sandwich/0000755000175400001440000000000013452220465012250 5ustar zeileisuserssandwich/inst/0000755000175400001440000000000013452214377013232 5ustar zeileisuserssandwich/inst/CITATION0000644000175400001440000000376713330034601014365 0ustar zeileisusersbibentry("Article", title = "Econometric Computing with {HC} and {HAC} Covariance Matrix Estimators", author = person(given = "Achim", family = "Zeileis", email = "Achim.Zeileis@R-project.org", comment = c(ORCID = "0000-0003-0918-3766")), journal = "Journal of Statistical Software", year = "2004", volume = "11", number = "10", pages = "1--17", doi = "10.18637/jss.v011.i10", header = "To cite sandwich in publications use:" ) bibentry("Article", title = "Object-Oriented Computation of Sandwich Estimators", author = person(given = "Achim", family = "Zeileis", email = "Achim.Zeileis@R-project.org", comment = c(ORCID = "0000-0003-0918-3766")), journal = "Journal of Statistical Software", year = "2006", volume = "16", number = "9", pages = "1--16", doi = "10.18637/jss.v016.i09", header = "If sandwich is applied to models other than lm(), please also cite:" ) bibentry("TechReport", title = "Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in {R}", author = c(person(given = "Susanne", family = "Berger"), person(given = "Nathaniel", family = "Graham"), person(given = "Achim", family = "Zeileis", email = "Achim.Zeileis@R-project.org", comment = c(ORCID = "0000-0003-0918-3766"))), institution = "Working Papers in Economics and Statistics, Research Platform Empirical and Experimental Economics, Universit\\\"at Innsbruck", year = "2017", type = "Working Paper", number = "2017-12", month = "July", url = "http://EconPapers.RePEc.org/RePEc:inn:wpaper:2017-12", header = "If clustered covariances are used, please also cite:" ) sandwich/inst/doc/0000755000175400001440000000000013452214377013777 5ustar zeileisuserssandwich/inst/doc/sandwich-OOP.pdf0000644000175400001440000046104413452214402016723 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 3650 /Filter /FlateDecode /N 61 /First 492 >> stream x[Ys۶~oMm yqF΢F,TI;Y"HYs[`6.s;L0L20\bl|3g:~i~̔ e` dcehv.̱$-rK/Dgd$b`Q8&<&=>s-g`HgoH< c`oh1_x`+%shȦ3ezƄ胦pJPix.ƀ0mT* MexMђ +l*HCGƂ AHwpEslNʶ "mc4bf@v1I ? `d6V~:[D0Ţ7y&d|5{ZYg8a/v&hɾ"җhAū ًbE|1e[lX?OE;"p6l$$q'Ӕ߃$ V-,Xfђ^N)֫ ?I$ Wb6ճxz_I<]OB0vtyʎq$ɘʒhp_>vz l/^ެu,`5&svP2ƽxHEIݣ9ԯ~obꋥ.^W-u/е)2(VX/zRC-g4 v;0I/^A <\ ^(f: Q;ˎ2zOAy~.nI< 3P'@:k)ÌFI@)ʞ>Q>y+p_bA|SgR9_Ue9AQ |yʙSL<3Y9 8*8LX5΄4g嬾vϠ|GO_%_ǫp(Wä{KMӧ2oxW*1dT3Q8T8R\{O) x egzB9&pg#z9[wO2O0 (>+x震+}Jse}UWC*'WxR%Qn opB5(ntuW=E~GdNcT7;Lu.nujYm]?ظCwcпjX*eh3RfT`Rо- T}[H#hiMk./,OY<,@ Y/KU8egPk^ 2GG7 ]}^{ϔe `XaL!!mϴ&,i8-z:jhHs /0Vq>H |~޿o"RL.TT‚`k>56s^kO):o&l{! hQi,N A(G~ 'IigY(KAF~UYCHQXhot\yFL,T*ٳV)Y_[%FS(cGR P!\3y*B7yaa@xaV:4BUiDIӫ7~[[r78C1ՁW"SɵוvPc}9JJq-tQ:NOi?e8KSB4Άhtt4-E)MG%M ?|tB&s_ƻ&CFa}6rS&VL)SH| *ҷ|IjV@"n?r{F$$$F`Q;_naTS>ݵmIP /W雍b׎Z}tD=BU\C4:y cMe\]_Ww_MVûԂ}u[w n@#]/4Jof k ormZr7P-lOo?Sntv;qU,~QĹźZ4ݠC7yM2ƒ|$UR*%뤛3:}Bk#9pL&uc֬t7dN6&^i7$N1CƦجkO.ϔO(6F_f-&҃cy?׿է[,ʛ-,&a&CB#$Ȩ~<% l.e,ؒG5\Eg,Ê^#aVcIQ>t(;H`<@Gw>\S+i0hfiF`vDǝyjh6krd~xGYGW_KVWeP.Hf虨aig"+3b!u^7O7>4b_zxG:lA8F:Z>zhq﹧hWaQuXݱf&NF2w*):ئlX=;>_f:O%iR,dA6 mKF!?`q.L# HdmD'y.7cm*2endstream endobj 63 0 obj << /Subtype /XML /Type /Metadata /Length 1685 >> stream GPL Ghostscript 9.26 covariance matrix estimators, estimating functions, object orientation, R 2019-04-06T23:24:17+02:00 2019-04-06T23:24:17+02:00 LaTeX with hyperref Object-Oriented Computation of Sandwich EstimatorsAchim Zeileis endstream endobj 64 0 obj << /Type /ObjStm /Length 2185 /Filter /FlateDecode /N 61 /First 503 >> stream xYRI}߯ǙpXu`5^1g6ЈzWHD|V,a͆,TuS'+3ְYDbuLJxe60s薂9Ck˜b84&d:*Q9&4{&,=&'LxL&"0/2 L*, 6e2;{h栐CH)t$~!'(4&BW/%I^(X/(H`FCH$3V + k$1CV2-q&h(Рu`Bɬ#6 f A` RIJR[cL i0 uzO%(@- _?%7 )2MɄ}aCPC⨏iM|*{Y^N22"~f:TXdMKC>/5^Sb\.l%^G [`[phǚshg[ giqnӲTќ.0=IFu mmn2vQdeza h߱%MUzV|S6K˞67޵Mu-(7j1lڇ+G 9pJtȵr*Z'-ZY%H^^9a,Z 'X~-nٖVp6-1s΄Y?.t f qk1yѧ|(ٜjQձӁTk6edxp/Pow}uLwGg5gyy:ovjin_`ң ey" @xthx 6j/Ǔ'lVzXV_ΒnxS mǵmpL˔uafP=u/DW\W\W "/uK~-nahqA|lVaLVvӦA>/γtpjxjxvIQO|"&-o *G0ePG2FJey3G7&|( tO4Ʊܛ(,a>O|yW}Ư>ͧ[9gٔ^2^~bs$A:Iz *A^QܯJ^(dȫ[1ގ$MA<`: yLzyi=7gvvW)H+=Z.P'lH++_@ SPRfnsT;Bx iH!Q8#0[5~hu+IBG@+ROEs-DI$qQOyEA}>7 ?S>gp.+(e0MfSM%G157n 9/[>3~= ~ϿoKHDO502*q uI7u |%LW$0W-YU$WPƋ+u.m'[cӋ2!Z%3?|t4:޼*7D,!^$EM̳BUz&^6vEe1۲Fh岑X߭鿕yEmWuwAߴ 2d4G5feV}|uU+Bَg{?%2U.C7 爾t'\ P+GOLbfnjd#XƶK򘮮6.ΡTwv;el$rW)Vem;q*MdD4~B^U%~! wendstream endobj 126 0 obj << /Filter /FlateDecode /Length 6997 >> stream x\Is\Gr##pPk_&g|h !@\jɪ~9tP^UVV._.~9W{}۳3u~}˙]ܞ uWg>:]v_7U{vo^nwAsNzwxlUFo;79g7Λ :lo0ɛ> Fs򛻭{ ^OmKFWZo3-4ʄͻG9PFS뜓ti:=aW8dzե`xB*d_L۰7!YÏiwHy͘ }Aǁ[r~gZlBR ~Go9 LTȼټ~ClFmJ1t@vHMS< xLD&2p(HN= v2&?P ag>l17n b9h!Q>dF 1|D`=RPnf΢opHNNK$xt0yo i_>@ο%P`nϼ17g?[tpAAX2MENHKXfyfub3_o!.Z9q_}E;yBLud#X g JSƒ29k"E݆016E $8ALwn]Nd2iywr@Q"58{u lME%vlnq$V$sNa uaK9j+K1M`\dE#.HsP. 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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.R0000644000175400001440000002125213452214377015724 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.Rnw0000644000175400001440000010530413452213324016714 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: <>= suppressWarnings(RNGversion("3.5.0")) 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:Therneau:2018} 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-CL.Rnw0000644000175400001440000027527413452213521016572 0ustar zeileisusers\documentclass[nojss]{jss} \usepackage{amsmath,amssymb,bm,thumbpdf,lmodern,soul} %% need no \usepackage{Sweave} \author{Susanne Berger\\Universit\"at Innsbruck \And Nathaniel Graham\\Texas A\&M\\International University \And Achim Zeileis\\Universit\"at Innsbruck} \title{Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in \proglang{R}} \Plainauthor{Susanne Berger, Nathaniel Graham, Achim Zeileis} \Plaintitle{Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R} \Shorttitle{Various Versatile Variances} \Keywords{clustered data, covariance matrix estimator, object orientation, simulation, \proglang{R}} \Plainkeywords{clustered data, covariance matrix estimator, object orientation, simulation, R} \Abstract{ Clustered covariances or clustered standard errors are very widely used to account for correlated or clustered data, especially in economics, political sciences, and other social sciences. They are employed to adjust the inference following estimation of a standard least-squares regression or generalized linear model estimated by maximum likelihood. Although many publications just refer to ``the'' clustered standard errors, there is a surprisingly wide variety of clustered covariances, particularly due to different flavors of bias corrections. Furthermore, while the linear regression model is certainly the most important application case, the same strategies can be employed in more general models (e.g., for zero-inflated, censored, or limited responses). % In \proglang{R}, functions for covariances in clustered or panel models have been somewhat scattered or available only for certain modeling functions, notably the (generalized) linear regression model. In contrast, an object-oriented approach to ``robust'' covariance matrix estimation -- applicable beyond \code{lm()} and \code{glm()} -- is available in the \pkg{sandwich} package but has been limited to the case of cross-section or time series data. Starting with \pkg{sandwich}~2.4.0, this shortcoming has been corrected: Based on methods for two generic functions (\code{estfun()} and \code{bread()}), clustered and panel covariances are provided in \code{vcovCL()}, \code{vcovPL()}, and \code{vcovPC()}. Moreover, clustered bootstrap covariances are provided in \code{vcovBS()}, using model \code{update()} on bootstrap samples. These are directly applicable to models from packages including \pkg{MASS}, \pkg{pscl}, \pkg{countreg}, and \pkg{betareg}, among many others. Some empirical illustrations are provided as well as an assessment of the methods' performance in a simulation study. } \Address{ Susanne Berger, Achim Zeileis\\ Department of Statistics\\ Faculty of Economics and Statistics\\ Universit\"at Innsbruck\\ Universit\"atsstr.~15\\ 6020 Innsbruck, Austria\\ E-mail: \email{Susanne.Berger@student.uibk.ac.at}, \email{Achim.Zeileis@R-project.org}\\ URL: \url{https://www.uibk.ac.at/statistics/personal/berger/},\\ \phantom{URL: }\url{https://eeecon.uibk.ac.at/~zeileis/}\\ Nathaniel Graham\\ Division of International Banking and Finance Studies\\ A.R. Sanchez, Jr.\ School of Business \\ Texas A\&M International University\\ 5201 University Boulevard\\ Laredo, Texas 78041, United States of America\\ E-mail: \email{npgraham1@npgraham1.com}\\ URL: \url{https://sites.google.com/site/npgraham1/} } \begin{document} \SweaveOpts{engine=R,eps=FALSE} %\VignetteIndexEntry{Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R} %\VignetteDepends{sandwich,geepack,lattice,lmtest,multiwayvcov,pcse,plm,pscl} %\VignetteKeywords{clustered data, clustered covariance matrix estimators, object orientation, simulation, R} %\VignettePackage{sandwich} <>= library("sandwich") library("geepack") library("lattice") library("lmtest") library("multiwayvcov") library("pcse") library("plm") library("pscl") panel.xyref <- function(x, y, ...) { panel.abline(h = 0.95, col = "slategray") panel.xyplot(x, y, ...) } se <- function(vcov) sapply(vcov, function(x) sqrt(diag(x))) options(prompt = "R> ", continue = "+ ", digits = 5) if(file.exists("sim-CL.rda")) { load("sim-CL.rda") } else { source("sim-CL.R") } @ \section{Introduction} \label{sec:intro} Observations with correlations between objects of the same group/cluster are often referred to as ``cluster-correlated'' observations. Each cluster comprises multiple objects that are correlated within, but not across, clusters, leading to a nested or hierarchical structure \citep{hac:Galbraith+Daniel+Vissel:2010}. Ignoring this dependency and pretending observations are independent not only across but also within the clusters, still leads to parameter estimates that are consistent (albeit not efficient) in many situations. However, the observations' information will typically be overestimated and hence lead to overstated precision of the parameter estimates and inflated type~I errors in the corresponding tests \citep{hac:Moulton:1986, hac:Moulton:1990}. Therefore, clustered covariances are widely used to account for clustered correlations in the data. Such clustering effects can emerge both in cross-section and in panel (or longitudinal) data. Typical examples for clustered cross-section data include firms within the same industry or students within the same school or class. In panel data, a common source of clustering is that observations for the same individual at different time points are correlated while the individuals may be independent \citep{hac:Cameron+Miller:2015}. This paper contributes to the literature particularly in two respects: % (1)~Most importantly, we discuss a set of computational tools for the \proglang{R} system for statistical computing \citep{hac:R:2018}, providing an object-oriented implementation of clustered covariances/standard errors in the \proglang{R} package \pkg{sandwich} \citep{hac:Zeileis:2004a,hac:Zeileis:2006}. Using this infrastructure, sandwich covariances for cross-section or time series data have been available for models beyond \code{lm()} or \code{glm()}, e.g., for packages \pkg{MASS} \citep{hac:Venables+Ripley:2002}, \pkg{pscl}/\pkg{countreg} \citep{hac:Zeileis+Kleiber+Jackman:2008}, and \pkg{betareg} \citep{hac:Cribari-Neto+Zeileis:2010,hac:Gruen+Kosmidis+Zeileis:2012}, among many others. However, corresponding functions for clustered or panel data had not been available in \pkg{sandwich} but have been somewhat scattered or available only for certain modeling functions. (2)~Moreover, we perform a Monte Carlo simulation study for various response distributions with the aim to assess the performance of clustered standard errors beyond \code{lm()} and \code{glm()}. This also includes special cases for which such a finite-sample assessment has not yet been carried out in the literature (to the best of our knowledge). The rest of this manuscript is structured as follows: Section~\ref{sec:overview} discusses the idea of clustered covariances and reviews existing \proglang{R} packages for sandwich as well as clustered covariances. Section~\ref{sec:methods} deals with the theory behind sandwich covariances, especially with respect to clustered covariances for cross-sectional and longitudinal data, clustered data, as well as panel data. Section~\ref{sec:software} then takes a look behind the scenes of the new object-oriented \proglang{R} implementation for clustered covariances, Section~\ref{sec:illu} gives an empirical illustration based on data provided from \cite{hac:Petersen:2009} and \cite{hac:Aghion+VanReenen+Zingales:2013}. The simulation setup and results are discussed in Section~\ref{sec:simulation}. \section{Overview} \label{sec:overview} In the statistics as well as in the econometrics literature a variety of strategies are popular for dealing with clustered dependencies in regression models. Here we give a a brief overview of clustered covariances methods, some background information, as well as a discussion of corresponding software implementations in \proglang{R} and \proglang{Stata}. \subsection{Clustered dependencies in regression models} In the statistics literature, random effects (especially random intercepts) are often introduced to capture unobserved cluster correlations (e.g., using the \pkg{lme4} package in \proglang{R}, \citealp{hac:Bates+Machler+Bolker:2015}). Alternatively, generalized estimating equations (GEE) can account for such correlations by adjusting the model's scores in the estimation \citep{hac:Liang+Zeger:1986}, also leading naturally to a clustered covariance (e.g., available in the \pkg{geepack} package for \proglang{R}, \citealp{hac:Halekoh+Hojsgaard+Yan:2002}). Feasible generalized least squares can be applied in a similar way and is frequently used in panel data econometrics (e.g., available in the \pkg{plm} package, \citealp{hac:Croissant+Millo:2008}). Another approach, widely used in econometrics and the social sciences, is to assume that the model's score function was correctly specified but that only the remaining likelihood was potentially misspecified, e.g., due to a lack of independence as in the case of clustered correlations (see \citealp{hac:White:1994}, for a classic textbook, and \citealp{hac:Freedman:2006}, for a criticial review). This approach leaves the parameter estimator unchanged and is known by different labels in different parts of the literature, e.g., quasi-maximum likelihood (QML), independence working model, or pooling model in ML, GEE, or panel econometrics jargon, respectively. In all of these approaches, only the covariance matrix is adjusted in the subsequent inference by using a sandwich estimator, especially in Wald tests and corresponding confidence intervals. Important special cases of this QML approach combined with sandwich covariances include: (1) independent but heteroscedastic observations necessitating heteroscedasticity-consistent (HC) covariances \citep[see e.g.,][]{hac:Long+Ervin:2000}, (2) autocorrelated time series of observations requiring heteroscedasticity- and autocorrelation-consistent (HAC) covariances \citep[such as][]{hac:Newey+West:1987,hac:Andrews:1991}, and (3) clustered sandwich covariances for clustered or panel data \citep[see e.g.,][]{hac:Cameron+Miller:2015}. \subsection{Clustered covariance methods} In the statistics literature, the basic sandwich estimator has been introduced first for cross-section data with independent but heteroscedastic observations by \cite{hac:Eicker:1963} and \cite{hac:Huber:1967} and was later popularized in the econometrics literature by \cite{hac:White:1980}. Early references for