A Primer on Spatial Regression Models: Applications to Poverty and Inequality of Indian Districts

The standard classical linear regression model (CLRM) assumption is that the observations are independent and identically distributed. This assumption is questionable whenever the units of analysis are spatially dependent. The first step for capturing spatial dependence is to define a spatial weight matrix, where the matrix elements capture pairwise spatial dependence between the units of analysis. Given the weight matrix, two main models that evolved in the literature are the spatial autoregressive (SAR) model and the spatial error model (SEM). SAR is much in line with the well-known autoregressive models frequently used in time series, where the outcomes of the dependent variable in a linear regression model depend on the outcomes of the neighboring regions. Estimating spatial autoregressive parameters and the parameters associated with other explanatory variables using least square methods will lead to an inconsistent estimator; hence needs to be taken care of with special attention. SEM, on the other hand, is applicable whenever the error term of the linear regression equation is spatially autocorrelated. If the error term exhibits such a pattern, then ignoring it would lead to a biased estimator of the standard errors of the parameters. Nevertheless, the coefficients will be consistent, which implies that, unlike SAR, if the sample size is large enough, then ignoring the SEM model and executing a linear regression will not be problematic. One can estimate the parameters of the SAR and SEM using either a maximum likelihood method or a generalized method of moments. In this chapter, we discuss the two methods at length. Furthermore, we also discuss how these models can be executed in the statistical software Stata 15.