# Evaluating one-way and two-way cluster-robust covariance matrix estimates

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1 Evaluating one-way and two-way cluster-robust covariance matrix estimates Christopher F Baum 1 Austin Nichols 2 Mark E Schaffer 3 1 Boston College and DIW Berlin 2 Urban Institute 3 Heriot Watt University and CEPR BOS 10 Stata Conference, July 2010 Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

2 Cluster-robust standard errors The importance of cluster-robust standard errors In working with linear regression models, researchers are increasingly likely to abandon the assumption of i.i.d. errors in favor of a more realistic error structure. The use of robust standard errors has become nearly ubiquitous in the applied literature. There are many settings where allowing for heteroskedasticity at the level of the observation is warranted, but that single deviation from an i.i.d. structure may not be sufficient to account for the behavior of the error process. In the context of time series data, one might naturally consider HAC standard errors: those robust to both heteroskedasticity and autocorrelation, familiar to economists as Newey West standard errors. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

3 Cluster-robust standard errors The importance of cluster-robust standard errors In working with linear regression models, researchers are increasingly likely to abandon the assumption of i.i.d. errors in favor of a more realistic error structure. The use of robust standard errors has become nearly ubiquitous in the applied literature. There are many settings where allowing for heteroskedasticity at the level of the observation is warranted, but that single deviation from an i.i.d. structure may not be sufficient to account for the behavior of the error process. In the context of time series data, one might naturally consider HAC standard errors: those robust to both heteroskedasticity and autocorrelation, familiar to economists as Newey West standard errors. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

4 Cluster-robust standard errors The importance of cluster-robust standard errors In working with linear regression models, researchers are increasingly likely to abandon the assumption of i.i.d. errors in favor of a more realistic error structure. The use of robust standard errors has become nearly ubiquitous in the applied literature. There are many settings where allowing for heteroskedasticity at the level of the observation is warranted, but that single deviation from an i.i.d. structure may not be sufficient to account for the behavior of the error process. In the context of time series data, one might naturally consider HAC standard errors: those robust to both heteroskedasticity and autocorrelation, familiar to economists as Newey West standard errors. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

5 Cluster-robust standard errors In this talk, we will consider how a broader set of assumptions on the error process may often be warranted, in the contexts of cross-sectional data of a hierarchical nature or in panel data. The key concept to be considered is that of the cluster-robust covariance matrix, or cluster VCE, which relaxes the i.i.d. assumption of independent errors, allowing for arbitrary correlation between errors within clusters of observations. These clusters may represent some hierarchical relationship in a cross-section, such as firms grouped by industries, or households grouped by neighborhood. Alternatively, they may be the observations associated with each unit (or time period) in a panel dataset. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

6 Cluster-robust standard errors In this talk, we will consider how a broader set of assumptions on the error process may often be warranted, in the contexts of cross-sectional data of a hierarchical nature or in panel data. The key concept to be considered is that of the cluster-robust covariance matrix, or cluster VCE, which relaxes the i.i.d. assumption of independent errors, allowing for arbitrary correlation between errors within clusters of observations. These clusters may represent some hierarchical relationship in a cross-section, such as firms grouped by industries, or households grouped by neighborhood. Alternatively, they may be the observations associated with each unit (or time period) in a panel dataset. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

7 Cluster-robust standard errors In this talk, we will consider how a broader set of assumptions on the error process may often be warranted, in the contexts of cross-sectional data of a hierarchical nature or in panel data. The key concept to be considered is that of the cluster-robust covariance matrix, or cluster VCE, which relaxes the i.i.d. assumption of independent errors, allowing for arbitrary correlation between errors within clusters of observations. These clusters may represent some hierarchical relationship in a cross-section, such as firms grouped by industries, or households grouped by neighborhood. Alternatively, they may be the observations associated with each unit (or time period) in a panel dataset. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