sandwich estimators accounting for correlated but homoscedastic observations are \cite{hac:Kloek:1981} and \cite{hac:Moulton:1990}, assuming a correlation structure equivalent to exchangeable working correlation introduced by \cite{hac:Liang+Zeger:1986} in the more general framework of generalized estimating equations. \cite{hac:Kiefer:1980} has implemented yet another clustered covariance estimator for panel data, which is robust to clustering but assumes homoscedasticity \citep[see also][]{hac:Baum+Nichols+Schaffer:2011}. A further generalization allowing for cluster correlations and heteroscedasticity is provided by the independence working model in GEE \citep{hac:Liang+Zeger:1986}, naturally leading to clustered covariances. In econometrics, \cite{hac:Arellano:1987} first proposed clustered covariances for the fixed effects estimator in panel models. Inference with clustered covariances with one or more cluster dimension(s) is examined in \cite{hac:Cameron+Gelbach+Miller:2011}. In a wider context, \cite{hac:Cameron+Miller:2015} include methods in the discussion where not only inference is adjusted but also estimation altered. This discussion focuses on the ``large $G$'' framework where not only the number of observations~$n$ but also the number of clusters $G$ is assumed to approach infinity. An alternative but less frequently-used asymptotic framework is the ``fixed $G$'' setup, where the number of clusters is assumed to be fixed \citep[see][]{hac:Bester+Conley+Hansen:2011,hac:Hansen+Lee:2017}. However, as this also requires non-standard (non-normal) inference it cannot be combined with standard tests in the same modular way as implemented in \pkg{sandwich} and is not pursued further here. In a recent paper, \cite{hac:Abadie+Athey+Imbens+Wooldridge:2017} survey state-of-the-art methods for clustered covariances with special focus on treatment effects in economic experiments with possibly multiple cluster dimensions. They emphasize that, in terms of effect on the covariance estimation, clustering in the treatment assignment is more important than merely clustering in the sampling. However, if a fixed cluster effect is included in the model, treatment heterogeneity across clusters is a requirement for clustered covariances to be necessary. \subsection[R packages for sandwich covariances]{\proglang{R} packages for sandwich covariances} Various kinds of sandwich covariances have already been implemented in several \proglang{R} packages, with the linear regression case receiving most attention. But some packages also cover more general models. The standard \proglang{R} package for sandwich covariance estimators is the \pkg{sandwich} package \citep{hac:Zeileis:2004a,hac:Zeileis:2006}, which provides an object-oriented implementation for the building blocks of the sandwich that rely only on a small set of extractor functions (\code{estfun()} and \code{bread()}) for fitted model objects. The function \code{sandwich()} computes a plain sandwich estimate \citep{hac:Eicker:1963,hac:Huber:1967,hac:White:1980} from a fitted model object, defaulting to what is known as HC0 or HC1 in linear regression models. \code{vcovHC()} is a wrapper to \code{sandwich()} combined with \code{meatHC()} and \code{bread()} to compute general HC covariances ranging from HC0 to HC5 \citep[see][and the references therein]{hac:Zeileis:2004a}. \code{vcovHAC()}, based on \code{sandwich()} with \code{meatHAC()} and \code{bread()}, computes HAC covariance matrix estimates. Further convenience interfaces \code{kernHAC()} for Andrews' kernel HAC \citep{hac:Andrews:1991} and \code{NeweyWest()} for Newey-West-style HAC \citep{hac:Newey+West:1987,hac:Newey+West:1994} are available. However, in versions prior to 2.4.0 of \pkg{sandwich} no similarly object-oriented approach to clustered sandwich covariances was available. Another \proglang{R} package that includes heteroscedasticity-consistent covariance estimators (HC0--HC4), for models produced by \code{lm()} only, is the \pkg{car} package \citep{hac:Fox+Weisberg:2011} in function \code{hccm()}. Like \code{vcovHC()} from \pkg{sandwich} this is limited to the cross-section case without clustering, though. Covariance estimators in \pkg{car} as well as in \pkg{sandwich} are constructed to insert the resulting covariance matrix estimate in different Wald-type inference functions, as \code{confint()} from \pkg{stats} \citep{hac:R:2018}, \code{coeftest()} and \code{waldtest()} from \pkg{lmtest} \citep{hac:Zeileis+Hothorn:2002}, \code{Anova()}, \code{Confint()}, \code{S()}, \code{linearHypothesis()}, and \code{deltaMethod()} from \pkg{car} \citep{hac:Fox+Weisberg:2011}. \subsection[R packages for clustered covariances]{\proglang{R} packages for clustered covariances} The lack of support for clustered sandwich covariances in standard packages like \pkg{sandwich} or \pkg{car} has led to a number of different implementations scattered over various packages. Typically, these are tied to either objects from \code{lm()} or dedicated model objects fitting certain (generalized) linear models for clustered or panel data. The list of packages includes: \pkg{multiwayvcov} \citep{hac:Graham+Arai+Hagstroemer:2016}, \pkg{plm} \citep{hac:Croissant+Millo:2008}, \pkg{geepack} \citep{hac:Halekoh+Hojsgaard+Yan:2002}, \pkg{lfe} \citep{hac:Gaure:2018}, \pkg{clubSandwich} \citep{hac:Pustejovsky:2018}, and \pkg{clusterSEs} \citep{hac:Esarey:2018}, among others. In \pkg{multiwayvcov}, the implementation was object-oriented, in many aspects building on \pkg{sandwich} infrastructure. However, certain details assumed `\code{lm}' or `\code{glm}'-like objects. In \pkg{plm} and \pkg{lfe} several types of sandwich covariances are available for the packages' \code{plm} (panel linear models) and \code{felm} (fixed-effect linear models), respectively. The \pkg{geepack} package can estimate independence working models for `\code{glm}'-type models, also supporting clustered covariances for the resulting `\code{geeglm}' objects. Finally, \pkg{clusterSEs} and \pkg{clubSandwich} focus on the case of ordinary or weighted least squares regression models. In a nutshell, there is good coverage of clustered covariances for (generalized) linear regression objects albeit potentially necessitating reestimating a certain model using a different model-fitting function/packages. However, there was no object-oriented implementation for clustered covariances in \proglang{R}, that enabled plugging in different model objects from in principle any class. Therefore, starting from the implementation in \pkg{multiwayvcov}, a new and object-oriented implementation was established and integrated in \pkg{sandwich}, allowing application to more general models, including zero-inflated, censored, or limited responses. \subsection[Stata software for clustered covariances]{\proglang{Stata} software for clustered covariances} Base \proglang{Stata} as well as contributed extension modules provide implementations for several types of clustered covariances. One-way clustered covariances are available in base \proglang{Stata} for a wide range of estimators (e.g., \code{reg}, \code{ivreg2}, \code{xtreg}, among others) by adding the \code{cluster} option. Furthermore, \code{suest} provides clustered covariances for seemingly unrelated regression models, \code{xtpcse} supports linear regression with panel-corrected standard errors, and the \code{svy} suite of commands can be employed in the presence of nested multi-level clustering \citep{hac:Nichols+Schaffer:2007}. The contributed \proglang{Stata} module \pkg{avar} \citep{hac:Baum+Schaffer:2013} allows to construct the ``meat'' matrix of various sandwich covariance estimators, including HAC, one- and two-way clustered, and panel covariances ({\`a} la \citealp{hac:Kiefer:1980} or \citealp{hac:Driscoll+Kraay:1998}). An alternative implementation for the latter is available in the contributed \proglang{Stata} module \pkg{xtscc} \citep{hac:Hoechle:2007}. While there is a wide variety of clustered covariances for many regression models in \proglang{Stata}, it is not always possible to fine-tune the flavor of covariance, e.g., with respect to the finite sample adjustment \citep[a common source of confusion and differences across implementations, see e.g.,][]{hac:StackOverflow:2014}. \section{Methods} \label{sec:methods} To establish the theoretical background of sandwich covariances for clustered as well as panel data the notation of \cite{hac:Zeileis:2006} is adopted. Here, the conceptual building blocks from \cite{hac:Zeileis:2006} are briefly repeated and then carried further for clustered covariances. \subsection{Sandwich covariances} Let $(y_{i},x_{i})$ for $i = 1, \ldots, n$ be data with some distribution controlled by a parameter vector $\theta$ with $k$ dimensions. For a wide range of models the (quasi-)maximum likelihood estimator $\hat \theta$ is governed by a central limit theorem \citep{hac:White:1994} so that $\hat \theta \approx \mathcal{N}(\theta, n^{-1} S(\theta))$. Moreover, the covariance matrix is of sandwich type with a meat matrix $M(\theta)$ between two slices of bread $B(\theta)$: \begin{eqnarray} \label{eq:sandwich} S(\theta) & = & B(\theta) \cdot M(\theta) \cdot B(\theta) \\ \label{eq:bread} B(\theta) & = & \left( \E\bigg[ - \frac{\partial \psi(y, x, \theta)}{\partial \theta} \bigg] \right)^{-1} \\ \label{eq:meat} M(\theta) & = & \VAR[ \psi(y, x, \theta) ]. \end{eqnarray} An estimating function \begin{eqnarray} \psi(y, x, \theta) \quad = \quad \frac{\partial \Psi(y, x, \theta)}{\partial \theta} \end{eqnarray} is defined as the derivative of an objective function $\Psi(y, x, \theta)$, typically the log-likelihood, with respect to the parameter vector $\theta$. Thus, an empirical estimating (or score) function evaluates an estimating function at the observed data and the estimated parameters such that an $n \times k$ matrix is obtained \citep{hac:Zeileis:2006}: \begin{eqnarray} \label{eq:estfun} \left( \begin{array}{c} \psi(y_1, x_1, \hat \theta)^\top \\ \vdots \\ \psi(y_n, x_n, \hat \theta)^\top \end{array} \right). \end{eqnarray} The estimate $\hat B$ for the bread is based on second derivatives, i.e., the empirical version of the inverse Hessian \begin{equation} \label{eq:Bhat} \hat B \quad = \quad \left( \frac{1}{n} \sum_{i = 1}^n - \frac{\partial \psi(y, x, \hat \theta)}{\partial \theta} \right)^{-1}, \end{equation} whereas $\hat M, \hat M_\mathrm{HAC}, \hat M_\mathrm{HC}$ compute outer product, HAC and HC estimators for the meat, respectively, \begin{eqnarray} \label{eq:meat-op} \hat M & = & \frac{1}{n} \sum_{i = 1}^n\psi(y_i, x_i, \hat \theta) \psi(y_i, x_i, \hat \theta)^\top \\ \label{eq:meat-hac} \hat M_\mathrm{HAC} & = & \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 \\ \label{eq:meat-hc} \hat M_\mathrm{HC} & = & \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_n, x_n^\top \theta)) \end{array} \right) X. \end{eqnarray} The outer product estimator in Equation~\ref{eq:meat-op} corresponds to the basic sandwich estimator \citep{hac:Eicker:1963,hac:Huber:1967,hac:White:1980}. $w_{|i-j|}$ in Equation~\ref{eq:meat-hac} is a vector of weights \citep{hac:Zeileis:2004a}. In Equation~\ref{eq:meat-hc}, functions $\omega(\cdot)$ derive estimates of the variance of the empirical working residuals $r(y_1, x_1^\top \hat \theta), \ldots, r(y_n, x_n^\top \hat \theta)$ and may also depend on hat values as well as degrees of freedom \citep{hac:Zeileis:2006}. To obtain the HC type estimator in Equation~\ref{eq:meat-hc} it is necessary that the score can be factorized into empirical working residuals times the regressor vector. \begin{equation} \label{eq:fact} \psi(y_i, x_i, \hat\theta) = r(y_i, x_i^\top \hat \theta) \cdot x_i \end{equation} This is, however, only possible in situations where the parameter of the response distribution depends on a single linear predictor (possibly through a link function). The building blocks for the calculation of the sandwich are provided by the \pkg{sandwich} package, where the \code{sandwich()} function calculates an estimator of the sandwich $S(\theta)$ (see Equation~\ref{eq:sandwich}) by multiplying estimators for the meat (from Equation~\ref{eq:meat}) between two slices of bread (from Equation~\ref{eq:bread}). A natural idea for an object-oriented implementation of these estimators is to provide common building blocks, namely a simple \code{bread()} extractor that computes $\hat B$ from Equation~\ref{eq:Bhat} and an \code{estfun()} extractor that returns the empirical estimating functions from Equation~\ref{eq:estfun}. On top of these extractors a number of meat estimators can be defined: \code{meat()} for $\hat M$ from Equation~\ref{eq:meat-op}, \code{meatHAC()} for $\hat M_\mathrm{HAC}$ from Equation~\ref{eq:meat-hac}, and \code{meatHC()} for $\hat M_\mathrm{HC}$ from Equation~\ref{eq:meat-hc}, respectively. In addition to the \code{estfun()} method a \code{model.matrix()} method is needed in \code{meatHC()} for the decomposition of the scores into empirical working residuals and regressor matrix. \subsection{Clustered covariances} For clustered observations, similar ideas as above can be employed but the data has more structure that needs to be incorporated into the meat estimators. Specifically, for one-way clustering there is not simply an observation $i$ from $1, \dots, n$ observations but an observation $(i,g)$ from $1, \dots, n_g$ observations within cluster/group $g$ (with $g = 1, \dots, G$ and $n = n_1 + \dots + n_G$). As only the $G$ groups can be assumed to be independent while there might be correlation withing the cluster/group, the empirical estimation function is summed up within each group prior to computing meat estimators. Thus, the core idea of many clustered covariances is to replace Equation~\ref{eq:estfun} with the following equation and then proceeding ``as usual'' in the computation of meat estimators afterwards: \begin{eqnarray} \label{eq:estfun-cl} \left( \begin{array}{c} \displaystyle \sum_{i = 1}^{n_{1}}\psi(y_{i,1}, x_{i,1}, \hat \theta)^\top\\ \vdots \\ \displaystyle \sum_{i = 1}^{n_{G}}\psi(y_{i,G}, x_{i,G}, \hat \theta)^\top \end{array} \right) = \left( \begin{array}{c} \displaystyle \vphantom{\sum_{i = 1}^{n_{1}}} \psi(y_{1,1}, x_{1,1}, \hat \theta)^\top + \dots + \psi(y_{n_1, 1}, x_{n_1, 1}, \hat \theta)^\top \\ \vdots \\ \displaystyle \vphantom{\sum_{i = 1}^{n_{G}}} \psi(y_{1,G}, x_{1,G}, \hat \theta)^\top + \dots + \psi(y_{n_G, G}, x_{n_G, G}, \hat \theta)^\top \end{array} \right). \end{eqnarray} The basic meat estimator based on the outer product from Equation~\ref{eq:meat-op} then becomes: % \begin{equation} \label{eq:meatCL} \hat M_\mathrm{CL} ~=~ \frac{1}{n} \sum_{g = 1}^G \bigg( \sum_{i = 1}^{n_{g}}\psi(y_{i,g}, x_{i,g}, \hat \theta) \bigg)\bigg( \sum_{i = 1}^{n_{g}} \psi(y_{i,g}, x_{i,g}, \hat \theta) \bigg)^\top. \end{equation} % In the case where observation is its own cluster, the clustered $\hat M_\mathrm{CL}$ corresponds to the basic~$\hat M$. The new function \code{meatCL()} in the \pkg{sandwich} package implements this basic trick along with several types of bias correction and the possibility for multi-way instead of one-way clustering. \subsubsection{Types of bias correction} The clustered covariance estimator controls for both heteroscedasticity across as well as within clusters, assuming that in addition to the number of observations $n$ the number of clusters $G$ also approaches infinity \citep{hac:Cameron+Gelbach+Miller:2008,hac:Cameron+Miller:2015}. Although many publications just refer to ``the'' clustered standard errors, there is a surprisingly wide variation in clustered covariances, particularly due to different flavors of bias corrections. The bias correction factor can be split in two parts, a ``cluster bias correction'' and an ``HC bias correction''. The