sqrt(diag(vcovHC(fm.ps, type = "const"))) sqrt(diag(vcovHC(fm.ps, type = "HC0"))) sqrt(diag(vcovHC(fm.ps, type = "HC3"))) sqrt(diag(vcovHC(fm.ps, type = "HC4"))) @ Test coefficient of quadratic term: <>= <> <> @ Visualization: %%non-dynamic for pretty printing \begin{Schunk} \begin{Sinput} plot(Expenditure ~ Income, data = ps, xlab = "per capita income", ylab = "per capita spending on public schools") inc <- seq(0.5, 1.2, by = 0.001) lines(inc, predict(fm.ps, data.frame(Income = inc)), col = 4, lty = 2) fm.ps2 <- lm(Expenditure ~ Income, data = ps) abline(fm.ps2, col = 4) text(ps[2,2], ps[2,1], rownames(ps)[2], pos = 2) \end{Sinput} \end{Schunk} \subsection{Testing coefficients in time-series data} Load investment equation data: <>= <> @ Fit regression model: <>= <> @ Test coefficients using Newey-West HAC estimator with user-defined and data-driven bandwidth and with Parzen kernel: %%non-dynamic for pretty printing \begin{Schunk} \begin{Sinput} coeftest(fm.inv, df = Inf, vcov = NeweyWest(fm.inv, lag = 4, prewhite = FALSE)) coeftest(fm.inv, df = Inf, vcov = NeweyWest) parzenHAC <- function(x, ...) kernHAC(x, kernel = "Parzen", prewhite = 2, adjust = FALSE, bw = bwNeweyWest, ...) coeftest(fm.inv, df = Inf, vcov = parzenHAC) \end{Sinput} \end{Schunk} Time-series visualization: <>= plot(Investment[, "RealInv"], type = "b", pch = 19, ylab = "Real investment") lines(ts(fitted(fm.inv), start = 1964), col = 4) @ 3-dimensional visualization: %%non-dynamic for pretty printing \begin{Schunk} \begin{Sinput} library("scatterplot3d") s3d <- scatterplot3d(Investment[,c(5,7,6)], type = "b", angle = 65, scale.y = 1, pch = 16) s3d$plane3d(fm.inv, lty.box = "solid", col = 4) \end{Sinput} \end{Schunk} \subsection[Testing and dating structural changes in the presence of heteroskedasticity and autocorrelation]{Testing and dating structural changes in the presence of\\ heteroskedasticity and autocorrelation} Load real interest series: <>= data("RealInt") @ OLS-based CUSUM test with quadratic spectral kernel HAC estimate: <>= <> plot(ocus, aggregate = FALSE) sctest(ocus) @ sup$F$ test with quadratic spectral kernel HAC estimate: <>= fs <- Fstats(RealInt ~ 1, vcov = kernHAC) plot(fs) sctest(fs) @ Breakpoint estimation and confidence intervals with quadratic spectral kernel HAC estimate: <>= <> plot(bp) @ Visualization: <>= plot(RealInt, ylab = "Real interest rate") lines(ts(fitted(bp), start = start(RealInt), freq = 4), col = 4) lines(confint(bp, vcov = kernHAC)) @ \subsection{Integrating covariance matrix estimators in other functions} If programmers want to allow for the same flexibility regarding the specification of covariance matrices in their own functions as illustrated in \code{coeftest}, only a few simple additions have to be made which are illustrated in the following. Say, a function \code{foo(lmobj, vcov = NULL, ...)} wants to compute some quantity involving the standard errors associated with the \code{"lm"} object \code{lmobj}. Then, \code{vcov} should use by default the standard \code{vcov} method for \code{"lm"} objects, otherwise \code{vcov} is assumed to be either a function returning the covariance matrix estimate or the estimate itself. The following piece of code is sufficient for computing the standard errors. \begin{Sinput} if(is.null(vcov)) { se <- vcov(lmobj) } else { if (is.function(vcov)) se <- vcov(lmobj) else se <- vcov } se <- sqrt(diag(se)) \end{Sinput} In the first step the default method is called: note, that \proglang{R} can automatically distinguish between the variable \code{vcov} (which is \code{NULL}) and the generic function \code{vcov} (from the \pkg{stats} package which dispatches to the \code{"lm"} method) that is called here. Otherwise, it is just distinguished between a function or non-function. In the final step the square root of the diagonal elements is computed and stored in the vector \code{se} which can subsequently used for further computation in \code{foo()}. \end{appendix} \end{document} sandwich/inst/doc/sandwich.R0000644000175400001440000002125212600735100015706 0ustar zeileisusers### R code from vignette source 'sandwich.Rnw' ################################################### ### code chunk number 1: preliminaries ################################################### library("zoo") library("sandwich") library("strucchange") library("lmtest") options(prompt = "R> ", continue = "+ ") ################################################### ### code chunk number 2: hac-kweights ################################################### curve(kweights(x, kernel = "Quadratic", normalize = TRUE), from = 0, to = 3.2, xlab = "x", ylab = "K(x)") curve(kweights(x, kernel = "Bartlett", normalize = TRUE), from = 0, to = 3.2, col = 2, add = TRUE) curve(kweights(x, kernel = "Parzen", normalize = TRUE), from = 0, to = 3.2, col = 3, add = TRUE) curve(kweights(x, kernel = "Tukey", normalize = TRUE), from = 0, to = 3.2, col = 4, add = TRUE) lines(c(0, 0.5), c(1, 1), col = 6) lines(c(0.5, 0.5), c(1, 0), lty = 3, col = 6) lines(c(0.5, 3.2), c(0, 0), col = 6) curve(kweights(x, kernel = "Quadratic", normalize = TRUE), from = 0, to = 3.2, col = 1, add = TRUE) text(0.5, 0.98, "Truncated", pos = 4) text(0.8, kweights(0.8, "Bartlett", normalize = TRUE), "Bartlett", pos = 4) text(1.35, kweights(1.4, "Quadratic", normalize = TRUE), "Quadratic Spectral", pos = 2) text(1.15, 0.29, "Parzen", pos = 4) arrows(1.17, 0.29, 1, kweights(1, "Parzen", normalize = TRUE), length = 0.1) text(1.3, 0.2, "Tukey-Hanning", pos = 4) arrows(1.32, 0.2, 1.1, kweights(1.1, "Tukey", normalize = TRUE), length = 0.1) ################################################### ### code chunk number 3: loadlibs1 ################################################### library("sandwich") library("lmtest") ################################################### ### code chunk number 4: hc-data ################################################### data("PublicSchools") ps <- na.omit(PublicSchools) ps$Income <- ps$Income * 0.0001 ################################################### ### code chunk number 5: hc-model ################################################### fm.ps <- lm(Expenditure ~ Income + I(Income^2), data = ps) ################################################### ### code chunk number 6: hc-test1 ################################################### coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC0")) ################################################### ### code chunk number 7: hc-test2 ################################################### coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC4")) ################################################### ### code chunk number 8: hc-plot ################################################### plot(Expenditure ~ Income, data = ps, xlab = "per capita income", ylab = "per capita spending on public schools") inc <- seq(0.5, 1.2, by = 0.001) lines(inc, predict(fm.ps, data.frame(Income = inc)), col = 4, lty = 2) fm.ps2 <- lm(Expenditure ~ Income, data = ps) abline(fm.ps2, col = 4) text(ps[2,2], ps[2,1], rownames(ps)[2], pos = 2) ################################################### ### code chunk number 9: hac-data ################################################### data("Investment") ################################################### ### code chunk number 10: hac-model ################################################### fm.inv <- lm(RealInv ~ RealGNP + RealInt, data = Investment) ################################################### ### code chunk number 11: hac-test1 ################################################### coeftest(fm.inv, df = Inf, vcov = NeweyWest(fm.inv, lag = 4, prewhite = FALSE)) ################################################### ### code chunk number 12: hac-test2 ################################################### coeftest(fm.inv, df = Inf, vcov = NeweyWest) ################################################### ### code chunk number 13: hac-test3 ################################################### parzenHAC <- function(x, ...) kernHAC(x, kernel = "Parzen", prewhite = 2, adjust = FALSE, bw = bwNeweyWest, ...) coeftest(fm.inv, df = Inf, vcov = parzenHAC) ################################################### ### code chunk number 14: hac-plot ################################################### library("scatterplot3d") s3d <- scatterplot3d(Investment[,c(5,7,6)], type = "b", angle = 65, scale.y = 1, pch = 16) s3d$plane3d(fm.inv, lty.box = "solid", col = 4) ################################################### ### code chunk number 15: loadlibs2 ################################################### library("strucchange") data("RealInt") ################################################### ### code chunk number 16: sc-ocus ################################################### ocus <- gefp(RealInt ~ 1, fit = lm, vcov = kernHAC) ################################################### ### code chunk number 17: sc-bp ################################################### bp <- breakpoints(RealInt ~ 1) confint(bp, vcov = kernHAC) ################################################### ### code chunk number 18: sc-plot ################################################### par(mfrow = c(1, 2)) plot(ocus, aggregate = FALSE, main = "") plot(RealInt, ylab = "Real interest rate") lines(ts(fitted(bp), start = start(RealInt), freq = 4), col = 4) lines(confint(bp, vcov = kernHAC)) ################################################### ### code chunk number 19: sandwich.Rnw:786-787 ################################################### options(prompt = " ") ################################################### ### code chunk number 20: sandwich.Rnw:805-807 (eval = FALSE) ################################################### ## library("sandwich") ## library("lmtest") ## library("strucchange") ################################################### ### code chunk number 21: sandwich.Rnw:814-815 (eval = FALSE) ################################################### ## data("PublicSchools") ## ps <- na.omit(PublicSchools) ## ps$Income <- ps$Income * 0.0001 ################################################### ### code chunk number 22: sandwich.Rnw:819-820 (eval = FALSE) ################################################### ## fm.ps <- lm(Expenditure ~ Income + I(Income^2), data = ps) ################################################### ### code chunk number 23: sandwich.Rnw:824-829 (eval = FALSE) ################################################### ## sqrt(diag(vcov(fm.ps))) ## sqrt(diag(vcovHC(fm.ps, type = "const"))) ## sqrt(diag(vcovHC(fm.ps, type = "HC0"))) ## sqrt(diag(vcovHC(fm.ps, type = "HC3"))) ## sqrt(diag(vcovHC(fm.ps, type = "HC4"))) ################################################### ### code chunk number 24: sandwich.Rnw:833-835 (eval = FALSE) ################################################### ## coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC0")) ## coeftest(fm.ps, df = Inf, vcov = vcovHC(fm.ps, type = "HC4")) ################################################### ### code chunk number 25: sandwich.Rnw:855-856 (eval = FALSE) ################################################### ## data("Investment") ################################################### ### code chunk number 26: sandwich.Rnw:860-861 (eval = FALSE) ################################################### ## fm.inv <- lm(RealInv ~ RealGNP + RealInt, data = Investment) ################################################### ### code chunk number 27: sandwich.Rnw:879-881 (eval = FALSE) ################################################### ## plot(Investment[, "RealInv"], type = "b", pch = 19, ylab = "Real investment") ## lines(ts(fitted(fm.inv), start = 1964), col = 4) ################################################### ### code chunk number 28: sandwich.Rnw:897-898 (eval = FALSE) ################################################### ## data("RealInt") ################################################### ### code chunk number 29: sandwich.Rnw:902-905 (eval = FALSE) ################################################### ## ocus <- gefp(RealInt ~ 1, fit = lm, vcov = kernHAC) ## plot(ocus, aggregate = FALSE) ## sctest(ocus) ################################################### ### code chunk number 30: sandwich.Rnw:909-912 (eval = FALSE) ################################################### ## fs <- Fstats(RealInt ~ 1, vcov = kernHAC) ## plot(fs) ## sctest(fs) ################################################### ### code chunk number 31: sandwich.Rnw:917-919 (eval = FALSE) ################################################### ## bp <- breakpoints(RealInt ~ 1) ## confint(bp, vcov = kernHAC) ## plot(bp) ################################################### ### code chunk number 32: sandwich.Rnw:923-926 (eval = FALSE) ################################################### ## plot(RealInt, ylab = "Real interest rate") ## lines(ts(fitted(bp), start = start(RealInt), freq = 4), col = 4) ## lines(confint(bp, vcov = kernHAC)) sandwich/inst/doc/sandwich-OOP.Rnw0000644000175400001440000010524412600735100016712 0ustar zeileisusers\documentclass[nojss]{jss} \usepackage{thumbpdf} %% need no \usepackage{Sweave} %% Symbols \newcommand{\darrow}{\stackrel{\mbox{\tiny \textnormal{d}}}{\longrightarrow}} \author{Achim Zeileis\\Universit\"at Innsbruck} \Plainauthor{Achim Zeileis} \title{Object-Oriented Computation of Sandwich Estimators} \Keywords{covariance matrix estimators, estimating functions, object orientation, \proglang{R}} \Plainkeywords{covariance matrix estimators, estimating functions, object orientation, R} \Abstract{ This introduction to the object-orientation features of the \proglang{R} package \pkg{sandwich} is a (slightly) modified version of \cite{hac:Zeileis:2006}, published in the \emph{Journal of Statistical Software}. Sandwich covariance matrix estimators are a popular tool in applied regression modeling for performing inference that is robust to certain types of model misspecification. Suitable implementations are available in the \proglang{R} system for statistical computing for certain model fitting functions only (in particular \code{lm()}), but not for other standard regression functions, such as \code{glm()}, \code{nls()}, or \code{survreg()}. Therefore, conceptual tools and their translation to computational tools in the package \pkg{sandwich} are discussed, enabling the computation of sandwich estimators in general parametric models. Object orientation can be achieved by providing a few extractor functions---most importantly for the empirical estimating functions---from which various types of sandwich estimators can be computed. } \Address{ Achim Zeileis\\ Department of Statistics\\ Faculty of Economics and Statistics\\ Universit\"at Innsbruck\\ Universit\"atsstr.~15\\ 6020 Innsbruck, Austria\\ E-mail: \email{Achim.Zeileis@R-project.org}\\ URL: \url{http://eeecon.uibk.ac.at/~zeileis/} } \begin{document} \SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Object-Oriented Computation of Sandwich Estimators} %\VignetteDepends{sandwich,zoo,AER,survival,MASS,lmtest} %\VignetteKeywords{covariance matrix estimators, estimating functions, object orientation, R} %\VignettePackage{sandwich} <>= library("AER") library("MASS") options(prompt = "R> ", continue = "+ ") @ \section{Introduction} \label{sec:intro} A popular approach to applied parametric regression modeling is to derive estimates of the unknown parameters via a set of estimating functions (including least squares and maximum likelihood scores). Inference for these models is typically based on a central limit theorem in which the covariance matrix is of a sandwich type: a slice of meat between two slices of bread, pictorially speaking. Employing estimators for the covariance matrix based on this sandwich form can make inference for the parameters more robust against certain model misspecifications (provided the estimating functions still hold and yield consistent estimates). Therefore, sandwich estimators such as heteroskedasticy consistent (HC) estimators for cross-section data and heteroskedasitcity and autocorrelation consistent (HAC) estimators for time-series data are commonly used in applied regression, in particular in linear regression models. \cite{hac:Zeileis:2004a} discusses a set of computational tools provided by the \pkg{sandwich} package for the \proglang{R} system for statistical computing \citep{hac:R:2008} which allows for computing HC and HAC estimators in linear regression models fitted by \code{lm()}. Here, we set out where the discussion of \cite{hac:Zeileis:2004a} ends and generalize the tools from linear to general parametric models fitted by estimating functions. This generalization is achieved by providing an object-oriented implementation for the building blocks of the sandwich that rely only on a small set of extractor functions for fitted model objects. The most important of these is a method for extracting the empirical estimating functions---based on this a wide variety of meat fillings for sandwiches is provided. The paper is organized as follows: Section~\ref{sec:model} discusses the model frame and reviews some of the underlying theory. Section~\ref{sec:R} presents some existing \proglang{R} infrastructure which can be re-used for the computation of sandwich covariance matrices in Section~\ref{sec:vcov}. Section~\ref{sec:illustrations} gives a brief illustration of the computational tools before Section~\ref{sec:disc} concludes the paper. { \section{Model