Heteroskedasticity, Autocorrelation, and Spatial Correlation Robust Inference in Linear Panel Models with Fixed-E¤ects

This paper develops an asymptotic theory for test statistics in linear panel models that are robust to heteroskedasticity, autocorrelation and/or spatial correlation. Two classes of standard errors are analyzed. Both are based on nonparametric heteroskedasticity autocorrelation (HAC) covariance matrix estimators. The …rst class is based on averages of HAC estimates across individuals in the cross-section, i.e. "averages of HACs". This class includes the well known cluster standard errors analyzed by Arellano (1987) as a special case. The second class is based on the HAC of cross-section averages and was proposed by Driscoll and Kraay (1998). The "HAC of averages" standard errors are robust to heteroskedasticity, serial correlation and spatial correlation but weak dependence in the time dimension is required. The "averages of HACs" standard errors are robust to heteroskedasticity and serial correlation including the nonstationary case but they are not valid in the presence of spatial correlation. The main contribution of the paper is to develop a …xed-b asymptotic theory for statistics based on both classes of standard errors in models with individual and possibly time …xed-e¤ects dummy variables. The asymptotics is carried out for large time sample sizes for both …xed and large cross-section sample sizes. Extensive simulations show that the …xed-b approximation is usually much better than the traditional normal or chi-square approximation especially for the DriscollKraay standard errors. The use of …xed-b critical values will lead to more reliable inference in practice especially for tests of joint hypotheses. Keywords: panel data, HAC estimator, kernel, bandwidth, …xed-b asymptotics I am grateful to Todd Elder, Emma Iglesias, Gary Solon and Je¤ Wooldridge for helpful conversations, and I thank Silvia Gonçalves and Chris Hansen for suggestions and comments on preliminary drafts. I thank seminar participants at Cornell, U. Michigan, Michigan State, U. Montreal, Purdue for helpful comments. Young Gui Kim provided excellent research assistance with the simulations and proofread earlier versions of the paper. Portions of this research were supported by the National Science Foundation through grant SES-0525707. Correspondence: Department of Economics, Michigan State University, 110 Marshall-Adams Hall, East Lansing, MI 48824-1038. Phone: 517-353-4582, Fax: 517–432-1068, email: [email protected] 1 Introduction Since the in‡uential work of White (1980) on heteroskedasticity robust standard errors 30 years ago, it has become standard practice in empirical work in economics to use standard errors that are robust to potentially unknown variance and covariance properties of the errors and data. In pure cross-section settings it is now so standard to use heteroskedasticity robust standard errors that authors often do not indicate they have used robust standard errors. In time series regression the use of heteroskedasticity and serial correlation robust standard errors is routine with authors usually indicating that they used Newey and West (1987) standard errors. In panel models where cross-section individuals are followed over time, the so-called panel cluster standard errors (see Arellano (1987)) are appealing because they are robust to heteroskedasticity in the cross-section and quite general forms of serial correlation over time including some nonstationary cases. Even though panel clustered standard errors are covered by graduate level textbooks (see Wooldridge (2002)), Bertrand, Du‡o and Mullainathan (2004) found that surprisingly few empirical studies based on panels with relatively many time series observations used clustered standard errors. The situation in the empirical …nance literature is similar as reported by Petersen (2009). The validity of panel clustered standard errors requires that individuals in the cross-section be uncorrelated with each other, i.e. no spatial correlation in the cross-section. Typically spatial correlation is ignored although sometimes the cross-section can be divided into groups or clusters where it is assumed that individuals within a cluster are correlated but individuals between clusters are uncorrelated. Then, standard errors can be con…gured that are robust to the cross-sectional clustering. See Wooldridge (2003) for a useful discussion of cluster methods and additional references. A recent paper by Bester, Conley and Hansen (2011) provides an interesting analysis of cluster standard errors in a general setting where the number of clusters is held …xed in the asymptotics. In some cases an empirical researcher may have a distance measure for pairs of individuals in the cross-section such that the spatial correlation is decreasing in distance. In a pure time series setting with stationarity, distance in time is the natural distance measure. When a distance measure is available (or can be estimated) in a spatial setting, robust standard errors can be obtained using the approaches of Conley (1999), Kelejian and Prucha (2007), Bester, Conley, Hansen and Vogelsang (2008) or Kim and Sun (2011) which are extensions of nonparametric kernel heteroskedasticity autocorrelation consistent (HAC) robust standard errors to the spatial context. For the case of linear panel models with individual and time dummy variables, a recent paper by Kim (2010) provides results on kernel HAC standard errors. A distance measure in the cross-section is needed to implement the approach of Kim (2010). Suppose that a distance measure is either not readily available or is unknown for the crosssection of the panel. If the time series dimension of the panel is stationary, then it is possible to