cluster bias correction captures the effect of having just a finite number of clusters $G$ and it is defined as \begin{equation} \label{eq:biasadjc} \frac{G}{G - 1}. \end{equation} The HC bias correction can be applied additionally in a manner similar to the corresponding cross-section data estimators. HC0 to HC3 bias corrections for cluster $g$ are defined as \begin{eqnarray} \label{eq:biasadj0} \mathrm{HC0:} & & 1 \\ \label{eq:biasadj1} \mathrm{HC1:} & & \frac{n}{n - k} \\ \label{eq:biasadj2} \mathrm{HC2:} & & \frac{G - 1}{G} \cdot (I_{n_{g}} - H_{gg})^{-1} \\ \label{eq:biasadj3} \mathrm{HC3:} & & \frac{G - 1}{G} \cdot (I_{n_{g}} - H_{gg})^{-2}, \end{eqnarray} where $n$ is the number of observations and $k$ is the number of estimated parameters, $I_{n_{g}}$ is an identity matrix of size $n_{g}$, $H_{gg}$ is the block from the hat matrix $H$ that pertains to cluster~$g$. In case of a linear model, the hat (or projection) matrix is defined as $H = X (X^\top X)^{-1} X^\top$, with model matrix $X$. A generalized hat matrix is available for generalized linear models \citep[GLMs, see Equation~12.3 in][]{hac:McCullagh+Nelder:1989}. To apply these factors to $\hat M_\mathrm{CL}$ (Equation~\ref{eq:meatCL}), their square root has to be applied to each of the $\psi$ factors in the outer product \citep[see][Section~VI.B for the linear regression case]{hac:Cameron+Miller:2015}. Thus, it is completely straightforward to incorporate the scalar factors for HC0 and HC1. However, the cluster generalizations of HC2 and HC3 \citep[due to][]{hac:Kauermann+Carroll:2001,hac:Bell+McCaffrey:2002} require some more work. More precisely, the square root of the correction factors from Equations~\ref{eq:biasadj2} and~\ref{eq:biasadj3} has to be applied to the (working) residuals prior to computing the clustered meat matrix. Thus, the empirical working residuals $r(y_{g}, x_{g}^\top \hat \theta)$ in group $g$ are adjusted via \begin{equation} \label{eq:fact-adj} \tilde r(y_{g}, x_{g}^\top \hat \theta) = \sqrt{\frac{G - 1}{G}} \cdot (I_{n_{g}} - H_{gg})^{-\alpha/2} \cdot r(y_{g}, x_{g}^\top \hat \theta) \end{equation} with $\alpha = 1$ for HC2 and $\alpha = 2$ for HC3, before obtaining the adjusted empirical estimating functions based on Equation~\ref{eq:fact} as \begin{equation} \label{eq:fact-reg} \tilde \psi(y_i, x_i, \hat\theta) = \tilde r(y_{i}, x_{i}^\top \hat \theta) \cdot x_i. \end{equation} Then these adjusted estimating functions can be employed ``as usual'' to obtain the $\hat M_\mathrm{CL}$. Note that, by default, adding the cluster adjustment factor $G/(G-1)$ from Equation~\ref{eq:biasadjc} simply cancels out the factor $(G-1)/G$ from the HC2/HC3 adjustment factor. Originally, \cite{hac:Bell+McCaffrey:2002} recommended to use the factor $(G-1)/G$ for HC3 (i.e., without cluster adjustment) but not for HC2 (i.e., with cluster adjustment). Here, we handle both cases in the same way so that both collapse to the traditional HC2/HC3 estimators when $G = n$ (i.e., where every observation is its own cluster). Note that in terms of methods in \proglang{R}, it is not sufficient for the cluster HC2/HC3 estimators to have just \code{estfun()} and \code{model.matrix()} extractors but an extractor for (blocks of) the full hat matrix are required as well. Currently, no such extractor method is available in base \proglang{R} (as \code{hatvalues()} just extracts $\mathrm{diag} H$) and hence HC2 and HC3 in \code{meatCL()} are just available for `\code{lm}' and `\code{glm}' objects. \subsubsection{Two-way and multi-way clustered covariances} Certainly, there can be more than one cluster dimension, as for example observations that are characterized by housholds within states or companies within industries. Therefore, it can sometimes be helpful that one-way clustered covariances can be extended to so-called multi-way clustering as shown by \cite{hac:Miglioretti+Heagerty:2007}, \cite{hac:Thompson:2011} and \cite{hac:Cameron+Gelbach+Miller:2011}. Multi-way clustered covariances comprise clustering on $2^{D} - 1$ dimensional combinations. Clustering in two dimensions, for example in $id$ and $time$, gives $D = 2$, such that the clustered covariance matrix is composed of $2^2 - 1 = 3$ one-way clustered covariance matrices that have to be added or subtracted, respectively. For two-way clustered covariances with cluster dimensions $id$ and $time$, the one-way clustered covariance matrices on $id$ and on $time$ are added, and the two-way clustered covariance matrix with clusters formed by the intersection of $id$ and $time$ is subtracted. Additionally, a cluster bias correction $\frac{\mathrm{G(\cdot)}}{\mathrm{G(\cdot)} - 1}$ is added, where $\mathrm{G}(id)$ is the number of clusters in cluster dimension $id$, $\mathrm{G}(time)$ is the number of clusters in cluster dimension $time$, and $\mathrm{G}(id \cap time)$ is the number of clusters formed by the intersection of $id$ and $time$. % \begin{equation} \label{eq:twoway} \hat M_{\mathrm{CL}} \quad = \quad \frac{\mathrm{G}(id)}{\mathrm{G}(id) - 1}\hat M_{id} + \frac{\mathrm{G}(time)}{\mathrm{G}(time) - 1}\hat M_{time} - \frac{\mathrm{G}(id \cap time)}{\mathrm{G}(id \cap time) - 1}\hat M_{id \cap time}. \end{equation} % The same idea is used for obtaining clustered covariances with more than two clustering dimensions: Meat parts with an odd number of cluster dimensions are added, whereas those with an even number are subtracted. Particularly when there is only one observation in each intersection of $id$ and $time$, \cite{hac:Petersen:2009}, \cite{hac:Thompson:2011} and \cite{hac:Ma:2014} suggest to subtract the standard sandwich estimator $\hat M$ from Equation (\ref{eq:meat-op}) instead of $\frac{\mathrm{G}(id \cap time)}{\mathrm{G}(id \cap time) - 1}\hat M_{(id \cap time)}$ in Equation (\ref{eq:twoway}). \subsection{Clustered covariances for panel data} The information in panel data sets is often overstated, as cross-sectional as well as temporal dependencies may occur \citep{hac:Hoechle:2007}. \citet[p.~702]{hac:Cameron+Trivedi:2005} noticed that ``$NT$ correlated observations have less information than $NT$ independent observations''. For panel data, the source of dependence in the data is crucial to find out what kind of covariance is optimal \citep{hac:Petersen:2009}. In the following, panel Newey-West standard errors as well as Driscoll and Kraay standard errors are examined \citep[see also][for a unifying view]{hac:Millo:2017}. To reflect that the data are now panel data with a natural time ordering within each \emph{group}~$g$ (as opposed to variables without a natural ordering, such as individuals, countries, or firms, \citealp{hac:Millo:2017}), we change our notation to an index $(g, t)$ for $g = 1, \dots, G$ \emph{groups} sampled at $t = 1, \ldots, n_g$ points in \emph{time} (with $n = n_1 + \dots + n_G$). Clustered covariances account for the average within-cluster covariance, where a cluster or group is any set of observations with can be identified by a variable to ``cluster on''. In the case of panel data, observations are frequently grouped by time and by one or more cross-sectional variables such as individuals or countries that have been observed repeatedly over time. \subsubsection{Panel Newey-West} \cite{hac:Newey+West:1987} proposed a heteroscedasticity and autocorrelation consistent standard error estimator that is traditionally used for time-series data, but can be modified for use in panel data \citep[see for example][]{hac:Petersen:2009}. A panel Newey-West estimator can be obtained by setting the cross-sectional as well as the cross-serial correlation to zero \citep{hac:Millo:2017}, effectively assuming that each cross-sectional group is a separate, autocorrelated time-series. The meat is composed of % \begin{equation} \label{eq:newey2} \hat M_\mathrm{PL}^{NW} \quad = \quad \frac{1}{n} \sum_{\ell = 0}^{L} w_{\ell} \, \bigg[\sum_{t = 1}^{n_{g}}\sum_{g = 1}^{G} \psi(y_{g,t}, x_{g,t}, \hat \theta) \psi(y_{g,t-\ell}, x_{g,t-\ell}, \hat \theta)^\top\bigg]. \end{equation} % \cite{hac:Newey+West:1987} employ a Bartlett kernel for obtaining the weights as $w_{\ell} = 1 - \frac{\ell}{L + 1}$ at lag $\ell$ up to lag $L$. As \cite{hac:Petersen:2009} noticed, the maximal lag length $L$ in a panel data set is $n_g - 1$, i.e., the maximum number of \emph{time} periods per \emph{group} minus one. \subsubsection{Driscoll and Kraay} \cite{hac:Driscoll+Kraay:1998} have adapted the Newey-West approach by using the aggregated estimating functions at each time point. This can be shown to be robust to spatial and temporal dependence of general form, but with the caveat that a long enough time dimension must be available. Thus, the idea is again to replace Equation~\ref{eq:estfun} by Equation~\ref{eq:estfun-cl} before computing $\hat M_\mathrm{HAC}$ from Equation~\ref{eq:meat-hac}. Note, however, that the aggregation is now done across \emph{groups}~$g$ within each time period $t$. This yields a panel sandwich estimator where the meat is computed as % \begin{equation} \label{eq:driscoll} \hat M_\mathrm{PL} \quad = \quad \frac{1}{n} \sum_{\ell = 0}^L w_\ell \, \bigg[ \sum_{t = 1}^{n_g} \bigg( \sum_{g = 1}^{G} \psi(y_{g,t}, x_{g,t}, \hat \theta) \bigg) \bigg( \sum_{g = 1}^{G} \psi(y_{g,t-\ell}, x_{g,t-\ell}, \hat \theta) \bigg)^\top \bigg], \end{equation} % The weights $w_{\ell}$ are usually again the Bartlett weights up to lag $L$. Note that for $L = 0$, $\hat M_\mathrm{PL}$ reduced to $\hat M_{\mathrm{CL}(\mathit{time})}$, i.e., the one-way covariance clustered by time. Also, for the special case that there is just one observation at each time point $t$, this panel covariance by \cite{hac:Driscoll+Kraay:1998} simply yields the panel Newey-West covariance. The new function \code{meatPL()} in the \pkg{sandwich} package implements this approach analogously to \code{meatCL()}. For the computation of the weights $w_\ell$ the same function is employed that \code{meatHAC()} uses. \subsection{Panel-corrected standard errors} \cite{hac:Beck+Katz:1995} proposed another form or panel-corrected covariances -- typically referred to as panel-corrected standard errors (PCSE). They are intended for panel data (also called time-series-cross-section data in this literature) with moderate dimensions of time and cross-section \citep{hac:Millo:2017}. They are robust against panel heteroscedasticity and contemporaneous correlation, with the crucial assumption that contemporaneous correlation across clusters follows a fixed pattern \citep{hac:Millo:2017, hac:Johnson:2004}. Autocorrelation within a cluster is assumed to be absent. \cite{hac:Hoechle:2007} argues that for the PCSE estimator the finite sample properties are rather poor if the cross-sectional dimension is large compared to the time dimension. This is in contrast to the panel covariance by \cite{hac:Driscoll+Kraay:1998} which relies on large-$t$ asymptotics and is robust to quite general forms of cross-sectional and temporal dependence and is consistent independently of the cross-sectional dimension. To emphasize again that both cross section \emph{and} and time ordering are considered, index $(g,t)$ is employed for the observation from group $g = 1, \dots, G$ at time $t = 1, \dots, n_g$. Here, $n_g$ denotes the last time period observed in cluster $g$, thus allowing for unbalanced data. In the balanced case (that we focus on below) $n_g = T$ for all groups~$g$ so that there are $n = G \cdot T$ observations overall. The basic idea for PCSE is to employ the outer product of (working) residuals within each cluster~$g$. Thus, the working residuals are split into $T \times 1$ vectors for each cluster $g$: $r(y_1, x_1^\top \hat \theta), \dots,$\linebreak $r(y_G, x_G^\top \hat \theta)$. For balanced data these can be arranged in a $T \times G$ matrix, \begin{equation} \label{eq:workres} R \quad = \quad [r(y_1, x_1^\top \hat \theta) \quad r(y_2, x_2^\top \hat \theta) \quad \ldots \quad r(y_G, x_G^\top \hat \theta)], \end{equation} and the meat of the panel-corrected covariance matrix can be computed using the Kronecker product as \begin{equation} \label{eq:pcse} \hat M_\mathrm{PC} \quad = \quad \frac{1}{n} X^\top \bigg[ \frac{(R^\top R)}{T} \otimes {\bm I}_T \bigg] X. \end{equation} The details for the unbalanced case are omitted here for brevity but are discussed in detail in \cite{hac:Bailey+Katz:2011}. The new function \code{meatPC()} in the \pkg{sandwich} package implements both the balanced and unbalanced case. As for \code{meatHC()} it is necessary to have a \code{model.matrix()} extractor in addition to the \code{estfun()} extractor for splitting up the empirical estimating functions into residuals and regressor matrix. \section{Software} \label{sec:software} As conveyed already in Section~\ref{sec:methods}, the \pkg{sandwich} package has been extended along the same lines it was originally established in \cite{hac:Zeileis:2006}. The new clustered and panel covariances require a new \code{meat*()} function that ideally only extracts the \code{estfun()} from a fitted model object. For the full sandwich covariance an accompanying \code{vcov*()} function is provided that couples the \code{meat*()} with the \code{bread()} estimate extracted from the model object. The new sandwich covariances \code{vcovCL()} for clustered data and \code{vcovPL()} for panel data, as well as \code{vcovPC()} for panel-corrected covariances all follow this structure and are introduced in more detail below. Model classes which provide the necessary building blocks include `\code{lm}', `\code{glm}', `\code{mlm}', and `\code{nls}' from \pkg{stats} \citep{hac:R:2018}, `\code{betareg}' from \pkg{betareg} \citep{hac:Cribari-Neto+Zeileis:2010,hac:Gruen+Kosmidis+Zeileis:2012}, `\code{clm}' from \pkg{ordinal} \citep{hac:Christensen:2018}, `\code{coxph}' and `\code{survreg}' from \pkg{survival} \citep{hac:Therneau:2018}, `\code{crch}' from \pkg{crch} \citep{hac:Messner+Mayr+Zeileis:2016}, `\code{hurdle}' and `\code{zeroinfl}' from \pkg{pscl}/\pkg{countreg} \citep{hac:Zeileis+Kleiber+Jackman:2008}, `\code{mlogit}' from \pkg{mlogit} \citep{hac:Croissant:2018}, and `\code{polr}' and `\code{rlm}' from \pkg{MASS} \citep{hac:Venables+Ripley:2002}. For all of these an \code{estfun()} method is available along with a \code{bread()} method (or the default method works). In case the models are based on a single linear predictor only, they also provide \code{model.matrix()} extractors so that the factorization from Equation~\ref{eq:fact} into working residuals and regressor matrix can be easily computed. \subsection{Clustered covariances} \label{sec:vcovcl} One-, two-, and multi-way clustered covariances with HC0--HC3 bias correction are implemented in \begin{verbatim} vcovCL(x, cluster = NULL, type = NULL, sandwich = TRUE, fix = FALSE, ...) \end{verbatim} for a fitted-model object \code{x} with the underlying meat estimator in \begin{verbatim} meatCL(x, cluster = NULL, type = NULL, cadjust = TRUE, multi0 = FALSE, ...) \end{verbatim} The essential idea is to aggregate the empirical estimating functions within each cluster and then compute a HC covariance analogous to \code{vcovHC()}. The \code{cluster} argument allows to supply either one cluster vector or a list (or data frame) of several cluster variables. If no cluster variable is supplied, each observation is its own cluster per default. Thus, by default, the clustered covariance estimator collapses to the basic sandwich estimator. The \code{cluster} specification may also be a \code{formula} if \code{expand.model.frame(x, cluster)} works for the model object \code{x}. The bias correction is composed of two parts that can be switched on and off separately: First, the cluster bias correction from Equation~\ref{eq:biasadjc} is controlled by \code{cadjust}. Second, the HC bias correction