frame} \label{sec:model} \nopagebreak To fix notations, let us assume we have data in a regression setup, i.e., $(y_i, x_i)$ for $i = 1, \dots, n$, that follow some distribution that is controlled by a $k$-dimensional parameter vector $\theta$. In many situations, an estimating function $\psi(\cdot)$ is available for this type of models such that $\E[\psi(y, x, \theta)] = 0$. Then, under certain weak regularity conditions \citep[see e.g.,][]{hac:White:1994}, $\theta$ can be estimated using an M-estimator $\hat \theta$ implicitely defined as \begin{equation} \label{eq:estfun} \sum_{i = 1}^n \psi(y_i, x_i, \hat \theta) \quad = \quad 0. \end{equation} This includes cases where the estimating function $\psi(\cdot)$ is the derivative of an objective function $\Psi(\cdot)$: \begin{equation} \label{eq:score} \psi(y, x, \theta) \quad = \quad \frac{\partial \Psi(y, x, \theta)}{\partial \theta}. \end{equation} } Examples for estimation techniques included in this framework are maximum likelihood (ML) and ordinary and nonlinear least squares (OLS and NLS) estimation, where the estimator is usually written in terms of the objective function as $\hat \theta = \mbox{argmin}_\theta \sum_i \Psi(y_i, x_i, \theta)$. Other techniques---often expressed in terms of the estimating function rather than the objective function---include quasi ML, robust M-estimation and generalized estimating equations (GEE). Inference about $\theta$ is typically performed relying on a central limit theorem (CLT) of type \begin{equation} \label{eq:clt} \sqrt{n} \, (\hat \theta - \theta) \quad \darrow \quad N(0, S(\theta)), \end{equation} where $\darrow$ denotes convergence in distribution. For the covariance matrix $S(\theta)$, a sandwich formula can be given \begin{eqnarray} \label{eq:sandwich} S(\theta) & = & B(\theta) \, M(\theta) \, B(\theta) \\ \label{eq:bread} B(\theta) & = & \left( \E[ - \psi'(y, x, \theta) ] \right)^{-1} \\ \label{obj} M(\theta) & = & \VAR[ \psi(y, x, \theta) ] \end{eqnarray} see Theorem~6.10 in \cite{hac:White:1994}, Chapter~5 in \cite{hac:Cameron+Trivedi:2005}, or \cite{hac:Stefanski+Boos:2002} for further details. The ``meat'' of the sandwich $M(\theta)$ is the variance of the estimating function and the ``bread'' is the inverse of the expectation of its first derivative $\psi'$ (again with respect to $\theta$). Note that we use the more evocative names $S$, $B$ and $M$ instead of the more conventional notation $V(\theta) = A(\theta)^{-1} B(\theta) A(\theta)^{-1}$. In correctly specified models estimated by ML (or OLS and NLS with homoskedastic errors), this sandwich expression for $S(\theta)$ can be simplified because $M(\theta) = B(\theta)^{-1}$, corresponding to the Fisher information matrix. Hence, the variance $S(\theta)$ in the CLT from Equation~\ref{eq:clt} is typically estimated by an empirical version of $B(\theta)$. However, more robust covariance matrices can be obtained by employing estimates for $M(\theta)$ that are consistent under weaker assumptions \citep[see e.g.,][]{hac:Lumley+Heagerty:1999} and plugging these into the sandwich formula for $S(\theta)$ from Equation~\ref{eq:sandwich}. Robustness can be achieved with respect to various types of misspecification, e.g., heteroskedasticity---however, consistency of $\hat \theta$ has to be assured, which implies that at least the estimating functions have to be correctly specified. Many of the models of interest to us, provide some more structure: the objective function $\Psi(y, x, \theta)$ depends on $x$ and $\theta$ in a special way, namely it does only depend on the univariate linear predictor $\eta = x^\top \theta$. Then, the estimating function is of type \begin{equation} \label{eq:estfunHC} \psi(y, x, \theta) \quad = \quad \frac{\partial \Psi}{\partial \eta} \cdot \frac{\partial \eta}{\partial \theta} \quad = \quad \frac{\partial \Psi}{\partial \eta} \cdot x. \end{equation} The partial derivative $r(y, \eta) = \partial \Psi(y, \eta) / \partial \eta$ is in some models also called ``working residual'' corresponding to the usual residuals in linear regression models. In such linear-predictor-based models, the meat of the sandwich can also be sloppily written as \begin{equation} \label{eq:objHC} M(\theta) \quad = \quad x \, \VAR[ r(y, x^\top \theta) ] \, x^\top. \end{equation} Whereas employing this structure for computing HC covariance matrix estimates is well-established practice for linear regression models \citep[see][among others]{hac:MacKinnon+White:1985,hac:Long+Ervin:2000}, it is less commonly applied in other regression models such as GLMs. \section[Existing R infrastructure]{Existing \proglang{R} infrastructure} \label{sec:R} To make use of the theory outlined in the previous section, some computational infrastructure is required translating the conceptual to computational tools. \proglang{R} comes with a multitude of model-fitting functions that compute estimates $\hat \theta$ and can be seen as special cases of the framework above. They are typically accompanied by extractor and summary methods providing inference based on the CLT from Equation~\ref{eq:clt}. For extracting the estimated parameter vector $\hat \theta$ and some estimate of the covariance matrix $S(\theta)$, there are usually a \code{coef()} and a \code{vcov()} method, respectively. Based on these estimates, inference can typically be performed by the \code{summary()} and \code{anova()} methods. By convention, the \code{summary()} method performs partial $t$ or $z$~tests and the \code{anova()} method performs $F$ or $\chi^2$~tests for nested models. The covariance estimate used in these tests (and returned by \code{vcov()}) usually relies on the assumption of correctly specified models and hence is simply an empirical version of the bread $B(\theta)$ only (divided by $n$). For extending these tools to inference based on sandwich covariance matrix estimators, two things are needed: 1.~generalizations of \code{vcov()} that enable computations of sandwich estimates, 2.~inference functions corresponding to the \code{summary()} and \code{anova()} methods which allow other covariance matrices to be plugged in. As for the latter, the package \pkg{lmtest} \citep{hac:Zeileis+Hothorn:2002} provides \code{coeftest()} and \code{waldtest()} and \pkg{car} \citep{hac:Fox:2002} provides \code{linear.hypothesis()}---all of these can perform model comparisons in rather general parametric models, employing user-specified covariance matrices. As for the former, only specialized solutions of sandwich covariances matrices are currently available in \proglang{R} packages, e.g., HAC estimators for linear models in previous versions of \pkg{sandwich} and HC estimators for linear models in \pkg{car} and \pkg{sandwich}. Therefore, we aim at providing a tool kit for plugging together sandwich matrices (including HC and HAC estimators and potentially others) in general parametric models, re-using the functionality that is already provided. \section{Covariance matrix estimators} \label{sec:vcov} In the following, the conceptual tools outlined in Section~\ref{sec:model} are translated to computational tools preserving their flexibility through the use of the estimating functions framework and re-using the computational infrastructure that is already available in \proglang{R}. Separate methods are suggested for computing estimates for the bread $B(\theta)$ and the meat $M(\theta)$, along with some convenience functions and wrapper interfaces that build sandwiches from bread and meat. \subsection{The bread} Estimating the bread $B(\theta)$ is usually relatively easy and the most popular estimate is the Hessian, i.e., the mean crossproduct of the derivative of the estimating function evaluated at the data and estimated parameters: \begin{equation} \label{eq:Bhat} \hat B \quad = \quad \left( \frac{1}{n} \sum_{i = 1}^n - \psi'(y_i, x_i, \hat \theta) \right)^{-1}. \end{equation} If an objective function $\Psi(\cdot)$ is used, this is the crossproduct of its second derivative, hence the name Hessian. This estimator is what the \code{vcov()} method is typically based on and therefore it can usually be extracted easily from the fitted model objects, e.g., for ``\code{lm}'' and ``\code{glm}'' it is essentially the \code{cov.unscaled} element returned by the \code{summary()} method. To unify the extraction of a suitable estimate for the bread, \pkg{sandwich} provides a new \code{bread()} generic that should by default return the bread estimate that is also used in \code{vcov()}. This will usually be the Hessian estimate, but might also be the expected Hessian \citep[Equation~5.36]{hac:Cameron+Trivedi:2005} in some models. The package \pkg{sandwich} provides \code{bread()} methods for ``\code{lm}'' (including ``\code{glm}'' by inheritance), ``\code{coxph}'', ``\code{survreg}'' and ``\code{nls}'' objects. All of them simply re-use the information provided in the fitted models (or their summaries) and perform hardly any computations, e.g., for ``\code{lm}'' objects: \begin{Schunk} \begin{Sinput} bread.lm <- function(obj, ...) { so <- summary(obj) so$cov.unscaled * as.vector(sum(so$df[1:2])) } \end{Sinput} \end{Schunk} \subsection{The meat} While the bread $B(\theta)$ is typically estimated by the Hessian matrix $\hat B$ from Equation~\ref{eq:Bhat}, various different types of estimators are available for the meat $M(\theta)$, usually offering certain robustness properties. Most of these estimators are based on the empirical values of estimating functions. Hence, a natural idea for object-oriented implementation of such estimators is the following: provide various functions that compute different estimators for the meat based on an \code{estfun()} extractor function that extracts the empirical estimating functions from a fitted model object. This is what \pkg{sandwich} does: the functions \code{meat()}, \code{meatHAC()} and \code{meatHC()} compute outer product, HAC and HC estimators for $M(\theta)$, respectively, relying on the existence of an \code{estfun()} method (and potentially a few other methods). Their design is described in the following. \subsubsection{Estimating functions} Whereas (different types of) residuals are typically available as discrepancy measure for a model fit via the \code{residuals()} method, the empirical values of the estimating functions $\psi(y_i, x_i, \hat \theta)$ are often not readily implemented in \proglang{R}. Hence, \pkg{sandwich} provides a new \code{estfun()} generic whose methods should return an $n \times k$ matrix with the empirical estimating functions: \[ \left( \begin{array}{c} \psi(y_1, x_1, \hat \theta) \\ \vdots \\ \psi(y_n, x_n, \hat \theta) \end{array} \right). \] Suitable methods are provided for ``\code{lm}'', ``\code{glm}'', ``\code{rlm}'', ``\code{nls}'', ``\code{survreg}'' and ``\code{coxph}'' objects. Usually, these can easily re-use existing methods, in particular \code{residuals()} and \code{model.matrix()} if the model is of type~(\ref{eq:estfunHC}). As a simple example, the most important steps of the ``\code{lm}'' method are \begin{Schunk} \begin{Sinput} estfun.lm <- function (obj, ...) { wts <- weights(obj) if(is.null(wts)) wts <- 1 residuals(obj) * wts * model.matrix(obj) } \end{Sinput} \end{Schunk} \subsubsection{Outer product estimators} A simple and natural estimator for the meat matrix $M(\theta) = \VAR[ \psi(y, x, \theta)]$ is the outer product of the empirical estimating functions: \begin{equation} \label{eq:meatOP} \hat M \quad = \quad \frac{1}{n} \sum_{i = 1}^n \psi(y_i, x_i, \hat \theta) \psi(y_i, x_i, \hat \theta)^\top \end{equation} This corresponds to the Eicker-Huber-White estimator \citep{hac:Eicker:1963,hac:Huber:1967,hac:White:1980} and is sometimes also called outer product of gradients estimator. In practice, a degrees of freedom adjustment is often used, i.e., the sum is scaled by $n-k$ instead of $n$, corresponding to the HC1 estimator from \cite{hac:MacKinnon+White:1985}. In non-linear models this has no theoretical justification, but has been found to have better finite sample performance in some simulation studies. In \pkg{sandwich}, these two estimators are provided by the function \code{meat()} which only relies on the existence of an \code{estfun()} method. A simplified version of the \proglang{R} code is \begin{Schunk} \begin{Sinput} meat <- function(obj, adjust = FALSE, ...) { psi <- estfun(obj) k <- NCOL(psi) n <- NROW(psi) rval <- crossprod(as.matrix(psi))/n if(adjust) rval <- n/(n - k) * rval rval } \end{Sinput} \end{Schunk} \subsubsection{HAC estimators} More elaborate methods for deriving consistent covariance matrix estimates in the presence of autocorrelation in time-series data are also available. Such HAC estimators $\hat M_\mathrm{HAC}$ are based on the weighted empirical autocorrelations of the empirical estimating functions: \begin{equation} \label{eq:meatHAC} \hat M_\mathrm{HAC} \quad = \quad \frac{1}{n} \sum_{i, j = 1}^n w_{|i-j|} \, \psi(y_i, x_i, \hat \theta) \psi(y_j, x_j, \hat \theta)^\top \end{equation} where different strategies are available for the choice of the weights $w_\ell$ at lag $\ell = 0, \dots, {n-1}$ \citep{hac:Andrews:1991,hac:Newey+West:1994,hac:Lumley+Heagerty:1999}. Again, an additional finite sample adjustment can be applied by multiplication with $n/(n-k)$. Once a vector of weights is chosen, the computation of $\hat M_\mathrm{HAC}$ in \proglang{R} is easy, the most important steps are given by \begin{Schunk} \begin{Sinput} meatHAC <- function(obj, weights, ...) { psi <- estfun(obj) n <- NROW(psi) rval <- 0.5 * crossprod(psi) * weights[1] for(i in 2:length(weights)) rval <- rval + weights[i] * crossprod(psi[1:(n-i+1),], psi[i:n,]) (rval + t(rval))/n } \end{Sinput} \end{Schunk} The actual function \code{meatHAC()} in \pkg{sandwich} is much more complex as it also interfaces different weighting and bandwidth selection functions. The details are the same compared to \cite{hac:Zeileis:2004a} where the selection of weights had been discussed for fitted ``\code{lm}'' objects. \subsubsection{HC estimators} In addition to the two HC estimators that can be written as outer product estimators (also called HC0 and HC1), various other HC estimators (usually called HC2--HC4) have been suggested, in particular for the linear regression model \citep{hac:MacKinnon+White:1985,hac:Long+Ervin:2000,hac:Cribari-Neto:2004}. In fact, they can be applied to more general models provided the estimating function depends on the parameters only through a linear predictor as described in Equation~\ref{eq:estfunHC}. Then, the meat matrix $M(\theta)$ is of type (\ref{eq:objHC}) which naturally leads to HC estimators of the form $\hat M_\mathrm{HC} = 1/n \, X^\top \hat \Omega X$, where $X$ is the regressor matrix and $\hat \Omega$ is a diagonal matrix estimating the variance of $r(y, \eta)$. Various functions $\omega(\cdot)$ have been suggested that derive estimates of the variances from the observed working residuals $(r(y_1, x_1^\top \hat \theta), \dots, r(y_n, x_n^\top \hat \theta))^\top$---possibly also depending on the hat values and the degrees of freedom. Thus, the HC estimators are of the form \begin{equation} \label{eq:meatHC} \hat M_\mathrm{HC} \quad = \quad \frac{1}{n} X^\top \left( \begin{array}{ccc} \omega(r(y_1, x_1^\top \theta)) & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \omega(r(y, x^\top \theta)) \end{array} \right) X. \end{equation} To transfer these tools into software in the function \code{meatHC()}, we need infrastructure for three elements in Equation~\ref{eq:meatHC}: 1.~the model matrix $X$, 2.~the function $\omega(\cdot)$, and 3.~the empirical working residuals $r(y_i, x_i^\top \hat \theta)$. As for 1, the model matrix $X$ can easily be accessed via the \code{model.matrix()} method. Concerning 2, the specification of $\omega(\cdot)$ is discussed in detail in \cite{hac:Zeileis:2004a}. Hence, we omit the details here and only assume that we have either a vector \code{omega} of diagonal elements or a function \code{omega} that computes the diagonal elements from the residuals, diagonal values of the hat matrix (provided by the \code{hatvalues()} method) and the degrees of freedom $n-k$. For 3, the working residuals, some fitted model classes provide infrastructure in their \code{residuals()} method. However, there is no unified interface available for this and instead of setting up a new separate generic, it is also possible to recover this information from the estimating function. As $\psi(y_i, x_i, \hat \theta) = r(y_i, x_i^\top \hat \theta) \cdot x_i$, we can simply divide the empirical estimating function by $x_i$ to obtain the working residual. Based on these functions, all necessary information can be extracted from fitted model objects and a condensed version of \code{meatHC()} can then be written as \begin{Schunk} \begin{Sinput} meatHC <- function(obj, omega, ...) { X <- model.matrix(obj) res <- rowMeans(estfun(obj)/X, na.rm = TRUE) diaghat <- hatvalues(obj) df <- NROW(X) - NCOL(X) if(is.function(omega)) omega <- omega(res, diaghat, df) rval <- sqrt(omega) * X crossprod(rval)/NROW(X) } \end{Sinput} \end{Schunk} \subsection{The sandwich} Based on the building blocks described in the previous sections, computing a sandwich estimate from a fitted model object is easy: the function \code{sandwich()} computes an estimate (by default the Eicker-Huber-White outer product estimate) for $1/n \, S(\theta)$ via \begin{Schunk} \begin{Sinput} sandwich <- function(obj, bread. = bread, meat. = meat, ...) { if(is.function(bread.)) bread. <- bread.