Spatial Dependence in Regressors and its Effect on Performance of Likelihood-Based and Instrumental Variable Estimators

Abstract

Most spatial econometrics work focuses on spatial dependence in the regressand or disturbances. However, Lesage and Pace (2009) as well as Pace and LeSage2009 showed that the bias in β 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 term involving the regressors. Therefore, high dependence in the regressor may affect the likelihood through this term. In addition, Bowden and Turkington (1984) showed that regressor temporal autocorrelation had a non-monotonic effect on instrumental variable estimators.

We provide empirical evidence that many common economic variables used as regressors (e.g., income, race, and employment) exhibit high levels of spatial dependence. Based on this observation, we conduct a Monte Carlo study of maximum likelihood (ML), REML and two instrumental variable specifications for spatial autoregressive (SAR) and spatial Durbin models (SDM) in the presence of spatially correlated regressors.

Findings indicate that as spatial dependence in the regressor rises, REML outperforms ML and that performance of the instrumental variable methods suffer. The combination of correlated regressors and the SDM specification provides a challenging environment for instrumental variable techniques.

We also examine estimates of marginal effects and show that these behave better than estimates of the underlying model parameters used to construct marginal effects estimates. Suggestions for improving design of Monte Carlo experiments are provided.

Citation

Kelley Pace, R., LeSage, J.P. and Zhu, S. (2012), "Spatial Dependence in Regressors and its Effect on Performance of Likelihood-Based and Instrumental Variable Estimators", Terrell, D. and Millimet, D. (Ed.) 30th Anniversary Edition (Advances in Econometrics, Vol. 30), Emerald Group Publishing Limited, Leeds, pp. 257-295. https://doi.org/10.1108/S0731-9053(2012)0000030014