The Cluster-Robust Variance-Covariance Estimator: A (Stata .

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Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesThe Cluster-Robust Variance-CovarianceEstimator: A (Stata) Practitioner’s GuideAustin Nichols and Mark Schaffer21 Sept 2007Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe problemSuppose we have a regression model likeymt xmt β νmtwhere the indexes arem 1.Mt 1.TmThe ms index “groups” of observations and the ts index individual observationswithin groups. The t suggests multiple observations over time, but the t indexcan represent any arbitrary index for observations grouped along twodimensions. The m subscript in Tm denotes that we may have groups ofdifferent sizes (“unbalanced” groups). It will also be convenient to have avariable id that identifies groups.We assume weak exogeneity, i.e., E (xmt νmt ) 0, so the OLS estimator isconsistent.However, the classical assumption that νmt is iid (independently and identicallydistributed) is clearly violated in many cases, making the classical OLScovariance estimator inconsistent.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsClustered ErrorsModel:ymt xmt β νmtA natural way of modeling the structure of the covariance of ν is toassume “clustered errors”: observations within group m are correlated insome unknown way, but groups m and j do not have correlated errors.ThusE (vmt νms ) 6 0E (vmt νjs ) 0and the variance-covariance matrix of ν is block-diagonal: zero acrossgroups, nonzero within groups.This kind of problem arises all the time.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsExamples of ClusteringExample:We have a survey in which blocks of observations are selected randomly,but there is no reason to suppose that observations within block haveuncorrelated errors. For example, consider a random sample of schoolsthat contain students whose response to some policy X might becorrelated (in which case m indexes school and t indexes student withinschool).Example:ymt xmt β um emtwhere we decompose the error νmt um emt and the eim are iid.This is the standard “error components” model in panel data. It istraditionally addressed using the fixed or random effects estimators. Ifthe emt are iid, the standard variance-covariance estimator is consistent.But.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsMore Examples of ClusteringExample:ymt ymt β um emtwhere now the the emt may be serially correlated, E (emt ems ) 6 0. If thisis the case, the fixed or random effects estimators will be consistent, butthe standard covariance estimators will not.Example:Observations are randomly sampled, but the explanatory variable X ismeasured at a higher level (see Moulton 1990; Bertrand, Duflo, andMullainathan 2004). For example, students might be randomly sampledto model test scores as a function of school characteristics, but this willresult in clustered errors at the school level. If students were randomlysampled to model test scores as a function of classes taken (measured atthe individual level), but classes taken and their effects on test scores arecorrelated within school, clustering of errors at the higher (school) levelmay result.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsStill More Examples of ClusteringExample: Systems of equationsSay we have a two-equation model:ym1 xm1 β1 νm1ym2 xm2 β2 νm2Say that νm1 and νm1 are both iid, so that each equation could beestimated separately and the standard covariance estimator would beconsistent. However, we want to test cross-equation restrictions involvingboth β1 and β2 . If we “stack” the data and estimate as a(seemingly-unrelated) system, the disturbances will be “clustered”:E (νm1 νm2 ) 6 0 because the error for the mth observation in equation 1will be correlated with the error for the mth observation in equation 2.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsStill More Examples of ClusteringExample: Spatial AutocorrelationWe have data on US counties or cities. We expect counties or cities thatare geographically close to each other will share some unobservableheterogeneity, but localities that are far apart will be less correlated; thatis, our data are spatially autocorrelated. However, the nature of theproblem we are investigating is such that US states can be regarded asessentially independent (e.g., they run separate legal, educational and taxsystems). Thus it is reasonable to assert that observations on localitiesare independent across states but dependent within states, i.e., they areclustered by state.NB: This is why the number 50 is of particular interest. Thus thecluster-robust covariance estimator relies on asymptotics where thenumber of clusters goes off to infinity. Is 50 far enough on the way toinfinity?Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsHow to Deal with Clustered Errors?Model: ymt xmt β νmtStructure of the disturbances is block-diagonal: Σ1 . . ΣmVar (ν) . .00 ΣMTwo questions: (1) Efficiency of parameter estimates. (2) Consistency ofstandard errors (var-cov matrix of β).Two approaches: (1) GLS, generalized least squares. Model theclustering. What is the structure of Σm , the within-group correlation?(2) “Robust” formulation. Allow for arbitrary forms of clustering. Obtainconsistent standard errors for any structure of Σm .Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsExample: Fixed Effects and ClustersExample: The fixed effects panel data modelDecompose the error term νmt um emt so that the model isymt xmt β ui emtand we assume the emt are iid. The structure of the within-group correlation isvery special: since all the observations in a group share the same um , everyobservation within a group is equally-well correlated with every otherobservation. The structure of Σm in the block-diagonal var(ν) is σu2 everywhereexcept the diagonal of the block, where it is σu2 σe2 .The GLS approach is to use this model of the error structure. With the FEestimator, we partial-out the um by demeaning, and we’re left with an iididiosyncratic error emt . Note that this approach addresses both (1) efficiency ofthe estimate of β and (2) consistency of the estimated standard errors.What’s wrong with this approach? Nothing, if we’ve modeled the structure ofthe disturbances correctly. But if we haven’t, then our estimated β isn’tefficient – not such a problem – and our estimated standard errors are wrong –big problem!Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard Errors“Clustered errors” are an example of Eicker-Huber-White-“sandwich”-robusttreatment of errors, i.e., make as few assumptions as possible. We keep theassumption of zero correlation across groups as with fixed effects, but allow thewithin-group correlation to be anything at all.Some notation:1 0X YN1E (xi0 xi ) Qxx Q̂xx X 0 XNCovariance matrix of orthogonality conditions (“GMM-speak”):E (xi0 yi ) Qxy Q̂xy S AVar (g (β)) limN 1E (X 0 νν 0 X )N“Sandwich” variance matrix of β: 1 1V QxxSQxxAustin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard Errors“Sandwich” variance matrix of β: 1 1V QxxSQxxQxx is estimated by Q̂xx . What will give V̂ its robustness is our choice of theestimator Ŝ.If errors are iid (no robustness), then S σ 2 Qxx , we estimate Ŝ with σ̂ 2 Q̂xxwhere σ̂ 2 is simply the root mean squared residual ν̂, and our estimate of thevariance of β reduces to V̂ σ̂ 2 Q̂xx , which is the standard, “classical” OLSvariance estimator.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard Errors“Sandwich” variance matrix of β: 1 1V QxxSQxxIf errors are independent but heteroskedastic, wePuse the02Eicker-Huber-White-“robust” approach. Ŝ N1 Ni 1 xi xi ν̂i or, in matrix01notation, Ŝ N X BX where B is a matrix with the squared residuals ν̂i2running down the diagonal and zeros elsewhere. This estimate of S is robust toarbitrary heteroskedasticity, and therefore so is our estimate of V . Theintuition is that although B (which looks like the covariance of ν) is NxN, S isKxK and fixed. We can’t get a consistent estimate of the covariance of ν – youcan’t estimate an nxn matrix with only n observations – but we don’t need it.We need only a consistent estimate of S, and with the number of observationsN going off to infinity, the asymptotics give us this.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsThe cluster-robust approach is a generalization of theEicker-Huber-White-“robust” to the case of observations that are correlatedwithin but not across groups. Instead of just summing across observations, wetake the crossproducts of x and ν̂ for each group m to get what looks like (butisn’t) a within-group correlation matrix, and sum these across all groups M:ŜCR M Tm Tm1 XXX 0xmt xms ν̂mt ν̂msN m 1 t 1 s 1Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsŜCR M Tm Tm1 XXX 0xmt xms ν̂mt ν̂msN m 1 t 1 s 1The intuition is similar to the heteroskedasticity-robust case. Since thewithin-group correlations are arbitrary and can vary from group to group, wecan’t estimate it with only one observation on each group. But we don’t needthis - we need only a consistent estimate of S, and if the number of groups Mgoes off to infinity, the asymptotics give us this.This ŜCR is consistent in the presence of arbitrary within-group correlation aswell as arbitrary heteroskedasticity. This is what “cluster-robust” means in thiscontext.