8 Cluster-robust standard errors As discussed in prior talks by Nichols and Schaffer (UKSUG 07) and in recent work by Cameron and Miller (UC Davis WP, 2010), estimation of the VCE without controlling for clustering can lead to understated standard errors and overstated statistical significance. Just as the use of the classical (i.i.d.) VCE is well known to yield biased estimates of precision in the absence of the i.i.d. assumptions, ignoring potential error correlations within groups, or clusters, may lead to erroneous statistical inference. The standard approach to clustering generalizes the White (robust/sandwich) approach to a VCE estimator robust to arbitrary heteroskedasticity: in fact, robust standard errors in Stata correspond to cluster-robust standard errors computed from clusters of size one. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

9 Cluster-robust standard errors As discussed in prior talks by Nichols and Schaffer (UKSUG 07) and in recent work by Cameron and Miller (UC Davis WP, 2010), estimation of the VCE without controlling for clustering can lead to understated standard errors and overstated statistical significance. Just as the use of the classical (i.i.d.) VCE is well known to yield biased estimates of precision in the absence of the i.i.d. assumptions, ignoring potential error correlations within groups, or clusters, may lead to erroneous statistical inference. The standard approach to clustering generalizes the White (robust/sandwich) approach to a VCE estimator robust to arbitrary heteroskedasticity: in fact, robust standard errors in Stata correspond to cluster-robust standard errors computed from clusters of size one. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

10 Cluster-robust standard errors Simple one-way clustering Simple one-way clustering In simple one-way clustering for a linear model, we consider that each observation (i = 1,..., N) is a member of one non-overlapping cluster, g (g = 1,..., G). y ig = x ig β + u ig Given the standard zero conditional mean assumption E[u ig x ig ] = 0, the error is assumed to be independent across clusters: for i j unless g = g. E[u ig u jg x ig, x jg ] = 0 How might this behavior of the error process arise? Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

11 Cluster-robust standard errors Simple one-way clustering Simple one-way clustering In simple one-way clustering for a linear model, we consider that each observation (i = 1,..., N) is a member of one non-overlapping cluster, g (g = 1,..., G). y ig = x ig β + u ig Given the standard zero conditional mean assumption E[u ig x ig ] = 0, the error is assumed to be independent across clusters: for i j unless g = g. E[u ig u jg x ig, x jg ] = 0 How might this behavior of the error process arise? Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

12 Common shocks Cluster-robust standard errors Common shocks The within-cluster correlation of errors can arise if the errors are not i.i.d., but rather contain a common shock component as well as an idiosyncratic component: u ig = ν g + ζ ig where ν g is a common shock, or cluster-specific error, itself i.i.d., and ζ ig is an i.i.d. idiosyncratic error. This is equivalent to the error representation in the random effects model of panel data, but may just as well arise in a cross-sectional context. As in random effects, Var[u ig ] = σ 2 ν + σ 2 ζ and Cov[u ig, u jg ] = σ 2 ν, i j. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

13 Common shocks Cluster-robust standard errors Common shocks The within-cluster correlation of errors can arise if the errors are not i.i.d., but rather contain a common shock component as well as an idiosyncratic component: u ig = ν g + ζ ig where ν g is a common shock, or cluster-specific error, itself i.i.d., and ζ ig is an i.i.d. idiosyncratic error. This is equivalent to the error representation in the random effects model of panel data, but may just as well arise in a cross-sectional context. As in random effects, Var[u ig ] = σ 2 ν + σ 2 ζ and Cov[u ig, u jg ] = σ 2 ν, i j. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

14 Cluster-robust standard errors Common shocks The intraclass correlation, common to all pairs of errors in a cluster, is ρ u = Corr[u ig, u jg ] = σ 2 ν (σ 2 ν + σ 2 ζ ) This constant within-cluster correlation is appropriate where observations within a cluster are exchangeable, in Stata parlance, with no implicit ordering. Individuals living in a household, families within a village or firms within an industry might follow this assumption. If common shocks are the primary cause of error clustering, classical OLS standard errors are biased downward, and should be inflated by a factor taking the intraclass correlation into account. The inflation factor for a particular regressor s coefficient is also an increasing function of the within-cluster correlation of the regressor. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

