Large Panels with Common Factors and Spatial Correlations

This paper considers the statistical analysis of large panel data sets where even after conditioning on common observed e¤ects the cross section units might remain dependently distributed. This could arise when the cross section units are subject to unobserved common e¤ects and/or if there are spill over e¤ects due to spatial or other forms of local dependencies. The paper provides an overview of the literature on cross section dependence, introduces the concepts of time-speci…c weak and strong cross section dependence and shows that the commonly used spatial models are examples of weak cross section dependence. It is then established that the Common Correlated E¤ects (CCE) estimator of panel data model with a multifactor error structure, recently advanced by Pesaran (2006), continues to provide consistent estimates of the slope coe¢ cient, even in the presence of spatial error processes. Small sample properties of the CCE estimator under various patterns of cross section dependence, including spatial forms, are investigated by Monte Carlo experiments. Results show that the CCE approach works well in the presence of weak and/or strong cross sectionally correlated errors. We also explore the role of certain characteristics of spatial processes in determining the performance of CCE estimators, such as the form and intensity of spatial dependence, and the sparseness of the spatial weight matrix. Keywords: Panels, Common Correlated E¤ects, Strong and Weak Cross Section Dependence. JEL Classi…cation: C10, C31, C33 We are grateful to Alexander Chudik and George Kapetanios for helpful comments and suggestions. 1 Introduction Over the past few years there has been a growing literature, both empirical and theoretical, on econometric analysis of panel data models with cross sectionally dependent error processes. Cross correlations can be due to omitted common e¤ects, spatial e¤ects, or could arise as a result of interactions within socio-economic networks. Conditioning on variables speci…c to the cross section units alone does not deliver cross section error independence; an assumption required by the standard literature on panel data models. In the presence of such dependence, conventional panel estimators such as …xed or random e¤ects can result in misleading inference and even inconsistent estimators. (Phillips and Sul, 2003). Further, conventional panel estimators will be inconsistent if regressors are correlated with the factors behind the error cross section dependence. (Pesaran, 2006). Correlation across units in panels has also serious drawbacks on commonly used panel unit root tests, since several of the existing tests assume independence (Levin, Lin and Chu, 2002; Im, Pesaran and Shin, 2003). As a result, when applied to cross sectional dependent panels, such unit root tests tend to have substantial size distortions (O’Connell, 1998). This potential problem has recently given major impetus to the research on panel unit root tests that allow for cross correlations (see, for example, Bai and Ng, 2004, Moon and Perron, 2004, and Pesaran, 2007). These and other related developments are reviewed in Breitung and Pesaran (2007). In the case of panel data models where the cross section dimension, N , is small and the time series dimension, T , is large, the standard approach to cross section dependence is to consider the equations from di¤erent cross section units as a system of seemingly unrelated regression equations (SURE), and then estimate it by Generalized Least Squares techniques (Zellner, 1962). If the time series dimension is not su¢ ciently large, and in particular if N > T , the SURE approach is not feasible. The approach also fails to yield consistent estimators if the cross dependence is due to an observed common factor which is correlated with the included observed regressors. Currently, there are two main strands in the literature for dealing with error cross section dependence in panels where N is large relative to T , namely the residual multifactor and the spatial econometric approaches. The multifactor approach assumes that the cross dependence can be characterized by a …nite number of unobserved common factors, possibly due to economy-wide shocks that a¤ect all units albeit with di¤erent intensities. Under this framework, the error term is a linear combination of few common time-speci…c e¤ects with heterogeneous factor loadings plus an idiosyncratic (individual-speci…c) error term. Estimation of a panel with such multifactor residual structure can be addressed by full maximum likelihood procedure (Robertson and Symons, 2000), or by principal component analysis (Coakley, Fuertes and Smith, 2002). A major shortcoming of these techniques is that they are not applicable when the regressors are correlated with the common shocks, as in this case they lead to inconsistent estimators. Recently, Pesaran (2006) has suggested an estimation method, known as Common Correlated E¤ects (CCE), that consists of approximating the linear combinations of the unobserved factors by cross section averages of the dependent and explanatory variables and then running standard panel regressions augmented with the cross section averages. An advantage of this approach is that it yields consistent estimates also when the regressors are correlated with the factors. Monte Carlo studies have also shown that,