(obj) if(is.function(meat.)) meat. <- meat.(obj, ...) 1/NROW(estfun(obj)) * (bread. %*% meat. %*% bread.) } \end{Sinput} \end{Schunk} For computing other estimates, the argument \code{meat.}~could also be set to \code{meatHAC} or \code{meatHC}. Therefore, all that an \proglang{R} user/developer would have to do to make a new class of fitted models, ``\code{foo}'' say, fit for this framework is: provide an \code{estfun()} method \code{estfun.}\emph{foo}\code{()} and a \code{bread()} method \code{bread.}\emph{foo}\code{()}. See also Figure~\ref{fig:sandwich}. Only for HC estimators (other than HC0 and HC1 which are available via \code{meat()}), it has to be assured in addition that \begin{itemize} \item the model only depends on a linear predictor (this cannot be easily checked by the software, but has to be done by the user), \item the model matrix $X$ is available via a \code{model.matrix.}\emph{foo}\code{()} method, \item a \code{hatvalues.}\emph{foo}\code{()} method exists (for HC2--HC4). \end{itemize} For both, HAC and HC estimators, the complexity of the meat functions was reduced for exposition in the paper: choosing the \code{weights} in \code{meatHAC} and the diagonal elements \code{omega} in \code{meatHC} can be controlled by a number of further arguments. To make these explicit for the user, wrapper functions \code{vcovHAC()} and \code{vcovHC()} with suitable default methods are provided in \pkg{sandwich} which work as advertised in \cite{hac:Zeileis:2004a} and are the recommended interfaces for computing HAC and HC estimators, respectively. Furthermore, the convenience interfaces \code{kernHAC()}, \code{NeweyWest()} and \code{weave()} setting the right defaults for \citep{hac:Andrews:1991}, \cite{hac:Newey+West:1994}, and \cite{hac:Lumley+Heagerty:1999}, respectively, continue to be provided by \pkg{sandwich}. \setkeys{Gin}{width=.85\textwidth} \begin{figure}[tbh] \begin{center} <>= par(mar = rep(0, 4)) plot(0, 0, xlim = c(0, 85), ylim = c(0, 110), type = "n", axes = FALSE, xlab = "", ylab = "") lgrey <- grey(0.88) dgrey <- grey(0.75) rect(45, 90, 70, 110, lwd = 2, col = dgrey) rect(20, 40, 40, 60, col = lgrey) rect(30, 40, 40, 60, col = dgrey) rect(20, 40, 40, 60, lwd = 2) rect(5, 0, 20, 20, lwd = 2, col = lgrey) rect(22.5, 0, 37.5, 20, lwd = 2, col = lgrey) rect(40, 0, 55, 20, lwd = 2, col = lgrey) rect(40, 0, 55, 20, lwd = 2, col = lgrey) rect(60, 0, 80, 20, col = lgrey) rect(70, 0, 80, 20, col = dgrey) rect(60, 0, 80, 20, lwd = 2) text(57.5, 100, "fitted model object\n(class: foo)") text(25, 50, "estfun") text(35, 50, "foo") text(12.5, 10, "meatHC") text(30, 10, "meatHAC") text(47.5, 10, "meat") text(65, 10, "bread") text(75, 10, "foo") arrows(57.5, 89, 70, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(57.5, 89, 30, 61, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 30, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 12.5, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 47.5, 21, lwd = 1.5, length = 0.15, angle = 20) @ \caption{\label{fig:sandwich} Structure of sandwich estimators} \end{center} \end{figure} \section{Illustrations} \label{sec:illustrations} This section briefly illustrates how the tools provided by \pkg{sandwich} can be applied to various models and re-used in other functions. Predominantly, sandwich estimators are used for inference, such as partial $t$ or $z$~tests of regression coefficients or restriction testing in nested regression models. As pointed out in Section~\ref{sec:R}, the packages \pkg{lmtest} \citep{hac:Zeileis+Hothorn:2002} and \pkg{car} \citep{hac:Fox:2002} provide some functions for this type of inference. The model for which sandwich estimators are employed most often is surely the linear regression model. Part of the reason for this is (together with the ubiquity of linear regression) that in linear regression mean and variance can be specified independently from each other. Thus, the model can be seen as a model for the conditional mean of the response with the variance left unspecified and captured only for inference by a robust sandwich estimator. \cite{hac:Zeileis:2004a} presents a collection of applications of sandwich estimators to linear regression, both for cross-section and time-series data. These examples are not affected by making \pkg{sandwich} object oriented, therefore, we do not present any examples for linear regression models here. To show that with the new object-oriented tools in \pkg{sandwich}, the functions can be applied as easily to other models we consider some models from microeconometrics: count data regression and probit and tobit models. In all examples, we compare the usual summary (coefficients, standard errors and partial $z$~tests) based on \code{vcov()} with the corresponding summary based on HC standard errors as provided by \code{sandwich()}. \code{coeftest()} from \pkg{lmtest} is always used for computing the summaries. \subsection{Count data regression} To illustrate the usage of sandwich estimators in count data regressions, we use artifical data simulated from a negative binomial model. The mean of the response \code{y} depends on a regressor \code{x} through a log link, the size parameter of the negative binomial distribution is 1, and the regressor is simply drawn from a standard normal distribution. After setting the random seed for reproducibility, we draw 250 observations from this model: <>= set.seed(123) x <- rnorm(250) y <- rnbinom(250, mu = exp(1 + x), size = 1) @ In the following, we will fit various count models to this data employing the overspecification \verb/y ~ x + I(x^2)/ and assessing the significance of \verb/I(x^2)/. First, we use \code{glm()} with \code{family = poisson} to fit a poisson regression as the simplest model for count data. Of course, this model is not correctly specified as \code{y} is from a negative binomial distribution. Hence, we are not surprised that the resulting test of \verb/I(x^2)/ is spuriously significant: <>= fm_pois <- glm(y ~ x + I(x^2), family = poisson) coeftest(fm_pois) @ However, the specification of the conditional mean of \code{y} is correct in this model which is reflected by the coefficient estimates that are close to their true value. Only the dispersion which is fixed at 1 in the \code{poisson} family is misspecified. In this situation, the problem can be alleviated by employing sandwich standard errors in the partial $z$~tests, capturing the overdispersion in \code{y}. <>= coeftest(fm_pois, vcov = sandwich) @ Clearly, sandwich standard errors are not the only way of dealing with this situation. Other obvious candidates would be to use a quasi-poisson or, of course, a negative binomial model \citep{hac:McCullagh+Nelder:1989}. The former is available through the \code{quasipoisson} family for \code{glm()} that leads to the same coefficient estimates as \code{poisson} but additionally estimates the dispersion for inference. The associated model summary is very similar to that based on the sandwich standard errors, leading to qualitatively identical results. <>= fm_qpois <- glm(y ~ x + I(x^2), family = quasipoisson) coeftest(fm_qpois) @ Negative binomial models can be fitted by \code{glm.nb()} from \pkg{MASS} \citep{hac:Venables+Ripley:2002}. <>= fm_nbin <- glm.nb(y ~ x + I(x^2)) coeftest(fm_nbin) @ Here, the estimated parameters are very similar to those from the poisson regression and the $z$~tests lead to the same conclusions as in the previous two examples. More details on various techniques for count data regression in \proglang{R} are provided in \cite{hac:Zeileis+Kleiber+Jackman:2008}. \subsection{Probit and tobit models} In this section, we consider an example from \citet[Section~22.3.6]{hac:Greene:2003} that reproduces the analysis of extramarital affairs by \citet{hac:Fair:1978}. The data, famously known as Fair's affairs, is available in the \pkg{AER} package \citep{hac:Kleiber+Zeileis:2008} and provides cross-section information on the number of extramarital affairs of 601 individuals along with several covariates such as \code{age}, \code{yearsmarried}, \code{religiousness}, \code{occupation} and a self-\code{rating} of the marriage. Table~22.3 in \cite{hac:Greene:2003} provides the parameter estimates and corresponding standard errors of a tobit model (for the number of affairs) and a probit model (for infidelity as a binary variable). In \proglang{R}, these models can be fitted using \code{tobit()} from \pkg{AER} \citep[a convenience interface to \code{survreg()} from