from Equations~\ref{eq:biasadj0}--\ref{eq:biasadj3} is specified via \code{type} with the default to use \code{"HC1"} for `\code{lm}' objects and \code{"HC0"} otherwise. Moreover, \code{type = "HC2"} and \code{"HC3"} are only available for `\code{lm}' and `\code{glm}' objects as they require computation of full blocks of the hat matrix (rather than just the diagonal elements as in \code{hatvalues()}). Hence, the hat matrices of (generalized) linear models are provided directly in \code{meatCL()} and are not object-oriented in the current implementation. The \code{multi0} argument is relevant only for multi-way clustered covariances with more than one clustering dimension. It specifies whether to subtract the basic cross-section HC0 covariance matrix as the last subtracted matrix in Equation~\ref{eq:twoway} instead of the covariance matrix formed by the intersection of groups \citep{hac:Petersen:2009,hac:Thompson:2011,hac:Ma:2014}. For consistency with \cite{hac:Zeileis:2004a}, the \code{sandwich} argument specifies whether the full sandwich estimator is computed (default) or only the meat. Finally, the \code{fix} argument specifies whether the covariance matrix should be fixed to be positive semi-definite in case it is not. This is achieved by converting any negative eigenvalues from the eigendecomposition to zero. \cite{hac:Cameron+Gelbach+Miller:2011} observe that this is most likely to be necessary in applications with fixed effects, especially when clustering is done over the same groups as the fixed effects. \subsection{Clustered covariances for panel data} For panel data, \begin{verbatim} vcovPL(x, cluster = NULL, order.by = NULL, kernel = "Bartlett", sandwich = TRUE, fix = FALSE, ...) \end{verbatim} based on \begin{verbatim} meatPL(x, cluster = NULL, order.by = NULL, kernel = "Bartlett", lag = "NW1987", bw = NULL, adjust = TRUE, ...) \end{verbatim} computes sandwich covariances for panel data, specificially including panel \cite{hac:Newey+West:1987} and \cite{hac:Driscoll+Kraay:1998}. The essential idea is to aggregate the empirical estimating functions within each time period and then compute a HAC covariance analogous to \code{vcovHAC()}. Again, \code{vcovPL()} returns the full sandwich if the argument \code{sandwich = TRUE}, and \code{fix = TRUE} forces a positive semi-definite result if necessary. The \code{cluster} argument allows the variable indicating the cluster/group/id variable to be specified, while \code{order.by} specifies the time variable. If only one of the two variables is provided, then it is assumed that observations are ordered within the other variable. And if neither is provided, only one cluster is used for all observations resulting in the standard \citep{hac:Newey+West:1987} estimator. Finally, \code{cluster} can also be a list or formula with both variables: the cluster/group/id and the time/ordering variable, respectively. (The formula specification requires again that \code{expand.model.frame(x, cluster)} works for the model object \code{x}.) The weights in the panel sandwich covariance are set up by means of a \code{kernel} function along with a bandwidth \code{bw} or the corresponding \code{lag}. All kernels described in \cite{hac:Andrews:1991} and implemented in \code{vcovHAC()} by \cite{hac:Zeileis:2004} are available, namely truncated, Bartlett, Parzen, Tukey-Hanning, and quadratic spectral. For the default case of the Bartlett kernel, the bandwidth \code{bw} corresponds to \code{lag + 1} and only one of the two arguments should be specified. The \code{lag} argument can either be an integer or one of three character specifications: \code{"max"}, \code{"NW1987"}, or \code{"NW1994"}. \code{"max"} (or equivalently, \code{"P2009"} for \citealp{hac:Petersen:2009}) indicates the maximum lag length $T - 1$, i.e., the number of time periods minus one. \code{"NW1987"} corresponds to \cite{hac:Newey+West:1987}, who have shown that their estimator is consistent if the number of lags increases with time periods $T$, but with speed less than $T^{1/4}$ \citep[see also][]{hac:Hoechle:2007}. \code{"NW1994"} sets the lag length to $\mathrm{floor}[4 \cdot (\frac{T}{100})^{2/9}]$ \citep{hac:Newey+West:1994}. The \code{adjust} argument allows to make a finite sample adjustment, which amounts to multiplication by $n/(n - k)$, where $n$ is the number of observations, and $k$ is the number of estimated parameters. \subsection{Panel-corrected covariance} Panel-corrected covariances and panel-corrected standard errors (PCSE) a la \cite{hac:Beck+Katz:1995} are implemented in \begin{verbatim} vcovPC(x, cluster = NULL, order.by = NULL, pairwise = FALSE, sandwich = TRUE, fix = FALSE, ...) \end{verbatim} based on \begin{verbatim} meatPC(x, cluster = NULL, order.by = NULL, pairwise = FALSE, kronecker = FALSE, ...) \end{verbatim} They are usually used for panel data or time-series-cross-section (TSCS) data with a large-enough time dimension. The arguments \code{sandwich}, \code{fix}, \code{cluster}, and \code{order.by} have the same meaning as in \code{vcovCL()} and \code{vcovPL()}. While estimation in balanced panels is straightforward, there are two alternatives to estimate the meat for unbalanced panels \citep{hac:Bailey+Katz:2011}. For \code{pairwise = TRUE}, a pairwise balanced sample is employed, whereas for \code{pairwise = FALSE}, the largest balanced subset of the panel is used. For details, see \cite{hac:Bailey+Katz:2011}. The argument \code{kronecker} relates to estimation of the meat and determines whether calculations are executed with the Kronecker product or element-wise. The former is typically computationally faster in moderately large data sets while the latter is less memory-intensive so that it can be applied to larger numbers of observations. \subsection{Further functionality: Bootstrap covariances} As an alternative to the asymptotic approaches to clustered covariances in \code{vcovCL()}, \code{vcovPL()}, and \code{vcovPC()}, the function \code{vcovBS()} provides a bootstrap solution. Currently, a default method is available as well as a dedicated method for `\code{lm}' objects with more refined bootstrapping options. Both methods take the arguments % \begin{verbatim} vcovBS(x, cluster = NULL, R = 250, ..., fix = FALSE, use = "pairwise.complete.obs", applyfun = NULL, cores = NULL) \end{verbatim} % where \code{cluster} and \code{fix} work the same as in \code{vcovCL()}. \code{R} is the number of bootstrap replications and \code{use} is passed on to the base \code{cov()} function. The \code{applyfun} is an optional \code{lapply()}-style function with arguments \code{function(X, FUN, ...)}. It is used for refitting the model to the bootstrap samples. The default is to use the basic \code{lapply()} function unless the \code{cores} argument is specified so that \code{parLapply()} (on Windows) or \code{mclapply()} (otherwise) from the basic \pkg{parallel} package are used with the desired number of \code{cores}. The default method samples clusters (where each observation is its own cluster by default), i.e., case-based ``xy'' (or ``pairs'') resampling \citep[see e.g.,][]{hac:Davison+Hinkley:1997}. For obtaining a covariance matrix estimate it is assumed that an \code{update(x, subset = ...)} of the model with the resampled subset can be obtained, the \code{coef()} extracted, and finally the covariance computed with \code{cov()}. In addition to the arguments listed above, it sends \code{...} to \code{update()}, and it provides an argument \code{start = FALSE}. If set to \code{TRUE}, the latter leads essentially to \code{update(x, start = coef(x), subset = ...)}, i.e., necessitates support for a \code{start} argument in the model's fitting function in addition to the \code{subset} argument. If available, this may reduce the computational burden of refitting the model. The \code{glm} method essentially proceeds in the same way but instead of using \code{update()} for estimating the coefficients on the ``xy'' bootstrap sample, \code{glm.fit()} is used which speeds up computations somewhat. Because the ``xy'' method makes the same assumptions as the asymptotic approach above, its results approach the asymptotic results as the number of bootstrap replications \code{R} becomes large, assuming the same bias adjustments and small sample corrections are applied -- see e.g., \cite{hac:Efron:1979} or \cite{hac:Mammen:1992} for a discussion of asymptotic convergence in bootstraps. Bootstrapping will often converge to slightly different estimates when the sample size is small due to a limited number of distinct iterations -- for example, there are 126 distinct ways to resample 5 groups with replacement, but many of them occur only rarely (such as drawing group 1 five times); see \cite{hac:Webb:2014} for more details. This makes ``xy'' bootstraps unnecessary if the desired asymptotic estimator is available and performs well (\code{vcovCL} may not be available if a software package has not implemented the \code{estfun()} function, for example). Bootstrapping also often improves inference when nonlinear models are applied to smaller samples. However, the ``xy'' bootstrap is not the only technique available, and the literature has found a number of alternative bootstrapping approaches -- which make somewhat different assumptions -- to be useful in practice, even for linear models applied to large samples. Hence, the \code{vcovBS()} method for `\code{lm}' objects provides several additional bootstrapping \code{type}s and computes the bootstrapped coefficient estimates somewhat more efficiently using \code{lm.fit()} or \code{qr.coef()} rather than \code{update()}. The default \code{type}, however, is also pair resampling (\code{type = "xy"}) as in the default method. Alternative \code{type} specifications are % \begin{itemize} \item \code{"residual"}. The residual cluster bootstrap resamples the residuals (as above, by cluster) which are subsequently added to the fitted values to obtain the bootstrapped response variable. Coefficients can then be estimated using \code{qr.coef()}, reusing the QR decomposition from the original fit. As \cite{hac:Cameron+Gelbach+Miller:2008} point out, the residual cluster bootstrap is not well-defined when the clusters are unbalanced as residuals from one cluster cannot be easily assigned to another cluster with different size. Hence a warning is issued in that case. \item \code{"wild"} (or equivalently \code{"wild-rademacher"} or \code{"rademacher"}). The wild cluster bootstrap does not actually resample the residuals but instead reforms the dependent variable by multiplying the residual by a randomly drawn value and adding the result to the fitted value \citep[see][]{hac:Cameron+Gelbach+Miller:2008}. By default, the factors are drawn from Rademacher distribution, i.e., selecting either $-1$ or $1$ with probability $0.5$. \item \code{"mammen"} (or \code{"wild-mammen"}). This draws the wild bootstrap factors as suggested by \cite{hac:Mammen:1993}. \item \code{"webb"} (or \code{"wild-webb"}). This implements the six-point distribution suggested by \cite{hac:Webb:2014}, which may improve inference when the number of clusters is small. \item \code{"norm"} (or \code{"wild-norm"}). The standard normal/Gaussian distribution is used for drawing the wild bootstrap factors. \item User-defined function. This should take a single argument \code{n}, the number of random values to produce, and return a vector of random factors of corresponding length. \end{itemize} \section{Illustrations} \label{sec:illu} The main motivation for the new object-oriented implementation of clustered covariances in \pkg{sandwich} was the applicability to models beyond \code{lm()} or \code{glm()}. Specifically when working on \cite{hac:Berger+Stocker+Zeileis:2017} -- an extended replication of \cite{hac:Aghion+VanReenen+Zingales:2013} -- clustered covariances for negative binomial hurdle models were needed to confirm reproducibility. After doing this ``by hand'' in \cite{hac:Berger+Stocker+Zeileis:2017}, we show in Section~\ref{ex-aghion} how the same results can now be conveniently obtained with the new general \code{vcovCL()} framework. Furthermore, to show that the new \pkg{sandwich} package can also replicate the classic linear regression results that are currently scattered over various packages, the benchmark data from \cite{hac:Petersen:2009} is considered in Section~\ref{ex-petersen}. This focuses on linear regression with model errors that are correlated within clusters. Section~\ref{ex-petersen} replicates a variety of clustered covariances from the \proglang{R} packages \pkg{multiwayvcov}, \pkg{plm}, \pkg{geepack}, and \pkg{pcse}. More specifically, one- and two-way clustered standard errors from \pkg{multiwayvcov} are replicated with \code{vcovCL()}. Furthermore, one-way clustered standard errors from \pkg{plm} and \pkg{geepack} can also be obtained by \code{vcovCL()}. The Driscoll and Kraay standard errors from \pkg{plm}'s \code{vcovSCC()} can also be computed with the new \code{vcovPL()}. Finally, panel-corrected standard errors can be estimated by function \code{vcovPC()} from \pkg{pcse} and are benchmarked against the new \code{vcovPC()} from \pkg{sandwich}. \subsection[Aghion et al. (2013) and Berger et al. (2017)]{\cite{hac:Aghion+VanReenen+Zingales:2013} and \cite{hac:Berger+Stocker+Zeileis:2017}} \label{ex-aghion} \cite{hac:Aghion+VanReenen+Zingales:2013} investigate the effect of institutional owners (these are, for example, pension funds, insurance companies, etc.) on innovation. The authors use firm-level panel data on innovation and institutional ownership from 1991 to 1999 over 803 firms, with the data clustered at company as well as industry level. To capture the differing value of patents, citation-weighted patent counts are used as a proxy for innovation, whereby the authors weight the patents by the number of future citations. This motivates the use of count data models. \cite{hac:Aghion+VanReenen+Zingales:2013} mostly employ Poisson and negative binomial models in a quasi-maximum likelihood approach and cluster standard errors by either companies or industries. \cite{hac:Berger+Stocker+Zeileis:2017} argue that zero responses should be treated separately both for statistical and economic reasons, as there is a difference in determinants of ``first innovation'' and ``continuing innovation''. Therefore, they employ two-part hurdle models with a binary part that models the decision to innovate at all, and a count part that models ongoing innovation, respectively. A basic negative binomial hurdle model is fitted with the \code{hurdle} function from the \pkg{pscl} package \citep{hac:Zeileis+Kleiber+Jackman:2008} using the \cite{hac:Aghion+VanReenen+Zingales:2013} data provided in the \pkg{sandwich} package. <>= data("InstInnovation", package = "sandwich") library("pscl") h_innov <- hurdle( cites ~ institutions + log(capital/employment) + log(sales), data = InstInnovation, dist = "negbin") @ The partial Wald tests for all coefficients based on clustered standard errors can be obtained by using \code{coeftest()} from \pkg{lmtest} \citep{hac:Zeileis+Hothorn:2002} and setting \code{vcov = vcovCL} and providing the company-level clustering (with a total of 803 clusters) via \code{cluster = ~ company}. <>= library("sandwich") library("lmtest") coeftest(h_innov, vcov = vcovCL, cluster = ~ company) @ This shows that institutional owners are have a small but positive impact in both submodels but that only the coefficient in the zero hurdle is significant. Below, the need for clustering is brought out through a comparison of clustered standard errors with ``standard'' standard errors and basic (cross-section) sandwich standard errors. As an additional reference, a simple clustered bootstrap covariance can be computed by \code{vcovBS()} (by