15 Cluster-robust standard errors Common shocks The intraclass correlation, common to all pairs of errors in a cluster, is ρ u = Corr[u ig, u jg ] = σ 2 ν (σ 2 ν + σ 2 ζ ) This constant within-cluster correlation is appropriate where observations within a cluster are exchangeable, in Stata parlance, with no implicit ordering. Individuals living in a household, families within a village or firms within an industry might follow this assumption. If common shocks are the primary cause of error clustering, classical OLS standard errors are biased downward, and should be inflated by a factor taking the intraclass correlation into account. The inflation factor for a particular regressor s coefficient is also an increasing function of the within-cluster correlation of the regressor. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

16 Cluster-robust standard errors Common shocks In fact, if we had a dataset containing a number of equal-sized clusters, and regressors taking on constant values within those clusters, OLS estimation on these data is equivalent to estimating the model ȳ g = x gβ + ū g where ȳ contains within-cluster averages of the dependent variable. There are really only G observations in the model, rather than N. This model also has a parallel to panel data: it is the between estimator (xtreg, be) applied in the special case where x values do not differ within-panel. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

17 Cluster-robust standard errors Common shocks In fact, if we had a dataset containing a number of equal-sized clusters, and regressors taking on constant values within those clusters, OLS estimation on these data is equivalent to estimating the model ȳ g = x gβ + ū g where ȳ contains within-cluster averages of the dependent variable. There are really only G observations in the model, rather than N. This model also has a parallel to panel data: it is the between estimator (xtreg, be) applied in the special case where x values do not differ within-panel. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

18 Cluster-robust standard errors Common shocks If in contrast OLS is applied to the individual data, for a constant regressor within-cluster, the true variance of an estimated coefficient is (1 + ρ u (N 1)) times larger than the classical OLS estimate, where ρ u is the intraclass correlation and N is the number of observations in each cluster. Moulton (REStat, 1990) demonstrated that in many settings this adjustment factor, and the consequent overstatement of precision, can be sizable even when ρ u is fairly small. In his example, with N = and G = 49 (US states), ˆρ u = 0.032: a quite modest intrastate error correlation. With average group size of 387, the correction factor is 13.3, so that cluster-corrected standard errors are 13.3 = 3.7 times larger for a state-level regressor than those computed by standard OLS techniques. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

19 Cluster-robust standard errors Common shocks If in contrast OLS is applied to the individual data, for a constant regressor within-cluster, the true variance of an estimated coefficient is (1 + ρ u (N 1)) times larger than the classical OLS estimate, where ρ u is the intraclass correlation and N is the number of observations in each cluster. Moulton (REStat, 1990) demonstrated that in many settings this adjustment factor, and the consequent overstatement of precision, can be sizable even when ρ u is fairly small. In his example, with N = and G = 49 (US states), ˆρ u = 0.032: a quite modest intrastate error correlation. With average group size of 387, the correction factor is 13.3, so that cluster-corrected standard errors are 13.3 = 3.7 times larger for a state-level regressor than those computed by standard OLS techniques. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

20 Cluster-robust standard errors Panel/longitudinal data Panel/longitudinal data In the context of panel or longitudinal data, correlated errors naturally arise within each panel unit, so that each panel unit or individual can be considered as a cluster. With a time dimension, the assumption of equi-correlated errors from the common shocks model is unlikely to be appropriate, as the strength of unit-specific autocorrelations will depend on their time difference. For instance, in the case of AR(1) errors u it = ρu i,t 1 + ζ it, the within-cluster error correlation becomes ρ t τ for observations dated t and τ, respectively. The decline in correlation for longer time spans implies that taking account of the presence of clustering will have a smaller effect than in the common shocks model. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