the \pkg{survival} package by][]{hac:Thernau+Lumley:2008} and \code{glm()}, respectively: <<>>= library("AER") data("Affairs", package = "AER") fm_tobit <- tobit(affairs ~ age + yearsmarried + religiousness + occupation + rating, data = Affairs) fm_probit <- glm(I(affairs > 0) ~ age + yearsmarried + religiousness + occupation + rating, data = Affairs, family = binomial(link = "probit")) @ Using \code{coeftest()}, we compare the usual summary based on the standard errors as computed by \code{vcov()} \citep[which reproduces the results in][]{hac:Greene:2003} and compare them to the HC standard errors provided by \code{sandwich()}. <<>>= coeftest(fm_tobit) coeftest(fm_tobit, vcov = sandwich) @ For the tobit model \code{fm_tobit}, the HC standard errors are only slightly different and yield qualitatively identical results. The picture is similar for the probit model \code{fm_probit} which leads to the same interpretations, both for the standard and the HC estimate. <<>>= coeftest(fm_probit) coeftest(fm_probit, vcov = sandwich) @ See \cite{hac:Greene:2003} for a more detailed discussion of these and other regression models for Fair's affairs data. \section{Discussion} \label{sec:disc} Object-oriented computational infrastructure in the \proglang{R} package \pkg{sandwich} for estimating sandwich covariance matrices in a wide class of parametric models is suggested. Re-using existing building blocks, all an \proglang{R} developer has to implement for adapting a new fitted model class to the sandwich estimators are methods for extracting a bread estimator and the empirical estimating functions (and possibly model matrix and hat values). Although the most important area of application of sandwich covariance matrices is inference, particularly restriction testing, the package \pkg{sandwich} does not contain any inference functions but rather aims at providing modular building blocks that can be re-used in or supplied to other computational tools. In this paper, we show how the \pkg{sandwich} functions can be plugged into some functions made available by other packages that implement tools for Wald tests. However, it should be pointed out that this is not the only strategy for employing sandwich covariances for restriction testing; recent research provides us with at least two other promising strategies: For cross-section data, \cite{hac:Godfrey:2006} shows that the finite sample performance of quasi $t$ or $z$~tests can be improved by computing HC estimators based on the residuals of the restricted model and assessing their significance based on their bootstrap distribution. For time-series data, \cite{hac:Kiefer+Vogelsang:2002} consider $t$-type statistics based on HAC estimators where the bandwidth is equal to the sample size, leading to a non-normal asymptotic distribution of the $t$ statistic. For both strategies, some tools from \pkg{sandwich} could be easily re-used but further infrastructure, in particular for the inference, is required. As this is beyond the scope of the \pkg{sandwich} package, we leave this for future developments in packages focused on inference in regression models. As the new tools in \pkg{sandwich} provide ``robust'' covariances for a wide class of parametric models, it is worth pointing out that this should \emph{not} encourage the user to employ them automatically for every model in every analysis. First, the use of sandwich estimators when the model is correctly specified leads to a loss of power. Second, if the model is not correctly specified, the sandwich estimators are only useful if the parameters estimates are still consistent, i.e., if the misspecification does not result in bias. Whereas it is well understood what types of misspecification can be dealt with in linear regression models, the situation is less obvious for general regression models. Some further expository discussion of this issue for ML and quasi ML estimators can be found in \cite{hac:Freedman:2006} and \cite{hac:Koenker:2006}. \section*{Acknowledgments} The extensions of \pkg{sandwich}, in particular to microeconometric models, was motivated by the joint work with Christian Kleiber on \cite{hac:Kleiber+Zeileis:2008}. We would also like to thank Henric Nilsson for helpful feedback and discussions that helped to improve and generalize the functions in the package. Furthermore, we gratefully acknowledge the valuable comments of the associate editor and two referees which led to an improvement of the paper. \bibliography{hac} \end{document} sandwich/inst/doc/sandwich-OOP.R0000644000175400001440000000647412600735100016352 0ustar zeileisusers### R code from vignette source 'sandwich-OOP.Rnw' ################################################### ### code chunk number 1: preliminaries ################################################### library("AER") library("MASS") options(prompt = "R> ", continue = "+ ") ################################################### ### code chunk number 2: sandwich ################################################### par(mar = rep(0, 4)) plot(0, 0, xlim = c(0, 85), ylim = c(0, 110), type = "n", axes = FALSE, xlab = "", ylab = "") lgrey <- grey(0.88) dgrey <- grey(0.75) rect(45, 90, 70, 110, lwd = 2, col = dgrey) rect(20, 40, 40, 60, col = lgrey) rect(30, 40, 40, 60, col = dgrey) rect(20, 40, 40, 60, lwd = 2) rect(5, 0, 20, 20, lwd = 2, col = lgrey) rect(22.5, 0, 37.5, 20, lwd = 2, col = lgrey) rect(40, 0, 55, 20, lwd = 2, col = lgrey) rect(40, 0, 55, 20, lwd = 2, col = lgrey) rect(60, 0, 80, 20, col = lgrey) rect(70, 0, 80, 20, col = dgrey) rect(60, 0, 80, 20, lwd = 2) text(57.5, 100, "fitted model object\n(class: foo)") text(25, 50, "estfun") text(35, 50, "foo") text(12.5, 10, "meatHC") text(30, 10, "meatHAC") text(47.5, 10, "meat") text(65, 10, "bread") text(75, 10, "foo") arrows(57.5, 89, 70, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(57.5, 89, 30, 61, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 30, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 12.5, 21, lwd = 1.5, length = 0.15, angle = 20) arrows(30, 39, 47.5, 21, lwd = 1.5, length = 0.15, angle = 20) ################################################### ### code chunk number 3: dgp ################################################### set.seed(123) x <- rnorm(250) y <- rnbinom(250, mu = exp(1 + x), size = 1) ################################################### ### code chunk number 4: poisson ################################################### fm_pois <- glm(y ~ x + I(x^2), family = poisson) coeftest(fm_pois) ################################################### ### code chunk number 5: poisson-sandwich ################################################### coeftest(fm_pois, vcov = sandwich) ################################################### ### code chunk number 6: quasipoisson ################################################### fm_qpois <- glm(y ~ x + I(x^2), family = quasipoisson) coeftest(fm_qpois) ################################################### ### code chunk number 7: negbin ################################################### fm_nbin <- glm.nb(y ~ x + I(x^2)) coeftest(fm_nbin) ################################################### ### code chunk number 8: sandwich-OOP.Rnw:626-631 ################################################### library("AER") data("Affairs", package = "AER") fm_tobit <- tobit(affairs ~ age + yearsmarried + religiousness + occupation + rating, data = Affairs) fm_probit <- glm(I(affairs > 0) ~ age + yearsmarried + religiousness + occupation + rating, data = Affairs, family = binomial(link = "probit")) ################################################### ### code chunk number 9: sandwich-OOP.Rnw:638-640 ################################################### coeftest(fm_tobit) coeftest(fm_tobit, vcov = sandwich) ################################################### ### code chunk number 10: sandwich-OOP.Rnw:648-650 ################################################### coeftest(fm_probit) coeftest(fm_probit, vcov = sandwich) sandwich/inst/doc/sandwich.pdf0000644000175400001440000064224112600735105016272 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4369 /Filter /FlateDecode /N 88 /First 725 >> stream x\ks۶ |;͜)A܁3g:8&;IvAh,)"9أI,v{,Yi&bieVX3g ,D΄#P>q&Dwä d S{<AYL`ڰ[oYD;Ǥ< 4o <:3@].5"r`P)zƃl01! 