default with \code{R = 250} bootstrap samples). <>= suppressWarnings(RNGversion("3.5.0")) set.seed(0) vc <- list( "standard" = vcov(h_innov), "basic" = sandwich(h_innov), "CL-1" = vcovCL(h_innov, cluster = ~ company), "boot" = vcovBS(h_innov, cluster = ~ company) ) se <- function(vcov) sapply(vcov, function(x) sqrt(diag(x))) se(vc) @ <>= se(vc_innov) @ This clearly shows that the usual standard errors greatly overstate the precision of the estimators and the basic sandwich covariances are able to improve but not fully remedy the situation. The clustered standard errors are scaled up by factors between about $1.5$ and $2$, even compared to the basic sandwich standard errors. Moreover, the clustered and bootstrap covariances agree very well -- thus highlighting the need for clustering -- with \code{vcovBS()} being computationally much more demanding than \code{vcovCL()} due to the resampling. \subsection[Petersen (2009)]{\cite{hac:Petersen:2009}} \label{ex-petersen} \cite{hac:Petersen:2009} provides simulated benchmark data (\url{http://www.kellogg.northwestern.edu/faculty/petersen/htm/papers/se/test_data.txt}) for assessing clustered standard error estimates in the linear regression model. This data set contains a dependent variable \code{y} and regressor \code{x} for 500 \code{firm}s over 10 \code{year}s. It is frequently used in illustrations of clustered covariances \citep[e.g., in \pkg{multiwayvcov}, see][]{hac:Graham+Arai+Hagstroemer:2016} and is also available in \pkg{sandwich}. The corresponding linear model is fitted with \code{lm()}. <>= data("PetersenCL", package = "sandwich") p_lm <- lm(y ~ x, data = PetersenCL) @ \subsubsection{One-way clustered standard errors} One-way clustered covariances for linear regression models are available in a number of different \proglang{R} packages. The implementations differ mainly in two aspects: (1) Whether a simple `\code{lm}' object can be supplied or (re-)estimation of a dedicated model object is necessary. (2) Which kind of bias adjustment is done by default (HC type and/or cluster adjustment). The function \code{cluster.vcov()} from \pkg{multiwayvcov} (whose implementation strategy \code{vcovCL()} follows) can use `\code{lm}' objects directly and then applies both the HC1 and cluster adjustment by default. In contrast, \pkg{plm} and \pkg{geepack} both require re-estimation of the model and then employ HC0 without cluster adjustment by default. In \pkg{plm}, a pooling model needs to be estimated with \code{plm()} and in \pkg{geepack} an independence working model needs to be fitted with \code{geeglm()}. The \pkg{multiwayvcov} results can be replicated as follows. <>= library("multiwayvcov") se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm), "multiwayvcov" = cluster.vcov(p_lm, cluster = ~ firm) )) @ And the \pkg{plm} and \pkg{geepack} results can be replicated with the following code. (Note that \pkg{geepack} does not provide a \code{vcov()} method for `\code{geeglm}' objects, hence the necessary code is included below.) <>= library("plm") p_plm <- plm(y ~ x, data = PetersenCL, model = "pooling", indexes = c("firm", "year")) library("geepack") vcov.geeglm <- function(object) { vc <- object$geese$vbeta rownames(vc) <- colnames(vc) <- names(coef(object)) return(vc) } p_gee <- geeglm(y ~ x, data = PetersenCL, id = PetersenCL$firm, corstr = "independence", family = gaussian) se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm, cadjust = FALSE, type = "HC0"), "plm" = vcovHC(p_plm, cluster = "group"), "geepack" = vcov(p_gee) )) @ \subsubsection{Two-way clustered standard errors} It would also be feasible to cluster the covariances with respect to both dimensions, \code{firm} and \code{year}, yielding similar but slightly larger standard errors. Again, \code{vcovCL()} from \pkg{sandwich} can replicate the results of \code{cluster.vcov()} from \pkg{multiwayvcov}. Only the default for the correction proposed by \cite{hac:Ma:2014} is different. <>= se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm + year, multi0 = TRUE), "multiwayvcov" = cluster.vcov(p_lm, cluster = ~ firm + year) )) @ However, note that the results should be regarded with caution as cluster dimension \code{year} has a total of only 10 clusters. Theory requires that each cluster dimension has many clusters \citep{hac:Petersen:2009,hac:Cameron+Gelbach+Miller:2011,hac:Cameron+Miller:2015}. \subsubsection{Driscoll and Kraay standard errors} The Driscoll and Kraay standard errors for panel data are available in \code{vcovSCC()} from \pkg{plm}, defaulting to a HC0-type adjustment. In \pkg{sandwich} the \code{vcovPL()} function can be used for replication, setting \code{adjust = FALSE} to match the HC0 (rather than HC1) adjustment. <>= se(list( "sandwich" = vcovPL(p_lm, cluster = ~ firm + year, adjust = FALSE), "plm" = vcovSCC(p_plm) )) @ \subsubsection{Panel-corrected standard errors} Panel-corrected covariances a la Beck and Katz are implemented in the package \pkg{pcse} -- providing the function that is also named \code{vcovPC()} -- which can handle both balanced and unbalanced panels. For the balanced Petersen data the two \code{vcovPC()} functions from \pkg{sandwich} and \pkg{pcse} agree. <>= library("pcse") se(list( "sandwich" = sandwich::vcovPC(p_lm, cluster = ~ firm + year), "pcse" = pcse::vcovPC(p_lm, groupN = PetersenCL$firm, groupT = PetersenCL$year) )) @ And also when omitting the last year for the first firm to obtain an unbalanced panel, the \pkg{pcse} results can be replicated. Both strategies for balancing the panel internally (pairwise vs.\ casewise) are illustrated in the following. <>= PU <- subset(PetersenCL, !(firm == 1 & year == 10)) pu_lm <- lm(y ~ x, data = PU) @ and again, panel-corrected standard errors from \pkg{sandwich} are equivalent to those from \pkg{pcse}. <>= se(list( "sandwichT" = sandwich::vcovPC(pu_lm, cluster = ~ firm + year, pairwise = TRUE), "pcseT" = pcse::vcovPC(pu_lm, PU$firm, PU$year, pairwise = TRUE), "sandwichF" = sandwich::vcovPC(pu_lm, cluster = ~ firm + year, pairwise = FALSE), "pcseF" = pcse::vcovPC(pu_lm, PU$firm, PU$year, pairwise = FALSE) )) @ \section{Simulation} \label{sec:simulation} For a more systematic analysis, a Monte Carlo simulation is carried out to assess the performance of clustered covariances beyond linear and generalized linear models. For the linear model, there are a number of simulation studies in the literature \citep[including][]{hac:Cameron+Gelbach+Miller:2008, hac:Arceneaux+Nickerson:2009,hac:Petersen:2009,hac:Cameron+Gelbach+Miller:2011, hac:Harden:2011,hac:Thompson:2011,hac:Cameron+Miller:2015,hac:Jin:2015}, far fewer for generalized linear models \citep[see for example][]{hac:Miglioretti+Heagerty:2007} and, to our knowledge, no larger systematic comparisons for models beyond. Therefore, we try to fill this gap by starting out with simulations of (generalized) linear models similar to the ones mentioned above and then moving on to other types of maximum likelihood regression models. \subsection{Simulation design} The main focus of the simulation is to assess the performance of clustered covariances (and related methods) at varying degrees of correlation within the clusters. The two most important parameters to control for this are the cluster correlation $\rho$, obviously, and the number of clusters $G$ as bias decreases with increasing number of clusters \citep{hac:Green+Vavreck:2008,hac:Arceneaux+Nickerson:2009,hac:Harden:2011}. More specifically, the cluster correlation varies from $0$ to $0.9$ and the number of clusters $G$ ranges from $10$ to $250$. In a first step, only balanced clusters with a low number of observations per cluster ($5$) are considered. All models are specified through linear predictors (with up to three regressors) but with different response distributions. A Gaussian copula is employed to introduce the cluster correlation $\rho$ for the different response distributions. The methods considered are the different clustered covariances implemented in \pkg{sandwich} as well as competing methods such as basic sandwich covariances (without clustering), mixed-effects models with a random intercept, or generalized estimating equations with an exchangeable correlation structure. \subsubsection{Linear predictor} Following \cite{hac:Harden:2011} the linear predictor considered for the simulation is composed of three regressors that are either \emph{correlated} with the clustering, \emph{clustered}, or \emph{uncorrelated}. The correlation of the first type of regressor is controlled by separate parameter $\rho_x$. Thus, in the terminology of \cite{hac:Abadie+Athey+Imbens+Wooldridge:2017}, the parameter $\rho$ above controls the correlation in the sampling process while the parameter $\rho_x$ controls the extent of the clustering in the ``treatment'' assignment. More formally, the model equation is given by % \begin{eqnarray} \label{eq:predictor} h(\mu_{i,g}) = \beta_{0} + \beta_{1} \cdot x_{1,i,g} + \beta_{2} \cdot x_{2,g} + \beta_{3} \cdot x_{3,i,g}, \end{eqnarray} % where $\mu_{i,g}$ is the expectation of the response for observation $i$ within cluster $g$ and the link function $h(\cdot)$ depends on the model type. The regressor variables are all drawn from standard normal distributions but at different levels (cluster vs.\ individual observation). \begin{eqnarray} \label{eq:regressors} x_{1,i,g} & \sim & \rho_{x} \cdot \mathcal{N}_{g}(0, 1) + (1 - \rho_{x}) \cdot \mathcal{N}_{i,g}(0, 1) \label{x1} \\ x_{2,g} & \sim & \mathcal{N}_{g}(0, 1) \label{x2} \\ x_{3,i,g} & \sim & \mathcal{N}_{i,g}(0, 1) \label{x3} \end{eqnarray} Regressor $x_{1,i,g}$ is composed of a linear combination of a random draw at cluster level ($\mathcal{N}_{g}$) and a random draw at individual level ($\mathcal{N}_{i,g}$) while regressors $x_{2,g}$ and $x_{3,i,g}$ are drawn only at cluster and individual level, respectively. Emphasis is given to the investigation of regressor $x_{1,i,g}$ with correlation (default: $\rho_{x} = 0.25$) which is probably the most common in practice. Furthermore, by considering the extremes $\rho_{x} = 1$ and $\rho_{x} = 0$ the properties of $x_{1,i,g}$ coincide with those of $x_{2,g}$ and $x_{3,i,g}$, respectively. The vector of coefficients $\beta = (\beta_{0}, \beta_{1}, \beta_{2}, \beta_{3})^\top$ is fixed to either one of \begin{eqnarray} \label{eq:coefs} \beta & = & (0, 0.85, 0.5, 0.7)^\top \label{beta1} \\ \beta & = & (0, 0.85, 0, 0)^\top \label{beta2} \end{eqnarray} which have been selected based on \cite{hac:Harden:2011}. \subsubsection{Response distributions} The response distributions encompass Gaussian (\code{gaussian}, with identity link) as the standard classical scenario as well as binary (\code{binomial}, with a size of one and a logit link) and Poisson (\code{poisson}, with log link) from the GLM exponential family. To move beyond the GLM, we also consider the beta (\code{betareg}, with logit link and fixed precision parameter $\phi = 10$), zero-truncated Poisson(\code{zerotrunc}, with log link), and zero-inflated Poisson (\code{zeroinfl}, with log link and fixed inflation probability $\pi = 0.3$) distributions. \subsubsection{Sandwich covariances} The types of covariances being compared include ``standard'' covariances (\emph{standard}, without considering any heteroscedasticity or clustering/correlations), basic sandwich covariances (\emph{basic}, without clustering), Driscoll and Kraay panel covariances (\emph{PL}), Beck and Katz panel-corrected covariances (\emph{PC}), and clustered covariances with HC0--HC3 adjustment (\emph{CL-0}--\emph{CL-3}). As further references, covariances from a clustered bootstrap (\emph{BS}), a mixed-effects model with random intercept (\emph{random}), and a GEE with exchangeable correlation structure (\emph{gee}) are assessed. \subsubsection{Outcome measure} In order to assess the validity of statistical inference based on clustered covariances, the empirical coverage rate of the 95\% Wald confidence intervals (from 10,000 replications) is the outcome measure of interest. If standard errors are estimated accurately, the empirical coverage should match the nominal rate of $0.95$. And empirical coverages falling short of $0.95$ are typically due to underestimated standard errors and would lead to inflated type~I errors in partial Wald tests of the coefficients. \subsubsection{Simulation code} \begin{table}[t!] \centering \begin{tabular}{llll} \hline Label & Model & Object & Variance-covariance matrix \\ \hline CL-0 & (\code{g})\code{lm} & \code{m} & \code{vcovCL(m, cluster = id, type = "HC0")} \\ CL-1 & (\code{g})\code{lm} & \code{m} & \code{vcovCL(m, cluster = id, type = "HC1")} \\ CL-2 & (\code{g})\code{lm} & \code{m} & \code{vcovCL(m, cluster = id, type = "HC2")} \\ CL-3 & (\code{g})\code{lm} & \code{m} & \code{vcovCL(m, cluster = id, type = "HC3")} \\ PL & (\code{g})\code{lm} & \code{m} & \code{vcovPL(m, cluster = id, adjust = FALSE)} \\ PC & (\code{g})\code{lm} & \code{m} & \code{vcovPC(m, cluster = id, order.by = round)} \\ \hline standard & (\code{g})\code{lm} & \code{m} & \code{vcov(m)} \\ basic & (\code{g})\code{lm} & \code{m} & \code{sandwich(m)} \\ \hline random & (\code{g})\code{lmer} & \code{m_re} & \code{vcov(m_re)} \\ gee & \code{geeglm} & \code{m_gee} & \code{m_gee\$geese\$vbeta} \\ \hline \end{tabular} \caption{Covariance matrices for responses from the exponential family in `\code{sim-CL.R}'. \label{tab:vcov}} \end{table} The supplementary \proglang{R} script `\code{sim-CL.R}' comprises the simulation code for the data generating process described above and includes functions \code{dgp()}, \code{fit()}, and \code{sim()}. While \code{dgp()} specifies the data generating process and generates a data frame with (up to) three regressors \code{x1}, \code{x2}, \code{x3} as well as cluster dimensions \code{id} and \code{round}, \code{fit()} is responsible for the model estimation as well as computation of the covariance matrix estimate and the empirical coverage. The function \code{sim()} sets up all factorial combinations of the specified scenarios and loops over the fits for each scenario (using multiple cores for parallelization). Table~\ref{tab:vcov} shows how the different types of covariances are calculated for responses from the exponential family. A pooled or marginal model (\code{m}), a random effects model (\code{m_re}, using \pkg{lme4}, \citealp{hac:Bates+Machler+Bolker:2015}), and a GEE with an exchangeable correlation structure (\code{m_gee}, using \pkg{geepack}, \citealp{hac:Halekoh+Hojsgaard+Yan:2002}) are fitted. For the other (non-GLM) responses, the functions \code{betareg()} (from \pkg{betareg}, \citealp{hac:Cribari-Neto+Zeileis:2010}), \code{zerotrunc()} (from \pkg{countreg}, \citealp{hac:Zeileis+Kleiber:2018}), and \code{zeroinfl()} (from \pkg{pscl}/\pkg{countreg}, \citealp{hac:Zeileis+Kleiber+Jackman:2008}) are used. \subsection{Results} Based on the design discussed above, the simulation study investigates the performance of clustered covariances for the following settings. % \begin{itemize} \item Experiment I: Different types of regressors for a Gaussian response distribution. \item Experiment II: Different GLM response distributions. \item Experiment III: Response distributions beyond the GLM. \item Experiment IV: GLMs with HC0--HC3 bias corrections. \end{itemize} \subsubsection{Experiment I} \setkeys{Gin}{width=\textwidth} \begin{figure}[t!] <>= my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s01$vcov <- factor(s01$vcov, levels(s01$vcov)[c(2,4,3,1,8,5,7,6)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8", "green","#006400", "#dc75ed", "darkred", "orange", "black", "grey") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 19, 19, 17, 25, 3, 8) xyplot(coverage ~ rho | par, groups = ~ factor(vcov), data = s01, subset = par != "(Intercept)", ylim = c(0.1, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 3), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Experiment I. Gaussian response with $G = 100$ (balanced) clusters of $5$ observations each. Regressor \code{x1} is correlated ($\rho_x = 0.25$), \code{x2} clustered, and \code{x3} uncorrelated. The coverage (from 10,000 replications) is plotted on the $y$-axis against the cluster correlation $\rho$ of the response distribution (from a Gaussian copula) on the $x$-axis. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-01} \end{figure} Figure~\ref{fig:sim-01} shows the results from Experiment~I and plots the empirical coverage probabilities (from 10,000 replications) on the $y$-axis for the coefficients of the correlated regressor \code{x1}, the clustered regressor \code{x2}, and the uncorrelated regressor \code{x3} against the cluster correlation $\rho$ on the $y$-axis. While for the uncorrelated regressor \code{x3} all methods -- except the Driscoll and Kraay PL estimator -- perform well and yield satisfactory coverage rates, the picture is different for the correlated and clustered regressors \code{x1} and \code{x2}. With increasing cluster correlation $\rho$ the performance deteriorates for those methods that either do not account for the clustering at all (i.e., ``standard'' covariance and basic sandwich covariance) or that treat the data as panel data (i.e., PL and PC). The reason for the poor performance of the panel data covariances is the low number of $5$~observations per cluster. This has already been documented in the literature: In a Monte-Carlo study, \cite{hac:Driscoll+Kraay:1998} use a minimum of 20--25 observations per cluster and \cite{hac:Hoechle:2007} notes that the PC estimator can be quite imprecise if the crosss-sectional dimension is large compared to the time dimension. As shown in Appendix~\ref{app:ar1}, the performance improves if an exponentially decaying AR(1) correlation structure is employed instead of the exchangeable structure and if the number of observations per cluster increases. As the effects of regressor \code{x1} are in between the effects of the clustered regressor \code{x2} and the uncorrelated regressor \code{x3}, the following simulation experiments focus on the situation with a single correlated regressor \code{x1}. \subsubsection{Experiment II} \begin{figure}[t!] <>= my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s02$dist <- factor(as.character(s02$dist), levels = c("gaussian", "binomial(logit)", "poisson")) s02$vcov <- factor(s02$vcov, levels(s02$vcov)[c(2,4,3,1,8,5,7,6)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8", "green","#006400", "#dc75ed", "darkred", "orange", "black", "grey") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 19, 19, 17, 25, 3, 8) xyplot(coverage ~ rho | dist, groups = ~ factor(vcov), data = s02, subset = par != "(Intercept)", ylim = c(0.5, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 3), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Experiment II. Exponential family response distributions with $G = 100$ (balanced) clusters of $5$ observations each. The only regressor \code{x1} is correlated ($\rho_x = 0.25$). The coverage (from 10,000 replications) is plotted on the $y$-axis against the cluster correlation $\rho$ of the response distribution (from a Gaussian copula) on the $x$-axis. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-02} \end{figure} Figure~\ref{fig:sim-02} illustrates the results from Experiment~II. The settings are mostly analogous to Experiment~I with two important differences: (1)~GLMs with Gaussian/binomial/Poisson response distribution are used. (2)~There is only one regressor (\code{x1}, correlated with $\rho_{x} = 0.25$). Overall, the results for binomial and Poisson response are very similar to the Gaussian case in Experiment~I. Thus, this confirms that clustered covariances also work well with GLMs. The only major difference between the linear Gaussian and nonlinear binomial/Poisson cases is the performance of the mixed-effects models with random intercept. While in linear models the marginal (or ``population-averaged'') approach employed with clustered covariances leads to analogous models compared to mixed-effects models, this is not the case in nonlinear models. With nonlinear links, mixed-effects models correspond to ``conditional'' rather than ``marginal'' models and for obtaining marginal expectations the random effects have to be integrated out \citep[see][]{hac:Molenberghs+Kenward+Verbeke+Iddi+Efendi:2013,hac:Fitzmaurice:2014}. Consequently, fixed effects have to be interpreted differently and their confidence intervals do not contain the population-averaged effects, thus leading to the results in Experiment~II. \subsubsection{Experiment III} \begin{figure}[t!] <>= s33 <- na.omit(s33) my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s33$vcov <- factor(s33$vcov, levels(s33$vcov)[c(2,1,4,3)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$fill <- c("#377eb8", "#000080", "darkred", "orange") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 17, 25) xyplot(coverage ~ rho | dist, groups = ~ factor(vcov), data = s33, subset = par == "x1", ylim = c(0.8, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Experiment III. Response distributions beyond the GLM (beta regression, zero-truncated Poisson, and zero-inflated Poisson) with $G = 100$ (balanced) clusters of $5$ observations each. The only regressor \code{x1} is correlated ($\rho_x = 0.25$). The coverage (from 10,000 replications for basic/standard/CL-0 and 1,000 replications for BS) is plotted on the $y$-axis against the cluster correlation $\rho$ of the response distribution (from a Gaussian copula) on the $x$-axis. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-03} \end{figure} Figure~\ref{fig:sim-03} shows the outcome of Experiment~III whose settings are similar to the previous Experiment~I. But now more general response distributions beyond the classic GLM framework are employed, revealing that the clustered covariances from \code{vcovCL()} indeed also work well in this setup. Again, the performance of the non-clustered covariances deteriorates with increasing cluster correlation. For the zero-truncated and zero-inflated Poisson distribution the empirical coverage rate is slightly lower than $0.95$. However, given that this does not depend on the extent of the correlation $\rho$ this is more likely due to the quality of the normal approximation in the Wald confidence interval. The clustered bootstrap covariance (BS) performs similarly to the clustered HC0 covariance but is computationally much more demanding due to the need for resampling and refitting the model (here with \code{R = 250} bootstrap samples). \subsubsection{Experiment IV} \begin{figure}[t!] <>= my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s04$dist <- factor(as.character(s04$dist), c("gaussian", "binomial(logit)", "poisson")) my.settings[["superpose.line"]]$col <- c("#377eb8", "#00E5EE", "#e41a1c", "#4daf4a", "#dc75ed") my.settings[["superpose.symbol"]]$col <- c("#377eb8", "#00E5EE","#e41a1c", "#4daf4a", "#dc75ed") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nid | dist, groups = ~ factor(vcov, levels = c(paste0("CL-", 0:3), "BS")), data = na.omit(s04), subset = par != "(Intercept)", type = "b", xlab = "G", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Experiment IV. Exponential family response distributions with cluster correlation $\rho = 0.25$ (from a Gaussian copula). The only regressor \code{x1} is correlated ($\rho_x = 0.25$). The coverage (from 10,000 replications) on the $y$-axis for different types of bias adjustment (HC0--HC3 and bootstrap) is plotted against the number of clusters $G$ on the $x$-axis. The number of clusters increases with $G = 10, 50, \dots, 250$ while the number of observations per cluster is fixed at~$5$. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-04} \end{figure} Figure~\ref{fig:sim-04} depicts the findings of Experiment~IV. The $y$-axis represents again the empirical coverage from 10,000 replications, but in contrast to the other simulation experiments, the number of clusters $G$ is plotted on the $x$-axis, ranging from 10 to 250 clusters. Gaussian, binomial and Poisson responses are compared with each other, with the focus on clustered standard errors with HC0--HC3 types of bias correction. (Recall that HC2 and HC3 require block-wise components of the full hat matrix and hence are at the moment only available for \code{lm()} and \code{glm()} fits, see Section~\ref{sec:vcovcl}.) Furthermore, clustered bootstrap standard errors are included for comparison (using \code{R = 250} bootstrap samples). In most cases, all of the standard errors are underestimated for $G = 10$ clusters (except clustered standard errors with HC3 bias correction for the binomial and Poisson response). As found in previous studies for clustered HC0/HC1 covariances \citep[][among others]{hac:Arceneaux+Nickerson:2009,hac:Petersen:2009,hac:Harden:2011,hac:Cameron+Miller:2015,hac:Pustejovsky+Tipton:2018}, the larger the number of clusters $G$, the better the coverage and the less standard errors are underestimated. In our study about 50--100 clusters are enough for sufficiently accurate coverage rates using clustered standard errors without a bias correction (CL-0 in the figure), which is consistent with other authors' findings. Additionally, it can be observed that the higher the number of clusters, the less the different types of HC bias correction differ. However, for a small number of clusters, the HC3 correction works best, followed by HC2, HC1 and HC0. This is also consistent with the findings for cross-sectional data, e.g., \cite{hac:Long+Ervin:2000} suggest to use HC3 in the linear model for small samples with fewer than 250 observations. Moreover, bootstrap covariances perform somewhat better than HC0/HC1 for a small number of clusters. However, \citet{hac:Pustejovsky+Tipton:2018} have argued that HC3 bias corrections for clustered covariances (CL-3 in the figure, sometimes referred to ``CR3'' or ``CR3VE'') tend to over-correct for small $G$, and recommend HC2 bias corrections (``CR2'', CL-2 in the figure) and $t$~tests using degrees of freedom calculated similar to \citet{hac:Satterthwaite:1946}. While Experiment IV finds that HC3 adjustments slightly over-correct for $G = 10$ in binomial models, overall, our findings are not consistent with the suggestion that HC3 is overly aggressive in correcting small $G$ bias in clustered standard errors. \section{Summary} While previous versions of the \pkg{sandwich} package already provided a flexible object-oriented implementation of covariances for cross-section and time series data, the corresponding functions for clustered and panel data have only been added recently (in version 2.4-0 of the package). Compared to previous implementations in \proglang{R} that were somewhat scattered over several packages, the implementation in \pkg{sandwich} offers a wide range of ``flavors'' of clustered covariances and, most importantly, is applicable to any model object that provides methods to extract the \code{estfun()} (estimating functions; observed score matrix) and \code{bread()} (inverse Hessian). Therefore, it is possible to apply the new functions \code{vcovCL()}, \code{vcovPL()}, and \code{vcovPC()} to models beyond linear regression. A thorough Monte Carlo study assesses the performance of these functions in regressions beyond the standard linear Gaussian scenario, e.g., for exponential family distributions and beyond and for the less frequently used HC2 and HC3 adjustments. This shows that clustered covariances work reasonably well in the models investigated but some care is needed when applying panel estimators (\code{vcovPL()} and \code{vcovPC()}) in panel data with ``short'' panels and/or non-decaying autocorrelations. \section*{Computational details} The packages \pkg{sandwich}, \pkg{boot}, \pkg{countreg}, \pkg{geepack}, \pkg{lattice}, \pkg{lme4}, \pkg{lmtest}, \pkg{multiwayvcov}, \pkg{plm} and \pkg{pscl} are required for the applications in this paper. For replication of the simulation study, the supplementary \proglang{R} script \code{sim-CL.R} is provided along with the corresponding results \code{sim-CL.rda}. \proglang{R} version \Sexpr{paste(R.Version()[6:7], collapse = ".")} has been used for computations. Package versions that have been employed are \pkg{sandwich} \Sexpr{gsub("-", "--", packageDescription("sandwich")$Version)}, \pkg{boot} \Sexpr{gsub("-", "--", packageDescription("boot")$Version)}, \pkg{countreg} 0.2-0, \pkg{geepack} \Sexpr{gsub("-", "--", packageDescription("geepack")$Version)}, \pkg{lattice} \Sexpr{gsub("-", "--", packageDescription("lattice")$Version)}, \pkg{lme4} \Sexpr{gsub("-", "--", packageDescription("lme4")$Version)}, \pkg{lmtest} \Sexpr{gsub("-", "--", packageDescription("lmtest")$Version)}, \pkg{multiwayvcov} \Sexpr{gsub("-", "--", packageDescription("multiwayvcov")$Version)}, \pkg{plm} \Sexpr{gsub("-", "--", packageDescription("plm")$Version)}, and \pkg{pscl} \Sexpr{gsub("-", "--", packageDescription("pscl")$Version)}. \proglang{R} itself and all packages (except \pkg{countreg}) used are available from CRAN at \url{https://CRAN.R-project.org/}. \pkg{countreg} is available from \url{https://R-Forge.R-project.org/projects/countreg/}. \section*{Acknowledgments} The authors are grateful to the editor and reviewers that helped to substantially improve manuscript and software, as well as to Keith Goldfeld (NYU School of Medicine) for providing insights and references regarding the differences of conditional and marginal models for clustered data. \bibliography{hac} \newpage \begin{appendix} \section{Simulation results for panel data with AR(1) correlations} \label{app:ar1} \begin{figure}[b!] <>= my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s0607$vcov <- factor(s0607$vcov, levels(s0607$vcov)[c(1,3,2)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8","green", "#006400") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nround | factor(par) + factor(copula), groups = ~ factor(vcov), data = na.omit(s0607), subset = par != "(Intercept)", type = "b", xlab = "Observations per cluster", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Supplementary simulation experiment. Gaussian response with $G = 100$ (balanced) clusters of $5$, $10$, $20$, or $50$ observations each. Regressor \code{x1} is correlated ($\rho_x = 0.25$), \code{x2} clustered, and \code{x3} uncorrelated. Either an exchangeable cluster correlation of $\rho = 0.25$ or an exponentially decaying AR(1) correlation structure with autoregressive coefficient $\rho = 0.25$ is used. The coverage (from 10,000 replications) is plotted on the $y$-axis against the number of observations per cluster on the $x$-axis. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-0607} \end{figure} As observed in Figures~\ref{fig:sim-01}--\ref{fig:sim-02}, the estimators for panel covariances ({PL} and \code{PC}) have problems with the ``short'' panels of only 5~observations per cluster. To assess whether the estimators perform correctly in those situations they were designed for, we take a closer look at (a)~``longer'' panels (with up to $50$~observations per cluster), and (b)~an exponentially decaying autoregressive (AR) correlation structure of order~1 instead of an exchangeable correlation structure. Figure~\ref{fig:sim-0607} shows the results from a supplementary simulation experiment that is analogous to Experiment~I. The two differences are: (1)~The number of observations per cluster is increased from $5$ up to $50$ and the cluster correlation is fixed at $\rho = 0.25$ (with higher values leading to qualitatively the same results). (2)~Additionally, an AR(1) correlation structure is considered. Somewhat surprisingly, the standard clustered HC0 covariance performs satisfactorily in all scenarios and better than the panel estimators (PL and PC). The latter approach the desired coverage of $0.95$ when the panels become longer (i.e., the number of observations per cluster increases) and the correlation structure is AR(1). However, in case of an exchangeable correlation structure and correlated/clustered regressors, the coverage even decreases for longer panels. The reason for this is that the panel covariance estimators are based on the assumption that correlations are dying out, which is the case for an AR(1) structure, but not for an exchangeable correlation structure. Additionally, Figure~\ref{fig:sim-08} brings out that the AR(1) findings are not limited to the Gaussian case but can be confirmed for binomial and Poisson GLMs as well. \begin{figure}[t!] <>= my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s08$vcov <- factor(s08$vcov, levels(s08$vcov)[c(1,3,2)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8","green", "#006400") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nround | factor(par) + factor(dist), groups = ~ factor(vcov), data = na.omit(s08), subset = par != "(Intercept)", type = "b", xlab = "Observations per cluster", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) @ \caption{Supplementary simulation experiment. Poisson and binomial response with $G = 100$ (balanced) clusters of $5$, $10$, $20$, or $50$ observations each. Regressor \code{x1} is correlated ($\rho_x = 0.25$), \code{x2} clustered, and \code{x3} uncorrelated. An exponentially decaying AR(1) correlation structure with autoregressive coefficient $\rho = 0.25$ is used. The coverage (from 10,000 replications) is plotted on the $y$-axis against the number of observations per cluster on the $x$-axis. The horizontal reference line indicates the nominal coverage of $0.95$.} \label{fig:sim-08} \end{figure} \end{appendix} \end{document} sandwich/inst/doc/sandwich-OOP.R0000644000175400001440000000654213452214374016361 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 ################################################### suppressWarnings(RNGversion("3.5.0")) 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:627-632 ################################################### 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:639-641 ################################################### coeftest(fm_tobit) coeftest(fm_tobit, vcov = sandwich) ################################################### ### code chunk number 10: sandwich-OOP.Rnw:649-651 ################################################### coeftest(fm_probit) coeftest(fm_probit, vcov = sandwich) sandwich/inst/doc/sandwich.pdf0000644000175400001440000062646313452214402016300 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4199 /Filter /FlateDecode /N 75 /First 620 >> stream x\Ys8~_Ij&\ܚJo;bEv[y%F#R3~" JM֖&H@FHH$0NDAdĉ$*D(HD|O1C TF\.a$p`paHT12zHr D@H1Ł0 @t{0p'a E  Pa 3 B>1ǩZ0P( 01~ `8( CAwЏ`z0T 8 0I@,Y>rGd" -e1B`AB!BxDXi292LB\LOnql>ȫ{ΠB:#o«p6MhrH+A{o6z|6Z Aϲ<ӇDo7N9y4Ck{wlf;$<)3+Nl1Qbq /+e<)7}s 'ax̕2o)7 6mgs6oE '**1iL m KFi={ O&g8QMY4.`aIg;L|1VC0&^l=s W|ݶT:učō7m}z==' z)=$w1}JYZ'zA?+-7f&`os[cO(|܎Bx £4'Y,kh2'&SsZ A/d}".~Y_ ?Z1M.@#X0^d@_f_M}eOdew&Ki6j ><,HUZown}z}oM1YwBjT5{ҍ>O?.g-q6LSrG ѭ̾8G:kO:)}\jx ޵++; 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}=C9(3=DA#YS\GR17(391O7$#57QOL-Tа())///K-/JQ(,PJ-N-*KMQp+QKMU9Tp-(-I-ROI-c``` b`0f`bdd3{m>d|]?ʿ_=5#vnaKoH=}ZlX'}wf7,ֽ[9+}*wJ=!/pq.3~.endstream endobj 203 0 obj << /Filter /FlateDecode /Length 608 >> stream xT=o@ +n<ՙ}"Keq|Nb;:l*/=PG^(- ?{&։mzt*|%> EjZFG֋vUM6AmA9GyOh!UXkN^ !e)9z,) 1F3d\R7*\pFC8EcJ!T^=I' Y `rSpY #aET,9j5$(6Rɟ\HǦr#H$0]c+BqRIvXʞJ,tUޡ$r%3lق^ný[vf~kr12zb1';EysMu䜜J':SyV-| &jj5̡1PSKa2Y[]TĄ \,y" uZ¨@)1gb^3\[K2@3 '  w (Àf{YrtV_]?yK[ w (\ zb]UaG4x`ݕ{c ۡa_Tkendstream endobj 204 0 obj << /Type /XRef /Length 224 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Info 3 0 R /Root 2 0 R /Size 205 /ID [<17f53e5d94cad4ed79b2cb011806ee64><9b3eae0ffd6b203acf8fc19c80a6d232>] >> stream x;aag#" AA(Tꩄ=P+$TT cم_Nn&WdI } OjicgkkȕvθV0 cr) rval$coverage <- as.numeric(abs(cf - rval$coef)/rval$se < cr) return(rval) } ## loop over simulation scenarios sim <- function(nrep = 1000, nid = 100L, nround = 5L, dist = "gaussian", rho = 0.5, xrho = 0.5, coef = c(0, 0.85, 0.5, 0.7), formula = response ~ x1 + x2 + x3, vcov = c("standard", "basic", "HC1", "HC2", "HC3", "CL-0", "CL-1", "CL-2", "CL-3", "fixed", "random", "gee", "PL", "PC", "BS"), ..., cores = NULL) { ## parallelization support applyfun <- if(is.null(cores)) { lapply } else { function(X, FUN, ...) parallel::mclapply(X, FUN, ..., mc.cores = cores) } ## all factorial combinations of experimental conditions par <- expand.grid(nid = nid, nround = nround, dist = dist, rho = rho, xrho = xrho, stringsAsFactors = FALSE) ## conduct all simulations rval <- lapply(1L:nrow(par), function(i) { rvali <- applyfun(1L:nrep, function(j) { d <- dgp(nid = par$nid[i], nround = par$nround[i], dist = par$dist[i], rho = par$rho[i], xrho = par$xrho[i], coef = coef, ...) ff <- formula try(fit(d, formula = ff, dist = par$dist[i], vcov = vcov)) }) rvali <- rvali[sapply(rvali, class) == "data.frame"] rvali[[1L]][, -(1L:2L)] <- Reduce("+", lapply(rvali, "[", , -(1:2)))/length(rvali) rvali <- rvali[[1L]] rvali$nid <- par$nid[i] rvali$nround <- par$nround[i] rvali$dist <- par$dist[i] rvali$rho <- par$rho[i] rvali$xrho <- par$xrho[i] return(rvali) }) rval <- do.call("rbind", rval) ## turn all experimental condition variables into factors rval$dist <- factor(rval$dist) rval$vcov <- factor(rval$vcov) rval$par <- factor(rval$par) rval$nid <- factor(rval$nid) rval$nround <- factor(rval$nround) rval$rho <- factor(rval$rho) rval$xrho <- factor(rval$xrho) return(rval) } ## Bootstrap for InstInnovation hurdle model ------------------------------------------------------- library("sandwich") library("pscl") data("InstInnovation", package = "sandwich") h_innov <- hurdle( cites ~ institutions + log(capital/employment) + log(sales), data = InstInnovation, dist = "negbin") suppressWarnings(RNGversion("3.5.0")) set.seed(0) vc_innov <- list( "standard" = vcov(h_innov), "basic" = sandwich(h_innov), "CL-1" = vcovCL(h_innov, cluster = InstInnovation$company), "boot" = vcovBS(h_innov, cluster = InstInnovation$company) ) ## Simulation study -------------------------------------------------------------------------------- library("copula") library("lme4") library("geepack") library("countreg") library("betareg") RNGkind(kind = "L'Ecuyer-CMRG") set.seed(1) s01 <- sim(nrep = 10000, nid = 100, nround = 5, dist = "gaussian", rho = seq(0, 0.9, by = 0.1), xrho = 0.25, coef = c(0, 0.85, 0.5, 0.7), formula = response ~ x1 + x2 + x3, vcov = c("standard", "basic", "CL-0", "random", "gee", "PC", "PL", "BS"), type = "copula", cores = 16) set.seed(2) s02 <- sim(nrep = 10000, nid = 100, nround = 5, dist = c("gaussian", "binomial(logit)", "poisson"), rho = seq(0, 0.9, by = 0.1), xrho = 0.25, coef = c(0, 0.85, 0, 0), formula = response ~ x1, vcov = c("standard", "basic", "CL-0", "random", "gee", "PC", "PL", "BS"), type = "copula", cores = 16) set.seed(3) s03 <- sim(nrep = 10000, nid = 100, nround = 5, dist = c("zerotrunc", "zeroinfl", "betareg"), rho = seq(0, 0.9, by = 0.1), xrho = 0.25, coef = c(0, 0.85, 0, 0), formula = response ~ x1, vcov = c("standard", "basic", "CL-0"), ## BS separately below (s33) type = "copula", cores = 16) set.seed(4) s04 <- sim(nrep = 10000, nid = c(10, seq(50, 250, by = 50)), nround = 5, dist = c("gaussian","poisson", "binomial(logit)"), rho = 0.25, xrho = 0.25, coef = c(0, 0.85, 0, 0), formula = response ~ x1, vcov = c("CL-0", "CL-1", "CL-2", "CL-3", "BS"), type = "copula", cores = 16) set.seed(6) s06 <- sim(nrep = 10000, nround = c(5, 10, 20, 50), nid = 100, dist = "gaussian", rho = 0.25, xrho = 0.25, coef = c(0, 0.85, 0.5, 0.7), formula = response ~ x1 + x2 + x3, vcov = c("CL-0", "PC", "PL"), type = "copula", cores = 16) set.seed(7) s07 <- sim(nrep = 10000, nround = c(5, 10, 20, 50), nid = 100, dist = "gaussian", rho = 0.25, xrho = 0.25, coef = c(0, 0.85, 0.5, 0.7), formula = response ~ x1 + x2 + x3, vcov = c("CL-0", "PC", "PL"), type = "copula-ar1", cores = 16) set.seed(8) s08 <- sim(nrep = 10000, nround = c(5, 10, 20, 50), nid = 100, dist = c("binomial(logit)", "poisson"), rho = 0.25, xrho = 0.25, coef = c(0, 0.85, 0.5, 0.7), formula = response ~ x1 + x2 + x3, vcov = c("CL-0", "PC", "PL"), type = "copula-ar1", cores = 16) set.seed(33) s33 <- sim(nrep = 10000, nid = 100, nround = 5, dist = c("zerotrunc", "zeroinfl", "betareg"), rho = seq(0, 0.9, by = 0.1), xrho = 0.25, coef = c(0, 0.85, 0, 0), formula = response ~ x1, vcov = "BS", type = "copula", cores = 16) s06$copula <- factor(rep.int("copula", nrow(s06)), levels = c("copula-ar1", "copula"), labels = c("AR(1)", "Exchangeable")) s07$copula <- factor(rep.int("copula-ar1", nrow(s07)), levels = c("copula-ar1", "copula"), labels = c("AR(1)", "Exchangeable")) s0607 <- rbind(s06, s07) s03$vcov <- as.character(s03$vcov) s33$vcov <- as.character(s33$vcov) s33 <- rbind(s03, s33) s33$vcov <- factor(s33$vcov) save(s01, s02, s03, s04, s06, s07, s0607, s08, vc_innov, s33, file = "sim-CL.rda") ## ------------------------------------------------------------------------------------------------- sandwich/inst/doc/sandwich-CL.R0000644000175400001440000002420613452214370016213 0ustar zeileisusers### R code from vignette source 'sandwich-CL.Rnw' ################################################### ### code chunk number 1: preliminaries ################################################### library("sandwich") library("geepack") library("lattice") library("lmtest") library("multiwayvcov") library("pcse") library("plm") library("pscl") panel.xyref <- function(x, y, ...) { panel.abline(h = 0.95, col = "slategray") panel.xyplot(x, y, ...) } se <- function(vcov) sapply(vcov, function(x) sqrt(diag(x))) options(prompt = "R> ", continue = "+ ", digits = 5) if(file.exists("sim-CL.rda")) { load("sim-CL.rda") } else { source("sim-CL.R") } ################################################### ### code chunk number 2: innovation-data ################################################### data("InstInnovation", package = "sandwich") library("pscl") h_innov <- hurdle( cites ~ institutions + log(capital/employment) + log(sales), data = InstInnovation, dist = "negbin") ################################################### ### code chunk number 3: innovation-coeftest ################################################### library("sandwich") library("lmtest") coeftest(h_innov, vcov = vcovCL, cluster = ~ company) ################################################### ### code chunk number 4: innovation-se (eval = FALSE) ################################################### ## suppressWarnings(RNGversion("3.5.0")) ## set.seed(0) ## vc <- list( ## "standard" = vcov(h_innov), ## "basic" = sandwich(h_innov), ## "CL-1" = vcovCL(h_innov, cluster = ~ company), ## "boot" = vcovBS(h_innov, cluster = ~ company) ## ) ## se <- function(vcov) sapply(vcov, function(x) sqrt(diag(x))) ## se(vc) ################################################### ### code chunk number 5: innovation-se2 ################################################### se(vc_innov) ################################################### ### code chunk number 6: petersen-model ################################################### data("PetersenCL", package = "sandwich") p_lm <- lm(y ~ x, data = PetersenCL) ################################################### ### code chunk number 7: petersen-comparison1 ################################################### library("multiwayvcov") se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm), "multiwayvcov" = cluster.vcov(p_lm, cluster = ~ firm) )) ################################################### ### code chunk number 8: petersen-comparison2 ################################################### library("plm") p_plm <- plm(y ~ x, data = PetersenCL, model = "pooling", indexes = c("firm", "year")) library("geepack") vcov.geeglm <- function(object) { vc <- object$geese$vbeta rownames(vc) <- colnames(vc) <- names(coef(object)) return(vc) } p_gee <- geeglm(y ~ x, data = PetersenCL, id = PetersenCL$firm, corstr = "independence", family = gaussian) se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm, cadjust = FALSE, type = "HC0"), "plm" = vcovHC(p_plm, cluster = "group"), "geepack" = vcov(p_gee) )) ################################################### ### code chunk number 9: petersen-twocl ################################################### se(list( "sandwich" = vcovCL(p_lm, cluster = ~ firm + year, multi0 = TRUE), "multiwayvcov" = cluster.vcov(p_lm, cluster = ~ firm + year) )) ################################################### ### code chunk number 10: petersen-comparison3 ################################################### se(list( "sandwich" = vcovPL(p_lm, cluster = ~ firm + year, adjust = FALSE), "plm" = vcovSCC(p_plm) )) ################################################### ### code chunk number 11: petersen-comparison4 ################################################### library("pcse") se(list( "sandwich" = sandwich::vcovPC(p_lm, cluster = ~ firm + year), "pcse" = pcse::vcovPC(p_lm, groupN = PetersenCL$firm, groupT = PetersenCL$year) )) ################################################### ### code chunk number 12: petersen-unbalanced1 ################################################### PU <- subset(PetersenCL, !(firm == 1 & year == 10)) pu_lm <- lm(y ~ x, data = PU) ################################################### ### code chunk number 13: petersen-unbalanced2 ################################################### se(list( "sandwichT" = sandwich::vcovPC(pu_lm, cluster = ~ firm + year, pairwise = TRUE), "pcseT" = pcse::vcovPC(pu_lm, PU$firm, PU$year, pairwise = TRUE), "sandwichF" = sandwich::vcovPC(pu_lm, cluster = ~ firm + year, pairwise = FALSE), "pcseF" = pcse::vcovPC(pu_lm, PU$firm, PU$year, pairwise = FALSE) )) ################################################### ### code chunk number 14: sim-01-figure ################################################### my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s01$vcov <- factor(s01$vcov, levels(s01$vcov)[c(2,4,3,1,8,5,7,6)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8", "green","#006400", "#dc75ed", "darkred", "orange", "black", "grey") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 19, 19, 17, 25, 3, 8) xyplot(coverage ~ rho | par, groups = ~ factor(vcov), data = s01, subset = par != "(Intercept)", ylim = c(0.1, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 3), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) ################################################### ### code chunk number 15: sim-02-figure ################################################### my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s02$dist <- factor(as.character(s02$dist), levels = c("gaussian", "binomial(logit)", "poisson")) s02$vcov <- factor(s02$vcov, levels(s02$vcov)[c(2,4,3,1,8,5,7,6)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8", "green","#006400", "#dc75ed", "darkred", "orange", "black", "grey") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 19, 19, 17, 25, 3, 8) xyplot(coverage ~ rho | dist, groups = ~ factor(vcov), data = s02, subset = par != "(Intercept)", ylim = c(0.5, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 3), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) ################################################### ### code chunk number 16: sim-03-figure ################################################### s33 <- na.omit(s33) my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s33$vcov <- factor(s33$vcov, levels(s33$vcov)[c(2,1,4,3)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- my.settings[["superpose.symbol"]]$fill <- c("#377eb8", "#000080", "darkred", "orange") my.settings[["superpose.symbol"]]$pch <- c(19, 19, 17, 25) xyplot(coverage ~ rho | dist, groups = ~ factor(vcov), data = s33, subset = par == "x1", ylim = c(0.8, 1), type = "b", xlab = expression(rho), ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) ################################################### ### code chunk number 17: sim-04-figure ################################################### my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s04$dist <- factor(as.character(s04$dist), c("gaussian", "binomial(logit)", "poisson")) my.settings[["superpose.line"]]$col <- c("#377eb8", "#00E5EE", "#e41a1c", "#4daf4a", "#dc75ed") my.settings[["superpose.symbol"]]$col <- c("#377eb8", "#00E5EE","#e41a1c", "#4daf4a", "#dc75ed") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nid | dist, groups = ~ factor(vcov, levels = c(paste0("CL-", 0:3), "BS")), data = na.omit(s04), subset = par != "(Intercept)", type = "b", xlab = "G", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) ################################################### ### code chunk number 18: sim-0607-figure ################################################### my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s0607$vcov <- factor(s0607$vcov, levels(s0607$vcov)[c(1,3,2)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8","green", "#006400") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nround | factor(par) + factor(copula), groups = ~ factor(vcov), data = na.omit(s0607), subset = par != "(Intercept)", type = "b", xlab = "Observations per cluster", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) ################################################### ### code chunk number 19: sim-08-figure ################################################### my.settings <- canonical.theme(color = TRUE) my.settings[["strip.background"]]$col <- "gray" my.settings[["strip.border"]]$col <- "black" my.settings[["superpose.line"]]$lwd <- 1 s08$vcov <- factor(s08$vcov, levels(s08$vcov)[c(1,3,2)]) my.settings[["superpose.line"]]$col <- my.settings[["superpose.symbol"]]$col <- c("#377eb8","green", "#006400") my.settings[["superpose.symbol"]]$pch <- 19 xyplot(coverage ~ nround | factor(par) + factor(dist), groups = ~ factor(vcov), data = na.omit(s08), subset = par != "(Intercept)", type = "b", xlab = "Observations per cluster", ylab = "Empirical coverage", auto.key = list(columns = 2), par.strip.text = list(col = "black"), par.settings = my.settings, panel = panel.xyref) sandwich/tests/0000755000175400001440000000000013334123677013420 5ustar zeileisuserssandwich/tests/vcovPL.R0000644000175400001440000000162313331707405014750 0ustar zeileisuserslibrary("sandwich") data("PetersenCL", package = "sandwich") m <- lm(y ~ x, data = PetersenCL) vcovPL(m, cluster = ~ firm + year, adjust = TRUE) vcovPL(m, cluster = ~ firm + year, adjust = FALSE) data("InstInnovation", package = "sandwich") n <- glm(cites ~ institutions, family = poisson, data = InstInnovation) vcovPL(n, cluster = ~ industry, adjust = TRUE) vcovPL(n, cluster = ~ industry, adjust = FALSE) ## vcovPL covariance matrix compared with vcovSCC from plm package pm <- plm::plm(y ~ x, data = PetersenCL, model = "pooling", indexes = c("firm", "year")) (pl1 <- vcovPL(m, cluster = ~ firm, adjust = FALSE)) (pl2 <- plm::vcovSCC(pm)) attr(pl2, "cluster") <- NULL all.equal(pl1, pl2) ## vcovPL compared with Stata's xtscc (xtscc y x, lag(1) ase) standard errors pl4 <- c(0.0243573, 0.0281633) names(pl4) <- c("(Intercept)", "x") pl4 (pl3 <- sqrt(diag(pl1))) all.equal(pl3, pl4, tolerance = 1e-6) sandwich/tests/vcovPC.Rout.save0000644000175400001440000000535313332070526016426 0ustar zeileisusers R version 3.5.1 (2018-07-02) -- "Feather Spray" Copyright (C) 2018 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. 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. > library("sandwich") > data("PetersenCL", package = "sandwich") > m <- lm(y ~ x, data = PetersenCL) > sandwich::vcovPC(m, cluster = ~ firm + year) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > > PU <- subset(PetersenCL, !(firm == 1 & year == 10)) > u_m <- lm(y ~ x, data = PU) > sandwich::vcovPC(u_m, cluster = ~ firm + year, pairwise = TRUE) (Intercept) x (Intercept) 4.870754e-04 -4.566605e-05 x -4.566605e-05 6.419998e-04 > sandwich::vcovPC(u_m, cluster = ~ firm + year, pairwise = FALSE) (Intercept) x (Intercept) 5.108853e-04 -8.787789e-05 x -8.787789e-05 6.371175e-04 > > > ## vcovPC compared with Stata's xtpcse (xtscc y x) standard errors > pc1 <- c(0.0222006 , 0.025276) > names(pc1) <- c("(Intercept)", "x") > pc1 (Intercept) x 0.0222006 0.0252760 > (pc2 <- sqrt(diag(sandwich::vcovPC(m, cluster = PetersenCL$firm, order.by = PetersenCL$year)))) (Intercept) x 0.02220064 0.02527598 > all.equal(pc1, pc2, tolerance = 1e-5) [1] TRUE > > > ## sandwich::vcovPC compared to pcse::vcovPC > (pc3 <- pcse::vcovPC(u_m, PU$firm, PU$year, pairwise = FALSE)) X.Intercept. x X.Intercept. 5.108853e-04 -8.787789e-05 x -8.787789e-05 6.371175e-04 > (pc4 <- sandwich::vcovPC(u_m, cluster = ~ firm + year, pairwise = FALSE)) (Intercept) x (Intercept) 5.108853e-04 -8.787789e-05 x -8.787789e-05 6.371175e-04 > (pc5 <- pcse::vcovPC(u_m, PU$firm, PU$year, pairwise = TRUE)) X.Intercept. x X.Intercept. 4.870754e-04 -4.566605e-05 x -4.566605e-05 6.419998e-04 > (pc6 <- sandwich::vcovPC(u_m, cluster = ~ firm + year, pairwise = TRUE)) (Intercept) x (Intercept) 4.870754e-04 -4.566605e-05 x -4.566605e-05 6.419998e-04 > rownames(pc3) <- colnames(pc3) <- rownames(pc5) <- colnames(pc5) <- c("(Intercept)", "x") > all.equal(pc3, pc4) [1] TRUE > all.equal(pc5, pc6) [1] TRUE > > proc.time() user system elapsed 2.726 0.995 3.707 sandwich/tests/Examples/0000755000175400001440000000000013335514117015170 5ustar zeileisuserssandwich/tests/Examples/sandwich-Ex.Rout.save0000644000175400001440000007564113452213254021165 0ustar zeileisusers R version 3.5.1 (2018-07-02) -- "Feather Spray" Copyright (C) 2018 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale 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') > base::assign(".old_wd", base::getwd(), pos = 'CheckExEnv') > cleanEx() > nameEx("InstInnovation") > ### * InstInnovation > > flush(stderr()); flush(stdout()) > > ### Name: InstInnovation > ### Title: Innovation and Institutional Ownership > ### Aliases: InstInnovation > ### Keywords: datasets > > ### ** Examples > > ## Poisson models from Table I in Aghion et al. (2013) > > ## load data set > data("InstInnovation", package = "sandwich") > > ## log-scale variable > InstInnovation$lograndd <- log(InstInnovation$randd) > InstInnovation$lograndd[InstInnovation$lograndd == -Inf] <- 0 > > ## regression formulas > f1 <- cites ~ institutions + log(capital/employment) + log(sales) + industry + year > f2 <- cites ~ institutions + log(capital/employment) + log(sales) + + industry + year + lograndd + drandd > f3 <- cites ~ institutions + log(capital/employment) + log(sales) + + industry + year + lograndd + drandd + dprecites + log(precites) > > ## Poisson models > tab_I_3_pois <- glm(f1, data = InstInnovation, family = poisson) > tab_I_4_pois <- glm(f2, data = InstInnovation, family = poisson) > tab_I_5_pois <- glm(f3, data = InstInnovation, family = poisson) > > ## one-way clustered covariances > vCL_I_3 <- vcovCL(tab_I_3_pois, cluster = ~ company) > vCL_I_4 <- vcovCL(tab_I_4_pois, cluster = ~ company) > vCL_I_5 <- vcovCL(tab_I_5_pois, cluster = ~ company) > > ## replication of columns 3 to 5 from Table I in Aghion et al. (2013) > cbind(coef(tab_I_3_pois), sqrt(diag(vCL_I_3)))[2:4, ] [,1] [,2] institutions 0.009687237 0.002406388 log(capital/employment) 0.482883549 0.135953255 log(sales) 0.820317600 0.041523405 > cbind(coef(tab_I_4_pois), sqrt(diag(vCL_I_4)))[c(2:4, 148), ] [,1] [,2] institutions 0.008460789 0.002242345 log(capital/employment) 0.346008637 0.165274677 log(sales) 0.349190437 0.117219737 lograndd 0.492667825 0.140473107 > cbind(coef(tab_I_5_pois), sqrt(diag(vCL_I_5)))[c(2:4, 148), ] [,1] [,2] institutions 0.007381543 0.002443707 log(capital/employment) 0.440056227 0.131984715 log(sales) 0.183853108 0.063364163 lograndd 0.008971905 0.107406681 > > > > 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.default 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 > > suppressWarnings(RNGversion("3.5.0")) > 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("vcovBS") > ### * vcovBS > > flush(stderr()); flush(stdout()) > > ### Name: vcovBS > ### Title: (Clustered) Bootstrap Covariance Matrix Estimation > ### Aliases: vcovBS vcovBS.default vcovBS.lm vcovBS.glm .vcovBSenv > ### Keywords: regression bootstrap > > ### ** Examples > > ## Petersen's data > data("PetersenCL", package = "sandwich") > m <- lm(y ~ x, data = PetersenCL) > > ## comparison of different standard errors > suppressWarnings(RNGversion("3.5.0")) > set.seed(1) > cbind( + "classical" = sqrt(diag(vcov(m))), + "HC-cluster" = sqrt(diag(vcovCL(m, cluster = ~ firm))), + "BS-cluster" = sqrt(diag(vcovBS(m, cluster = ~ firm))) + ) classical HC-cluster BS-cluster (Intercept) 0.02835932 0.06701270 0.07067533 x 0.02858329 0.05059573 0.04878784 > > ## two-way wild cluster bootstrap with Mammen distribution > vcovBS(m, cluster = ~ firm + year, type = "wild-mammen") (Intercept) x (Intercept) 0.003535619 0.000488708 x 0.000488708 0.002589785 > > > > cleanEx() > nameEx("vcovCL") > ### * vcovCL > > flush(stderr()); flush(stdout()) > > ### Name: vcovCL > ### Title: Clustered Covariance Matrix Estimation > ### Aliases: vcovCL meatCL > ### Keywords: regression > > ### ** Examples > > ## Petersen's data > data("PetersenCL", package = "sandwich") > m <- lm(y ~ x, data = PetersenCL) > > ## clustered covariances > ## one-way > vcovCL(m, cluster = ~ firm) (Intercept) x (Intercept) 4.490702e-03 -6.473517e-05 x -6.473517e-05 2.559927e-03 > vcovCL(m, cluster = PetersenCL$firm) ## same (Intercept) x (Intercept) 4.490702e-03 -6.473517e-05 x -6.473517e-05 2.559927e-03 > ## one-way with HC2 > vcovCL(m, cluster = ~ firm, type = "HC2") (Intercept) x (Intercept) 4.494487e-03 -6.592912e-05 x -6.592912e-05 2.568236e-03 > ## two-way > vcovCL(m, cluster = ~ firm + year) (Intercept) x (Intercept) 4.233313e-03 -2.845344e-05 x -2.845344e-05 2.868462e-03 > vcovCL(m, cluster = PetersenCL[, c("firm", "year")]) ## same (Intercept) x (Intercept) 4.233313e-03 -2.845344e-05 x -2.845344e-05 2.868462e-03 > > ## comparison with cross-section sandwiches > ## HC0 > all.equal(sandwich(m), vcovCL(m, type = "HC0", cadjust = FALSE)) [1] TRUE > ## HC2 > all.equal(vcovHC(m, type = "HC2"), vcovCL(m, type = "HC2")) [1] TRUE > ## HC3 > all.equal(vcovHC(m, type = "HC3"), vcovCL(m, type = "HC3")) [1] TRUE > > ## Innovation data > data("InstInnovation", package = "sandwich") > > ## replication of one-way clustered standard errors for model 3, Table I > ## and model 1, Table II in Berger et al. (2016) > > ## count regression formula > f1 <- cites ~ institutions + log(capital/employment) + log(sales) + industry + year > > ## model 3, Table I: Poisson model > ## one-way clustered standard errors > tab_I_3_pois <- glm(f1, data = InstInnovation, family = poisson) > vcov_pois <- vcovCL(tab_I_3_pois, InstInnovation$company) > sqrt(diag(vcov_pois))[2:4] institutions log(capital/employment) log(sales) 0.002406388 0.135953255 0.041523405 > > ## coefficient tables > if(require("lmtest")) { + coeftest(tab_I_3_pois, vcov = vcov_pois)[2:4, ] + } Loading required package: lmtest Loading required package: zoo Attaching package: ‘zoo’ The following objects are masked from ‘package:base’: as.Date, as.Date.numeric Estimate Std. Error z value Pr(>|z|) institutions 0.009687237 0.002406388 4.025634 5.682195e-05 log(capital/employment) 0.482883549 0.135953255 3.551835 3.825545e-04 log(sales) 0.820317600 0.041523405 19.755548 7.187199e-87 > > ## Not run: > ##D ## model 1, Table II: negative binomial hurdle model > ##D ## (requires "pscl" or alternatively "countreg" from R-Forge) > ##D library("pscl") > ##D library("lmtest") > ##D tab_II_3_hurdle <- hurdle(f1, data = InstInnovation, dist = "negbin") > ##D # dist = "negbin", zero.dist = "negbin", separate = FALSE) > ##D vcov_hurdle <- vcovCL(tab_II_3_hurdle, InstInnovation$company) > ##D sqrt(diag(vcov_hurdle))[c(2:4, 149:151)] > ##D coeftest(tab_II_3_hurdle, vcov = vcov_hurdle)[c(2:4, 149:151), ] > ## End(Not run) > > > > cleanEx() detaching ‘package:lmtest’, ‘package:zoo’ > 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("vcovPC") > ### * vcovPC > > flush(stderr()); flush(stdout()) > > ### Name: vcovPC > ### Title: Panel-Corrected Covariance Matrix Estimation > ### Aliases: vcovPC meatPC > ### Keywords: regression > > ### ** Examples > > ## Petersen's data > data("PetersenCL", package = "sandwich") > m <- lm(y ~ x, data = PetersenCL) > > ## Beck and Katz (1995) standard errors > ## balanced panel > sqrt(diag(vcovPC(m, cluster = ~ firm + year))) (Intercept) x 0.02220064 0.02527598 > > ## unbalanced panel > PU <- subset(PetersenCL, !(firm == 1 & year == 10)) > pu_lm <- lm(y ~ x, data = PU) > sqrt(diag(vcovPC(pu_lm, cluster = ~ firm + year, pairwise = TRUE))) (Intercept) x 0.02206979 0.02533772 > sqrt(diag(vcovPC(pu_lm, cluster = ~ firm + year, pairwise = FALSE))) (Intercept) x 0.02260277 0.02524119 > > ## the following specifications of cluster/order.by are equivalent > vcovPC(m, cluster = ~ firm + year) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > vcovPC(m, cluster = PetersenCL[, c("firm", "year")]) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > vcovPC(m, cluster = ~ firm, order.by = ~ year) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > vcovPC(m, cluster = PetersenCL$firm, order.by = PetersenCL$year) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > > ## these are also the same when observations within each > ## cluster are already ordered > vcovPC(m, cluster = ~ firm) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > vcovPC(m, cluster = PetersenCL$firm) (Intercept) x (Intercept) 4.928685e-04 -4.396037e-05 x -4.396037e-05 6.388754e-04 > > > > cleanEx() > nameEx("vcovPL") > ### * vcovPL > > flush(stderr()); flush(stdout()) > > ### Name: vcovPL > ### Title: Clustered Covariance Matrix Estimation for Panel Data > ### Aliases: vcovPL meatPL > ### Keywords: regression > > ### ** Examples > > ## Petersen's data > data("PetersenCL", package = "sandwich") > m <- lm(y ~ x, data = PetersenCL) > > ## Driscoll and Kraay standard errors > ## lag length set to: T - 1 (maximum lag length) > ## as proposed by Petersen (2009) > sqrt(diag(vcovPL(m, cluster = ~ firm + year, lag = "max", adjust = FALSE))) (Intercept) x 0.01618977 0.01426121 > > ## lag length set to: floor(4 * (T / 100)^(2/9)) > ## rule of thumb proposed by Hoechle (2007) based on Newey & West (1994) > sqrt(diag(vcovPL(m, cluster = ~ firm + year, lag = "NW1994"))) (Intercept) x 0.02289115 0.02441980 > > ## lag length set to: floor(T^(1/4)) > ## rule of thumb based on Newey & West (1987) > sqrt(diag(vcovPL(m, cluster = ~ firm + year, lag = "NW1987"))) (Intercept) x 0.02436219 0.02816896 > > ## the following specifications of cluster/order.by are equivalent > vcovPL(m, cluster = ~ firm + year) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > vcovPL(m, cluster = PetersenCL[, c("firm", "year")]) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > vcovPL(m, cluster = ~ firm, order.by = ~ year) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > vcovPL(m, cluster = PetersenCL$firm, order.by = PetersenCL$year) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > > ## these are also the same when observations within each > ## cluster are already ordered > vcovPL(m, cluster = ~ firm) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > vcovPL(m, cluster = PetersenCL$firm) (Intercept) x (Intercept) 5.935164e-04 2.222292e-05 x 2.222292e-05 7.934905e-04 > > > > 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 > > > > ### *