21 Cluster-robust standard errors Panel/longitudinal data Panel/longitudinal data In the context of panel or longitudinal data, correlated errors naturally arise within each panel unit, so that each panel unit or individual can be considered as a cluster. With a time dimension, the assumption of equi-correlated errors from the common shocks model is unlikely to be appropriate, as the strength of unit-specific autocorrelations will depend on their time difference. For instance, in the case of AR(1) errors u it = ρu i,t 1 + ζ it, the within-cluster error correlation becomes ρ t τ for observations dated t and τ, respectively. The decline in correlation for longer time spans implies that taking account of the presence of clustering will have a smaller effect than in the common shocks model. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

22 Cluster-robust standard errors Panel/longitudinal data In the context of a fixed-t, large N panel in which the common fixed-effects estimator (xtreg, fe) is applied, Stock and Watson (Econometrica, 2008) showed that the conventional robust or sandwich VCE estimator is inconsistent if T > 2. This result applies even in the presence of serially uncorrelated errors. They present a bias-adjusted estimator that circumvents this problem, and illustrate how the asymptotically equivalent one-way cluster-robust VCE estimator we discuss next will provide consistent (although not fully efficient) estimates. Given Stock and Watson s critique, Stata s xtreg, fe command was modified in version 10.0 (25feb2008) to automatically apply vce(cluster id), where id is the panel unit identifier variable, if the robust option is specified. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

23 Cluster-robust standard errors Panel/longitudinal data In the context of a fixed-t, large N panel in which the common fixed-effects estimator (xtreg, fe) is applied, Stock and Watson (Econometrica, 2008) showed that the conventional robust or sandwich VCE estimator is inconsistent if T > 2. This result applies even in the presence of serially uncorrelated errors. They present a bias-adjusted estimator that circumvents this problem, and illustrate how the asymptotically equivalent one-way cluster-robust VCE estimator we discuss next will provide consistent (although not fully efficient) estimates. Given Stock and Watson s critique, Stata s xtreg, fe command was modified in version 10.0 (25feb2008) to automatically apply vce(cluster id), where id is the panel unit identifier variable, if the robust option is specified. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

24 The cluster-robust VCE estimator The cluster-robust VCE estimator Cluster-robust VCE estimates are generalizations of the sandwich method used to compute heteroskedasticity-robust standard errors (Stata s robust option), as developed by White (Asymptotic Theory for Econometricians, 1984). The cluster-robust estimate takes the sandwich form VCE( ˆβ) = (X X) 1 ˆΩ(X X) 1 where ˆΩ = G X gûgû gx g g=1 with û g = y g X g ˆβ and g indicating membership in the g th cluster. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

25 The cluster-robust VCE estimator This formula is derived from a more general specification where we consider the population moments, E(x i u i ), where u i are the error terms. The corresponding sample moments are ḡ( ˆβ) = 1 N N x i û i i=1 where û i are the residuals computed from point estimates ˆβ. The VCE of ˆβ is then V = ˆQ 1 xx ˆΩ ˆQ 1 xx where ˆΩ is the estimated VCE of ḡ( ˆβ), with its form based upon our assumptions about the properties of the error process u. ˆQ xx 1 is merely (X X) 1. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

26 The cluster-robust VCE estimator This formula is derived from a more general specification where we consider the population moments, E(x i u i ), where u i are the error terms. The corresponding sample moments are ḡ( ˆβ) = 1 N N x i û i i=1 where û i are the residuals computed from point estimates ˆβ. The VCE of ˆβ is then V = ˆQ 1 xx ˆΩ ˆQ 1 xx where ˆΩ is the estimated VCE of ḡ( ˆβ), with its form based upon our assumptions about the properties of the error process u. ˆQ xx 1 is merely (X X) 1. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

