Spatial Dependence in Regressors and its Effect on Estimator Performance

Abstract

Most spatial econometrics work focuses on spatial dependence in the regressand or disturbances. However, LeSage and Pace (2009) show that the bias from applying OLS to a regressand generated from a spatial autoregressive process was exacerbated by spatial dependence in the regressor. Also, the marginal likelihood function or restricted maximum likelihood (REML) function includes a determinant of a function of the regressors. Therefore, high dependence in the regressor may affect the likelihood through this term. Finally, the notion of effective sample size for dependent data suggests that the loss of information from dependence may have implications for the information content of various variables. Empirically, many common economic regressors such as income, race, and employment show high levels of spatial autocorrelation. Based on these empirical results, we conduct a Monte Carlo study using maximum likelihood, restricted maximum likelihood, and two instrumental variable specifications for the lag y model (SAR) and spatial Durbin model (SDM) in the presence of correlated regressors while varying signal-to-noise, spatial dependence, and weight matrix specifications. We find that REML outperforms ML in the presence of correlated regressors and that instrumental variable performance is affected by such dependence. The combination of correlated regressors and the SDM provides a challenging environment for instrumental variable techniques. In addition, we examine the estimation of marginal effects and show that this can behave better than estimation of component parameters. We also make suggestions for improving Monte Carlo experiments.

Keywords: regressor autocorrelation, spatial Durbin model, REML, spatial autoregression, maximum likelihood, spatial econometrics

JEL Classification: C11, O31, R11