Variable selection in spatial lag models using the focussed information criterion

Spatial regression models have a variety of applications in several fields ranging from economics to public health. Typically, it is of interest to select important exogenous predictors of the spatially autocorrelated response variable. In this paper, we propose variable selection in linear spatial lag models by means of the focussed information criterion (FIC). The FIC-based variable selection involves the minimization of the asymptotic risk in the estimation of a certain parametric focus function of interest under potential model misspecification. We systematically investigate the key asymptotics of the maximum likelihood estimators under the sequence of locally perturbed mutually contiguous probability models. Using these results, we obtain the expressions for the bias and the variance of the estimated focus leading to the desired FIC formula. We provide practically useful focus functions that account for various spatial characteristics such as mean response, variability in the estimation and spatial spillover effects. Furthermore, we develop an averaged version of the FIC that incorporates varying covariate levels while evaluating the models. The empirical performance of the proposed methodology is demonstrated through simulations and real data analysis.