27 The cluster-robust VCE estimator Under the classical assumptions of i.i.d. errors: independence and conditional homoskedasticity, Ω = σ 2 ui, and the estimate of V is merely s 2 (X X) 1, where s 2 is a consistent estimate of σ 2 u. If we relax the assumption of conditional homoskedasticity, N ˆΩ = 1 N i=1 x i x iû2 i with the VCE estimator as ˆV = ˆQ 1 xx ˆΩ ˆQ 1 xx the Huber sandwich White robust estimator of the VCE, as invoked by the robust option in Stata. The expression for ˆΩ is a single sum over observations as we are maintaining the assumption of independence of each pair of errors. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

28 The cluster-robust VCE estimator Under the classical assumptions of i.i.d. errors: independence and conditional homoskedasticity, Ω = σ 2 ui, and the estimate of V is merely s 2 (X X) 1, where s 2 is a consistent estimate of σ 2 u. If we relax the assumption of conditional homoskedasticity, N ˆΩ = 1 N i=1 x i x iû2 i with the VCE estimator as ˆV = ˆQ 1 xx ˆΩ ˆQ 1 xx the Huber sandwich White robust estimator of the VCE, as invoked by the robust option in Stata. The expression for ˆΩ is a single sum over observations as we are maintaining the assumption of independence of each pair of errors. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

29 The cluster-robust VCE estimator In the cluster-robust case, where we allow for dependence between errors belonging to the same cluster, the VCE of ḡ( ˆβ) becomes ˆΩ = 1 N N i=1 ĝ i N i=1 ĝ i where the double sum will include within-cluster cross products and between-cluster cross products of (x i û i ). By the assumption of independence across clusters, all terms involving errors in different clusters will be dropped, as they have zero expectation. The remaining within-cluster terms involve only the sums (x i û i ) for observations in each cluster, giving rise to the formula for ˆΩ which we presented earlier. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

30 The cluster-robust VCE estimator In the special case where errors are heteroskedastic but still independently distributed, the number of clusters G is equal to the number of observations, N, and each cluster is of size one. In this case the cluster-robust formula becomes the standard heteroskedasticity-robust White formula implemented by Stata s robust option, as presented above. When clusters with N g > 1 are considered, the number of clusters G should be compared to the number of parameters to be estimated, k. The rank of VCE( ˆβ) is at most G, as the meat in the sandwich contains only G super-observations. This implies that we cannot test more than G restrictions on the parameter vector, possibly invalidating a test for overall significance of the model, while tests on smaller subsets of the parameters are possible. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

31 The cluster-robust VCE estimator In the special case where errors are heteroskedastic but still independently distributed, the number of clusters G is equal to the number of observations, N, and each cluster is of size one. In this case the cluster-robust formula becomes the standard heteroskedasticity-robust White formula implemented by Stata s robust option, as presented above. When clusters with N g > 1 are considered, the number of clusters G should be compared to the number of parameters to be estimated, k. The rank of VCE( ˆβ) is at most G, as the meat in the sandwich contains only G super-observations. This implies that we cannot test more than G restrictions on the parameter vector, possibly invalidating a test for overall significance of the model, while tests on smaller subsets of the parameters are possible. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

32 The cluster-robust VCE estimator Bias in the cluster-robust estimator Bias in the cluster-robust estimator While the formula for ˆΩ is appropriate as the number of clusters G goes to infinity, finite-sample corrections are usually applied to deal with downward bias in the cluster-robust standard errors. Stata uses cûg in computing ˆΩ, with c G G 1. Simulations have shown that the bias is larger when clusters are unbalanced: for instance, in a dataset with 50 clusters, in which half the data are in a single cluster and the other 49 contain about one percent of the data. A further finite-sample adjustment factor N 1 N K can also be applied. As a rule of thumb, Nichols and Schaffer (2007) suggest that the data should have at least 20 balanced clusters or 50 reasonably balanced clusters. Rogers seminal work (Stata Tech.Bull., 1993) suggested that no cluster should contain more than five per cent of the data. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