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsHere’s an alternative exposition that highlights the parallels with the standardand heteroskedastic-robust covariance estimators.General case: Covariance matrix of orthogonality conditions (“GMM-speak”):S AVar (g (β)) limN 1E (X 0 νν 0 X )NIndependently-distributed observations means cov (νi , νj ) 0, and S becomesS E (xi0 xi νi2 ).Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsHomoskedasticity means observations are identically as well as independentlydistributed (iid), and so S E (xi0 xi νi2 ) E (xi0 xi )E (νi2 ). The standardestimator of S under the iid assumption isŜhomo N1 X 0xi xiN i 1N1 X 2ν̂iN i 1where the second term is just the estimated error variance σ̂ 2 and the first termis just Q̂xx .Heteroskedasticity means observations are not identically distributed, and weuse instead the Eicker-Huber-White-robust estimator of S:Ŝhet N1 X 0 2xi xi ν̂iN i 1Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsNow we consider “clustered” data. Observations are independent acrossclusters, but dependent within clusters. Denote by Xm. the Tm xK matrix ofobservations on X for the mth cluster, and νm. the Tm x1 vector of disturbances000for cluster m. Then we can write S as S E (Xm.νm. νm.Xm. ), where E (νm. νm.)is just Σm , the covariance matrix of the disturbance ν for cluster m.The cluster-robust covariance estimator for S isŜCR M1 X 00Xm. ν̂m. ν̂m.Xm.M m 1Note the parallels with Ŝhet . We are summing over clusters instead of individualobservations; the X s inside the summation are all the observations on a clusterinstead of a single row of data; the term inside the X s looks like (but isn’t) theautocovariance of the disturbance for the cluster instead of what looks like (butisn’t) the variance for the observation in the heteroskedastic-robust approach.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsIn fact, there is another covariance estimator due to Kiefer (1980) that isrobust to clustering but assumes homoskedasticity. To keep thing simple,assume that the dataset is a balanced panel, so that Tm T m. If the dataare homoskedastic, the TxT matrix Σm Σ m and we can estimate Σ̂ byΣ̂ M1 X0ν̂m. ν̂m.M m 1The Kiefer covariance estimator isŜKiefer M1 X 0Xm. Σ̂Xm.M m 1Again, note the parallels, this time with the usual homoskedastic estimatorŜhomo . With Ŝhomo , we weight each observation with a scalar σ̂ 2 , but since it’s ascalar it can be pulled out of the summation; With ŜKiefer , we weight eachcluster with the matrix Σ̂.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe “Robust” Approach: Cluster-Robust Standard ErrorsWhat about efficiency of the OLS β̂?If the model is just-identified, OLS is still the efficient estimator. Why? We’veassumed no structure at all for the intra-group correlations, and we have noextra information to bring to the estimate of S. With no additionalassumptions or information, OLS is the best we can do.If the model is overidentified, however, the cluster-robust approach can be usedto obtain more efficient estimates of β via two-step or CUE(continuously-updated) GMM. This is the generalization of “heteroskedasticOLS” (Cragg 1983) to the case of clustered errors. “Overidentified” meansthat there are variables (instruments) that are not regressors, that areuncorrelated with ν, but that are “correlated” with the form of within-groupclustering and/or heteroskedasticity in the data. The resulting β̂ will be bothconsistent and efficient in the presence of arbitrary clustering andheteroskedasticity. In Stata, use ivreg2 with the cluster(id) gmm2s options.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsExample: Estimation of a System of EquationsWe have a system of T equations. For each equation, we have M observations.Regressors are all exogenous. We want to be able to test cross-equationrestrictions. “Clustering” arises because we use the same dataset to estimateall the equations, and the error νmt for the mth observation in equation t willbe correlated with the error νms for the mth observation in equation s.The GLS approach is Zellner’s ”seemingly-unrelated regressions estimator”(SURE ). We model the covariances of the νmt , estimate them, and constructthe variance-covariance matrix that incorporates these off-diagonal elements.This lets us perform tests across equations, and obtain more efficient estimatesof β in a second step. In Stata, this is the sureg command. but if we model the covariances incorrectly, all our inferences and testingwill be wrong.