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You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > pkgname <- "sandwich" > source(file.path(R.home("share"), "R", "examples-header.R")) > options(warn = 1) > library('sandwich') > > base::assign(".oldSearch", base::search(), pos = 'CheckExEnv') > cleanEx() > nameEx("Investment") > ### * Investment > > flush(stderr()); flush(stdout()) > > ### Name: Investment > ### Title: US Investment Data > ### Aliases: Investment > ### Keywords: datasets > > ### ** Examples > > ## Willam H. Greene, Econometric Analysis, 2nd Ed. > ## Chapter 15 > ## load data set, p. 411, Table 15.1 > data(Investment) > > ## fit linear model, p. 412, Table 15.2 > fm <- lm(RealInv ~ RealGNP + RealInt, data = Investment) > summary(fm) Call: lm(formula = RealInv ~ RealGNP + RealInt, data = Investment) Residuals: Min 1Q Median 3Q Max -34.987 -6.638 0.180 10.408 26.288 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -12.53360 24.91527 -0.503 0.622 RealGNP 0.16914 0.02057 8.224 3.87e-07 *** RealInt -1.00144 2.36875 -0.423 0.678 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 17.21 on 16 degrees of freedom (1 observation deleted due to missingness) Multiple R-squared: 0.8141, Adjusted R-squared: 0.7908 F-statistic: 35.03 on 2 and 16 DF, p-value: 1.429e-06 > > ## visualize residuals, p. 412, Figure 15.1 > plot(ts(residuals(fm), start = 1964), + type = "b", pch = 19, ylim = c(-35, 35), ylab = "Residuals") > sigma <- sqrt(sum(residuals(fm)^2)/fm$df.residual) ## maybe used df = 26 instead of 16 ?? > abline(h = c(-2, 0, 2) * sigma, lty = 2) > > if(require(lmtest)) { + ## Newey-West covariances, Example 15.3 + coeftest(fm, vcov = NeweyWest(fm, lag = 4)) + ## Note, that the following is equivalent: + coeftest(fm, vcov = kernHAC(fm, kernel = "Bartlett", bw = 5, prewhite = FALSE, adjust = FALSE)) + + ## Durbin-Watson test, p. 424, Example 15.4 + dwtest(fm) + + ## Breusch-Godfrey test, p. 427, Example 15.6 + bgtest(fm, order = 4) + } Loading required package: lmtest Loading required package: zoo Attaching package: 'zoo' The following objects are masked from 'package:base': as.Date, as.Date.numeric Breusch-Godfrey test for serial correlation of order up to 4 data: fm LM test = 12.07, df = 4, p-value = 0.01684 > > ## visualize fitted series > plot(Investment[, "RealInv"], type = "b", pch = 19, ylab = "Real investment") > lines(ts(fitted(fm), start = 1964), col = 4) > > > ## 3-d visualization of fitted model > if(require(scatterplot3d)) { + s3d <- scatterplot3d(Investment[,c(5,7,6)], + type = "b", angle = 65, scale.y = 1, pch = 16) + s3d$plane3d(fm, lty.box = "solid", col = 4) + } Loading required package: scatterplot3d > > > > cleanEx() detaching 'package:scatterplot3d', 'package:lmtest', 'package:zoo' > nameEx("NeweyWest") > ### * NeweyWest > > flush(stderr()); flush(stdout()) > > ### Name: NeweyWest > ### Title: Newey-West HAC Covariance Matrix Estimation > ### Aliases: bwNeweyWest NeweyWest > ### Keywords: regression ts > > ### ** Examples > > ## fit investment equation > data(Investment) > fm <- lm(RealInv ~ RealGNP + RealInt, data = Investment) > > ## Newey & West (1994) compute this type of estimator > NeweyWest(fm) (Intercept) RealGNP RealInt (Intercept) 594.1004817 -0.5617817294 36.04992496 RealGNP -0.5617817 0.0005563172 -0.04815937 RealInt 36.0499250 -0.0481593694 13.24912546 > > ## The Newey & West (1987) estimator requires specification > ## of the lag and suppression of prewhitening > NeweyWest(fm, lag = 4, prewhite = FALSE) (Intercept) RealGNP RealInt (Intercept) 359.4170681 -0.3115505035 -4.089319305 RealGNP -0.3115505 0.0002805888 -0.005355931 RealInt -4.0893193 -0.0053559312 11.171472998 > > ## bwNeweyWest() can also be passed to kernHAC(), e.g. > ## for the quadratic spectral kernel > kernHAC(fm, bw = bwNeweyWest) (Intercept) RealGNP RealInt (Intercept) 794.986166 -0.7562570101 48.19485118 RealGNP -0.756257 0.0007537517 -0.06485461 RealInt 48.194851 -0.0648546058 17.58798679 > > > > cleanEx() > nameEx("PublicSchools") > ### * PublicSchools > > flush(stderr()); flush(stdout()) > > ### Name: PublicSchools > ### Title: US Expenditures for Public Schools > ### Aliases: PublicSchools > ### Keywords: datasets > > ### ** Examples > > ## Willam H. Greene, Econometric Analysis, 2nd Ed. > ## Chapter 14 > ## load data set, p. 385, Table 14.1 > data(PublicSchools) > > ## omit NA in Wisconsin and scale income > ps <- na.omit(PublicSchools) > ps$Income <- ps$Income * 0.0001 > > ## fit quadratic regression, p. 385, Table 14.2 > fmq <- lm(Expenditure ~ Income + I(Income^2), data = ps) > summary(fmq) Call: lm(formula = Expenditure ~ Income + I(Income^2), data = ps) Residuals: Min 1Q Median 3Q Max -160.709 -36.896 -4.551 37.290 109.729 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 832.9 327.3 2.545 0.01428 * Income -1834.2 829.0 -2.213 0.03182 * I(Income^2) 1587.0 519.1 3.057 0.00368 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 56.68 on 47 degrees of freedom Multiple R-squared: 0.6553, Adjusted R-squared: 0.6407 F-statistic: 44.68 on 2 and 47 DF, p-value: 1.345e-11 > > ## compare standard and HC0 standard errors > ## p. 391, Table 14.3 > library(sandwich) > coef(fmq) (Intercept) Income I(Income^2) 832.9144 -1834.2029 1587.0423 > sqrt(diag(vcovHC(fmq, type = "const"))) (Intercept) Income I(Income^2) 327.2925 828.9855 519.0768 > sqrt(diag(vcovHC(fmq, type = "HC0"))) (Intercept) Income I(Income^2) 460.8917 1243.0430 829.9927 > > > if(require(lmtest)) { + ## compare t ratio + coeftest(fmq, vcov = vcovHC(fmq, type = "HC0")) + + ## White test, p. 393, Example 14.5 + wt <- lm(residuals(fmq)^2 ~ poly(Income, 4), data = ps) + wt.stat <- summary(wt)$r.squared * nrow(ps) + c(wt.stat, pchisq(wt.stat, df = 3, lower = FALSE)) + + ## Bresch-Pagan test, p. 395, Example 14.7 + bptest(fmq, studentize = FALSE) + bptest(fmq) + + ## Francisco Cribari-Neto, Asymptotic Inference, CSDA 45 + ## quasi z-tests, p. 229, Table 8 + ## with Alaska + coeftest(fmq, df = Inf)[3,4] + coeftest(fmq, df = Inf, vcov = vcovHC(fmq, type = "HC0"))[3,4] + coeftest(fmq, df = Inf, vcov = vcovHC(fmq, type = "HC3"))[3,4] + coeftest(fmq, df = Inf, vcov = vcovHC(fmq, type = "HC4"))[3,4] + ## without Alaska (observation 2) + fmq1 <- lm(Expenditure ~ Income + I(Income^2), data = ps[-2,]) + coeftest(fmq1, df = Inf)[3,4] + coeftest(fmq1, df = Inf, vcov = vcovHC(fmq1, type = "HC0"))[3,4] + coeftest(fmq1, df = Inf, vcov = vcovHC(fmq1, type = "HC3"))[3,4] + coeftest(fmq1, df = Inf, vcov = vcovHC(fmq1, type = "HC4"))[3,4] + } Loading required package: lmtest Loading required package: zoo Attaching package: 'zoo' The following objects are masked from 'package:base': as.Date, as.Date.numeric [1] 0.8923303 > > ## visualization, p. 230, Figure 1 > plot(Expenditure ~ Income, data = ps, + xlab = "per capita income", + ylab = "per capita spending on public schools") > inc <- seq(0.5, 1.2, by = 0.001) > lines(inc, predict(fmq, data.frame(Income = inc)), col = 4) > fml <- lm(Expenditure ~ Income, data = ps) > abline(fml) > text(ps[2,2], ps[2,1], rownames(ps)[2], pos = 2) > > > > cleanEx() detaching 'package:lmtest', 'package:zoo' > nameEx("bread") > ### * bread > > flush(stderr()); flush(stdout()) > > ### Name: bread > ### Title: Bread for Sandwiches > ### Aliases: bread bread.lm bread.mlm bread.survreg bread.coxph bread.gam > ### bread.nls bread.rlm bread.hurdle bread.zeroinfl bread.mlogit > ### bread.polr bread.clm > ### Keywords: regression > > ### ** Examples > > ## linear regression > x <- sin(1:10) > y <- rnorm(10) > fm <- lm(y ~ x) > > ## bread: n * (x'x)^{-1} > bread(fm) (Intercept) x (Intercept) 1.0414689 -0.2938577 x -0.2938577 2.0823419 > solve(crossprod(cbind(1, x))) * 10 x 1.0414689 -0.2938577 x -0.2938577 2.0823419 > > > > cleanEx() > nameEx("estfun") > ### * estfun > > flush(stderr()); flush(stdout()) > > ### Name: estfun > ### Title: Extract Empirical Estimating Functions > ### Aliases: estfun estfun.lm estfun.glm estfun.mlm estfun.rlm estfun.polr > ### estfun.clm estfun.survreg estfun.coxph estfun.nls estfun.hurdle > ### estfun.zeroinfl estfun.mlogit > ### Keywords: regression > > ### ** Examples > > ## linear regression > x <- sin(1:10) > y <- rnorm(10) > fm <- lm(y ~ x) > > ## estimating function: (y - x'beta) * x > estfun(fm) (Intercept) x 1 -0.68507480 -0.57647056 2 0.13214846 0.12016225 3 -0.96783127 -0.13658036 4 1.36873882 -1.03586495 5 0.08173006 -0.07837294 6 -0.99685418 0.27853651 7 0.40942540 0.26898700 8 0.69524135 0.68784276 9 0.47205088 0.19454089 10 -0.50957471 0.27721940 > residuals(fm) * cbind(1, x) x [1,] -0.68507480 -0.57647056 [2,] 0.13214846 0.12016225 [3,] -0.96783127 -0.13658036 [4,] 1.36873882 -1.03586495 [5,] 0.08173006 -0.07837294 [6,] -0.99685418 0.27853651 [7,] 0.40942540 0.26898700 [8,] 0.69524135 0.68784276 [9,] 0.47205088 0.19454089 [10,] -0.50957471 0.27721940 > > > > cleanEx() > nameEx("isoacf") > ### * isoacf > > flush(stderr()); flush(stdout()) > > ### Name: isoacf > ### Title: Isotonic Autocorrelation Function > ### Aliases: isoacf pava.blocks > ### Keywords: regression ts > > ### ** Examples > > x <- filter(rnorm(100), 