33 The cluster-robust VCE estimator Bias in the cluster-robust estimator Bias in the cluster-robust estimator While the formula for ˆΩ is appropriate as the number of clusters G goes to infinity, finite-sample corrections are usually applied to deal with downward bias in the cluster-robust standard errors. Stata uses cûg in computing ˆΩ, with c G G 1. Simulations have shown that the bias is larger when clusters are unbalanced: for instance, in a dataset with 50 clusters, in which half the data are in a single cluster and the other 49 contain about one percent of the data. A further finite-sample adjustment factor N 1 N K can also be applied. As a rule of thumb, Nichols and Schaffer (2007) suggest that the data should have at least 20 balanced clusters or 50 reasonably balanced clusters. Rogers seminal work (Stata Tech.Bull., 1993) suggested that no cluster should contain more than five per cent of the data. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

34 The cluster-robust VCE estimator Cluster-robust t and F tests Cluster-robust t and F tests When a cluster-robust VCE has been calculated, Wald t or F test statistics should take account of the number of clusters, rather than relying on the asymptotically behavior of the statistic as N. The approach that Stata follows involves using the t distribution with G 1 degrees of freedom rather than N k degrees of freedom. If the number of clusters is small, this will substantially increase the critical values relative to those computed from the standard Normal (t with large d.f.). Some authors (e.g., Donald and Lang (Rev.Ec.Stat., 2007) recommend using t G L, where L is the number of regressors constant within cluster, as an even more conservative approach. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

35 The cluster-robust VCE estimator Cluster-robust t and F tests Cluster-robust t and F tests When a cluster-robust VCE has been calculated, Wald t or F test statistics should take account of the number of clusters, rather than relying on the asymptotically behavior of the statistic as N. The approach that Stata follows involves using the t distribution with G 1 degrees of freedom rather than N k degrees of freedom. If the number of clusters is small, this will substantially increase the critical values relative to those computed from the standard Normal (t with large d.f.). Some authors (e.g., Donald and Lang (Rev.Ec.Stat., 2007) recommend using t G L, where L is the number of regressors constant within cluster, as an even more conservative approach. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

36 The cluster-robust VCE estimator Fixed effects models with clustering Fixed effects models with clustering In any context where we identify clusters, we could consider including a fixed-effect parameter for each cluster, as in y ig = α g + x ig β + u ig As is well known from analysis of this model in the special case of longitudinal or panel data, the inclusion of the α g parameters centers each cluster s residuals around zero. However, as the inclusion of these fixed-effect parameters does nothing to deal with potential intra-cluster correlation of errors, it is always advisable to question the i.i.d. error assumption and produce cluster-robust estimates of the VCE. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

37 The cluster-robust VCE estimator Fixed effects models with clustering By analogy to the panel-data fixed effects model, we may note: The β parameters may be consistently estimated, but the coefficients of cluster-invariant regressors are not identified. For instance, if household data are clustered by state, state-level variables cannot be included. For G, the α g parameters cannot be consistently estimated due to the incidental parameter problem. The cluster-specific fixed effects, α g, may be correlated with elements of x. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

38 The cluster-robust VCE estimator Fixed effects models with clustering By analogy to the panel-data fixed effects model, we may note: The β parameters may be consistently estimated, but the coefficients of cluster-invariant regressors are not identified. For instance, if household data are clustered by state, state-level variables cannot be included. For G, the α g parameters cannot be consistently estimated due to the incidental parameter problem. The cluster-specific fixed effects, α g, may be correlated with elements of x. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

39 The cluster-robust VCE estimator Fixed effects models with clustering By analogy to the panel-data fixed effects model, we may note: The β parameters may be consistently estimated, but the coefficients of cluster-invariant regressors are not identified. For instance, if household data are clustered by state, state-level variables cannot be included. For G, the α g parameters cannot be consistently estimated due to the incidental parameter problem. The cluster-specific fixed effects, α g, may be correlated with elements of x. Baum, Nichols, Schaffer (BC / UI / HWU) Cluster-Robust Covariance Matrices BOS 10, July / 42

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