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsExample: Estimation of a System of EquationsThe “robust” approach is to allow for arbitrary correlation of the νmt acrossequations. This uses the cluster-robust covariance estimator, where eachobservation m in the dataset defines a group or cluster. This is automated inStata with the suest (“seemingly-unrelated estimations”) command. Itgenerates standard errors that are robust to heteroskedasticity as well asallowing cross-equation tests, but leaves the point estimates unchanged.Alternatively, we can “stack” the equations “by hand” and use thecluster-robust covariance estimator. Estimation with OLS generates the sameresults as suest. However, if the model is overidentified (some regressorsappear in one equation and not in others), we can do two-step GMM withivreg2 and obtain efficiency gains in our estimate of β.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsCombining the GLS and Cluster-Robust ApproachesIt is possible – and in some literatures, standard – to combine the GLS andcluster-robust approaches.Consider again the fixed effects model. Say we consider it to be a good firstapproximation to within-group correlation, but there may be remainingwithin-group correlation even after accounting for fixed effects. For example,the emt could be serially correlated. One possibility would be to model the serialcorrelation, GLS-style. This is possible with the Stata command xtregar.Alternatively, we could partial out the fixed effects in the usual way, and thenuse the cluster-robust covariance estimator. The only difference is that insteadof using ν̂mt as residuals, we are using êmt . In effect, we are using cluster-robustto address any within-group correlation remaining after the fixed effects areremoved. In Stata, use xtreg,fe with cluster(id).Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsCombining the GLS and Cluster-Robust ApproachesFirst-differencing (FD) can be similarly motivated. With lagged dependentvariables, we have to FD to get rid of the fixed effects (Arellano-Bond et al.).We then use cluster-robust errors to mop up the remaining and/or introducedserial correlation.For some reason, combining the GLS and robust approaches is absolutelystandard in the panel/serial correlation literature, and almost completelyignored in cross-section/heteroskedasticity practice. It’s perfectly reasonable todo feasible GLS on a cross-section to get improvements in efficiency and thenuse robust SEs to address any remaining heteroskedasticity, but nobody seemsto do this (GLS is too old-fashioned, perhaps).Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsNumber of ClustersThe cluster-robust standard error estimator converges to the true standarderror as the number of clusters M (not the number of observations N)approaches infinity.Kézdi (2003) shows that 50 clusters (with roughly equal cluster sizes) is oftenclose enough to infinity for accurate inference, and further that, even in theabsence of clustering, there is little to no cost of using the CR estimator, aslong as the number of clusters is large. 50 is an interesting number because ofthe many studies that use US state-level data.With a small number of clusters (M 50), or very unbalanced cluster sizes,the cure can be worse than the disease, i.e., inference using the cluster-robustestimator may be incorrect more often than when using the classical covarianceestimator.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsRank of VCVThe rank of the variance-covariance matrix produced by the cluster-robustestimator has rank no greater than the number of clusters M, which meansthat at most M linear constraints can appear in a hypothesis test (so we cantest for joint significance of at most M coefficients).In a fixed-effect model, where there are a large number of parameters, thisoften means that test of overall model significance is feasible. However, testingfewer than M linear constraints is perfectly feasible in these models, thoughwhen fixed effects and clustering are specified at the same level, tests thatinvolve the fixed effects themselves are inadvisable (the standard errors on fixedeffects are likely to be substantially underestimated, though this will not affectthe other variance estimates in general).Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsEstimates and their VCVNote that the heteroskedasticity-robust and cluster-robust estimators forstandard errors have no impact whatsoever on point estimates.One could use information about the within-cluster correlation of errors toobtain more efficient estimates in many cases (see e.g. Diggle et al. 2002).There are also a variety of multi-level methods of parameterizing thedistribution of errors to obtain more efficient estimates (using e.g. xtmixedand other model types—see Rabe-Hesketh and Skrondal 2005 for more). Wewill focus however on models where the point estimates are unchanged andonly the estimated variance of our point estimates is affected by changingassumptions about errors.In addition to improving the efficiency of the point estimates in regressions,modeling intra-cluster correlations can also result in improvements inmeta-analysis, both in correctly modeling the variance of individual estimatesand computing effect sizes. See Hedges (2006) for details.