In this paper, we propose an improved inference procedure for the interactive fixed effects model in the presence of cross-sectional dependence and heteroskedasticity. It is well known in the literature that the LS estimator in this model by Bai (2009) is asymptotically biased when the error term is cross-sectionally dependent, and we address this problem. Our procedure involves two parts, correcting the asymptotic bias of the LS estimator and employing the cross-sectional dependence robust covariance matrix estimator. We prove the validity of the proposed procedure in the asymptotic sense. Since our approach is based on the spatial HAC estimation, e.g., Conley (1999), Kelejian and Prucha (2007) and Kim and Sun (2011), we need a distance measure that characterizes the dependence structure. Such a distance may not be available in practice and we address this by considering a data-driven distance that does not rely on prior information. We also develop a bandwidth selection procedure based on a cluster wild bootstrap method. Monte Carlo simulations show our procedure work well in finite samples. As empirical illustrations, we apply the proposed approach to study the effects of divorce law reforms on U.S. divorce rates Wolfers (2006) and the impacts of clean water and sewerage interventions on U.S. child mortality Alsan and Goldin (2019).
Our procedure can be applied to the broad empirical literature in economics. We illustrate this with two empirical examples. The first one is the well-known problem of the U.S. divorce rates affected by divorce law reforms around the 1970s. Using the standard fixed-effects model with weighted least squares (WLS) estimation, Wolfers (2006) identifies the rise of divorce rates in the first eight years after the law reform. However, the robustness of Wolfers (2006)’s results is doubted in two regards. First, the model he uses may not be flexible enough to capture the factors varying over time and across states (e.g., the stigma of divorce; religious belief). This may lead to the observed large discrepancy between the ordinary least squares (OLS) and WLS estimates found by later studies. Second, the idiosyncratic errors in his model are assumed to be cross-sectionally independent, which does not seem to be appropriate in practice. Kim and Oka (2013) employ the IFE model for the study. Their results confirm the findings of Wolfers (2006) and are robust to the weighting schemes. However, their bias correction procedure and standard error estimation do not take the cross-sectional dependence into account. We apply the proposed approach and provide inference results for this model. We find the IFE model with the proposed procedure yields smaller estimates with wider confidence intervals than Kim and Oka (2013)’s results.
The second example studies the effects of clean water and effective sewerage systems on U.S. child mortality. An essential question in public health is the cause of the sharp decrease in the U.S. and Massachusetts infant mortality from 1870 to 1930. Alsan and Goldin (2019) exploit the independent and combined effects of clean water and effective sewerage systems on under-5 mortality in Massachusetts, 1880-1920. For empirical strategy, they employ a standard fixed-effects model, which identifies the two interventions together account for approximately one-third of the decline in the log of child mortality during the 41 years. Since they use the municipality-level data, the potential unobserved time-varying heterogeneity and cross-sectional correlation in the idiosyncratic errors may affect the results. To check the robustness of their results, we employ the IFE model with the proposed inference procedure for the study. We find that the combined impacts of sewerage and safe water treatments on child mortality are significantly decreased by using the IFE model with the proposed procedure.