0.9, "recursive") > isoacf(x) [1] 1.00000000 0.75620784 0.52668286 0.31877074 0.17874234 0.10451987 [7] 0.07597397 0.07597397 0.07054562 0.03324149 -0.02266489 -0.02266489 [13] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [19] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [25] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [31] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [37] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [43] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [49] -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 -0.02266489 [55] -0.03242424 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 [61] -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 [67] -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 [73] -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 [79] -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 [85] -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03500610 -0.03924011 [91] -0.03924011 -0.03924011 -0.03924011 -0.03924011 -0.03924011 -0.03924011 [97] -0.03924011 -0.03924011 -0.03924011 -0.03924011 > acf(x, plot = FALSE)$acf , , 1 [,1] [1,] 1.00000000 [2,] 0.75620784 [3,] 0.52668286 [4,] 0.31877074 [5,] 0.17874234 [6,] 0.10451987 [7,] 0.06774750 [8,] 0.08420043 [9,] 0.07054562 [10,] 0.03324149 [11,] -0.02547696 [12,] -0.08386780 [13,] -0.12702588 [14,] -0.15733924 [15,] -0.22570274 [16,] -0.27858103 [17,] -0.32634007 [18,] -0.31457877 [19,] -0.32132555 [20,] -0.32323138 [21,] -0.28412580 > > > > cleanEx() > nameEx("kweights") > ### * kweights > > flush(stderr()); flush(stdout()) > > ### Name: kweights > ### Title: Kernel Weights > ### Aliases: kweights > ### Keywords: regression ts > > ### ** Examples > > curve(kweights(x, kernel = "Quadratic", normalize = TRUE), + from = 0, to = 3.2, xlab = "x", ylab = "k(x)") > curve(kweights(x, kernel = "Bartlett", normalize = TRUE), + from = 0, to = 3.2, col = 2, add = TRUE) > curve(kweights(x, kernel = "Parzen", normalize = TRUE), + from = 0, to = 3.2, col = 3, add = TRUE) > curve(kweights(x, kernel = "Tukey", normalize = TRUE), + from = 0, to = 3.2, col = 4, add = TRUE) > curve(kweights(x, kernel = "Truncated", normalize = TRUE), + from = 0, to = 3.2, col = 5, add = TRUE) > > > > cleanEx() > nameEx("lrvar") > ### * lrvar > > flush(stderr()); flush(stdout()) > > ### Name: lrvar > ### Title: Long-Run Variance of the Mean > ### Aliases: lrvar > ### Keywords: regression ts > > ### ** Examples > > set.seed(1) > ## iid series (with variance of mean 1/n) > ## and Andrews kernel HAC (with prewhitening) > x <- rnorm(100) > lrvar(x) [1] 0.007958048 > > ## analogous multivariate case with Newey-West estimator (without prewhitening) > y <- matrix(rnorm(200), ncol = 2) > lrvar(y, type = "Newey-West", prewhite = FALSE) [,1] [,2] [1,] 0.0097884718 0.0005978738 [2,] 0.0005978738 0.0073428222 > > ## AR(1) series with autocorrelation 0.9 > z <- filter(rnorm(100), 0.9, method = "recursive") > lrvar(z) [1] 0.4385546 > > > > cleanEx() > nameEx("meat") > ### * meat > > flush(stderr()); flush(stdout()) > > ### Name: meat > ### Title: A Simple Meat Matrix Estimator > ### Aliases: meat > ### Keywords: regression > > ### ** Examples > > x <- sin(1:10) > y <- rnorm(10) > fm <- lm(y ~ x) > > meat(fm) (Intercept) x (Intercept) 0.54308202 -0.06199868 x -0.06199868 0.21823310 > meatHC(fm, type = "HC") (Intercept) x (Intercept) 0.54308202 -0.06199868 x -0.06199868 0.21823310 > meatHAC(fm) (Intercept) x (Intercept) 0.32259620 0.08446047 x 0.08446047 0.37529225 > > > > cleanEx() > nameEx("sandwich") > ### * sandwich > > flush(stderr()); flush(stdout()) > > ### Name: sandwich > ### Title: Making Sandwiches with Bread and Meat > ### Aliases: sandwich > ### Keywords: regression > > ### ** Examples > > x <- sin(1:10) > y <- rnorm(10) > fm <- lm(y ~ x) > > sandwich(fm) (Intercept) x (Intercept) 0.06458514 -0.04395562 x -0.04395562 0.10690628 > vcovHC(fm, type = "HC") (Intercept) x (Intercept) 0.06458514 -0.04395562 x -0.04395562 0.10690628 > > > > cleanEx() > nameEx("vcovHAC") > ### * vcovHAC > > flush(stderr()); flush(stdout()) > > ### Name: vcovHAC > ### Title: Heteroskedasticity and Autocorrelation Consistent (HAC) > ### Covariance Matrix Estimation > ### Aliases: vcovHAC vcovHAC.default meatHAC > ### Keywords: regression ts > > ### ** Examples > > x <- sin(1:100) > y <- 1 + x + rnorm(100) > fm <- lm(y ~ x) > vcovHAC(fm) (Intercept) x (Intercept) 0.008125428 -0.002043239 x -0.002043239 0.018939164 > vcov(fm) (Intercept) x (Intercept) 8.124921e-03 2.055475e-05 x 2.055475e-05 1.616308e-02 > > > > cleanEx() > nameEx("vcovHC") > ### * vcovHC > > flush(stderr()); flush(stdout()) > > ### Name: vcovHC > ### Title: Heteroskedasticity-Consistent Covariance Matrix Estimation > ### Aliases: vcovHC vcovHC.default vcovHC.mlm meatHC > ### Keywords: regression ts > > ### ** Examples > > ## generate linear regression relationship > ## with homoskedastic variances > x <- sin(1:100) > y <- 1 + x + rnorm(100) > ## model fit and HC3 covariance > fm <- lm(y ~ x) > vcovHC(fm) (Intercept) x (Intercept) 0.008318070 -0.002037159 x -0.002037159 0.019772693 > ## usual covariance matrix > vcovHC(fm, type = "const") (Intercept) x (Intercept) 8.124921e-03 2.055475e-05 x 2.055475e-05 1.616308e-02 > vcov(fm) (Intercept) x (Intercept) 8.124921e-03 2.055475e-05 x 2.055475e-05 1.616308e-02 > > sigma2 <- sum(residuals(lm(y ~ x))^2)/98 > sigma2 * solve(crossprod(cbind(1, x))) x 8.124921e-03 2.055475e-05 x 2.055475e-05 1.616308e-02 > > > > cleanEx() > nameEx("vcovOPG") > ### * vcovOPG > > flush(stderr()); flush(stdout()) > > ### Name: vcovOPG > ### Title: Outer Product of Gradients Covariance Matrix Estimation > ### Aliases: vcovOPG > ### Keywords: regression ts > > ### ** Examples > > ## generate poisson regression relationship > x <- sin(1:100) > y <- rpois(100, exp(1 + x)) > ## compute usual covariance matrix of coefficient estimates > fm <- glm(y ~ x, family = poisson) > vcov(fm) (Intercept) x (Intercept) 0.004526581 -0.003679570 x -0.003679570 0.008110051 > vcovOPG(fm) (Intercept) x (Intercept) 0.005183615 -0.003086646 x -0.003086646 0.009584083 > > > > cleanEx() > nameEx("weightsAndrews") > ### * weightsAndrews > > flush(stderr()); flush(stdout()) > > ### Name: weightsAndrews > ### Title: Kernel-based HAC Covariance Matrix Estimation > ### Aliases: weightsAndrews bwAndrews kernHAC > ### Keywords: regression ts > > ### ** Examples > > curve(kweights(x, kernel = "Quadratic", normalize = TRUE), + from = 0, to = 3.2, xlab = "x", ylab = "k(x)") > curve(kweights(x, kernel = "Bartlett", normalize = TRUE), + from = 0, to = 3.2, col = 2, add = TRUE) > curve(kweights(x, kernel = "Parzen", normalize = TRUE), + from = 0, to = 3.2, col = 3, add = TRUE) > curve(kweights(x, kernel = "Tukey", normalize = TRUE), + from = 0, to = 3.2, col = 4, add = TRUE) > curve(kweights(x, kernel = "Truncated", normalize = TRUE), + from = 0, to = 3.2, col = 5, add = TRUE) > > ## fit investment equation > data(Investment) > fm <- lm(RealInv ~ RealGNP + RealInt, data = Investment) > > ## compute quadratic spectral kernel HAC estimator > kernHAC(fm) (Intercept) RealGNP RealInt (Intercept) 788.6120652 -0.7502080996 49.78912814 RealGNP -0.7502081 0.0007483977 -0.06641343 RealInt 49.7891281 -0.0664134303 17.71735491 > kernHAC(fm, verbose = TRUE) Bandwidth chosen: 1.744749 (Intercept) RealGNP RealInt (Intercept) 788.6120652 -0.7502080996 49.78912814 RealGNP -0.7502081 0.0007483977 -0.06641343 RealInt 49.7891281 -0.0664134303 17.71735491 > > ## use Parzen kernel instead, VAR(2) prewhitening, no finite sample > ## adjustment and Newey & West (1994) bandwidth selection > kernHAC(fm, kernel = "Parzen", prewhite = 2, adjust = FALSE, + bw = bwNeweyWest, verbose = TRUE) Bandwidth chosen: 2.814444 (Intercept) RealGNP RealInt (Intercept) 608.3101258 -0.5089107386 -64.93690203 RealGNP -0.5089107 0.0004340803 0.04689293 RealInt -64.9369020 0.0468929322 15.58251456 > > ## compare with estimate under assumption of spheric errors > vcov(fm) (Intercept) RealGNP RealInt (Intercept) 620.7706170 -0.5038304429 8.47475285 RealGNP -0.5038304 0.0004229789 -0.01145679 RealInt 8.4747529 -0.0114567949 5.61097245 > > > > cleanEx() > nameEx("weightsLumley") > ### * weightsLumley > > flush(stderr()); flush(stdout()) > > ### Name: weightsLumley > ### Title: Weighted Empirical Adaptive Variance Estimation > ### Aliases: weightsLumley weave > ### Keywords: regression ts > > ### ** Examples > > x <- sin(1:100) > y <- 1 + x + rnorm(100) > fm <- lm(y ~ x) > weave(fm) (Intercept) x (Intercept) 0.007957440 -0.001936926 x -0.001936926 0.018775226 > vcov(fm) (Intercept) x (Intercept) 8.124921e-03 2.055475e-05 x 2.055475e-05 1.616308e-02 > > > > ### *