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsSandwich Estimators and Other RobustificationsEicker (1967) and Huber (1967) introduced these sandwich estimators, but White(1980; 1982), Liang and Zeger (1986), Arellano (1987), Newey and West (1987),Froot (1989), Gail, Tan, and Piantodosi (1988), Kent (1982), Royall (1986), and Linand Wei (1989), Rogers (1993), Williams (2000), and others explicated and extendedaspects of the method in a non-survey context, so these are often cited as sources inspecific applications. In the context of clustering induced by survey design, Kish andFrankel (1974), Fuller (1975), and Binder (1983), and Binder and Patak (1994), alsoderived results on cluster-robust estimators with broad applicability.Stock and Watson (2006) point out that with fixed effects, both the standardheteroskedasticity-robust and HAC-robust covariance estimators are inconsistent for Tfixed and T 2, but the cluster-robust estimator does not suffer from this problem.One of their conclusions is that if serial correlation is expected, the cluster-robustestimator is the preferred choice.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsFinite-Sample AdjustmentsThe cluster-robust covariance estimator is often used with a finite-sampleadjustment qc . The most common three forms are:qc 1N 1 MN K M 1Mqc M 1The Stata manual entry “Methods and Formulas” of [R] regress calls thesethe regression-like formula and the asymptotic-like formula, respectively. Fulleret al. (1986) and Mackinnon and White (1985) discuss finite-sampleadjustments in more detail.qc Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsThe Nature of the CR CorrectionThe heteroskedasticity-robust SE estimator scales not by the sum of squaredresiduals, but by the sum of “squared” products of residuals and the Xvariables, and the CR estimator further sums the products within cluster (if theproducts are negatively correlated within cluster, the CR standard errors will besmaller than the HR standard errors, and if positively correlated, larger). If thetraditional OLS model is true, the residuals should, of course, be uncorrelatedwith the X variables, but this is rarely the case in practice.The correlation may arise not from correlations in the residuals within acorrectly specified model, but from specification error (such as omittedvariables), so one should always be alert to that possibility.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsMisspecification and the CR CorrectionAs Sribney (1998) points out: When CR estimates are smaller than standardSE estimates,[S]ince what you are seeing is an effect due to (negative) correlationof residuals, it is important to make sure that the model isreasonably specified and that it includes suitable within-clusterpredictors. With the right predictors, the correlation of residualscould disappear, and certainly this would be a better model.[S]uppose that you measured the number of times each monththat individuals took out the garbage, with the data clustered byhousehold. There should be a strong negative correlation here.Adding a gender predictor to the model should reduce the residualcorrelations.The CR estimator will do nothing about bias in β̂ when E (X 0 e) 6 0.Austin Nichols and Mark SchafferThe Cluster-Robust Variance-Covariance Estimator: A (Stata) Practitioner’s

Overview of ProblemPotential Problems with CR Standard ErrorsTest for ClusteringSome Specific Examples with SimulationsReferencesClustering of ErrorsMore DimensionsApproximating the CR CorrectionAs Cameron, Gelbach, and Miller (2006a, p.5) note, if the primary source of clustering is due togroup-level common shocks, a useful approximation is that for the kth regressor the default OLSvariance estimate based on s 2 (X 0 X

The \Robust" Approach: Cluster-Robust Standard Errors The cluster-robust approach is a generalization of the Eicker-Huber-White-\robust" to the case of observations that are correlated within but not across groups. Instead of just summing across observations, we take the crossproducts of x and for each group m to get what looks like (but S .

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