Endogeneity in a Spatial Context: Properties of Estimators

Endogeneity is a pervasive problem in applied econometrics, and this is no less true in spatial econometrics. However, while the appropriate treatment and estimation of the endogenous spatial lag has received a good deal of attention (Cliff and Ord 1981; Upton and Fingleton 1985; Anselin 1988, 2006), the analysis of the effects of other endogenous variables has been rather neglected so far. Nevertheless, it is known that the consistent estimation of spatial lag models with additional endogenous variables is straightforward since it can be accomplished by two-stage least squares, with the lower orders of the spatial lags of the exogenous variables as instruments (see Anselin and Lozano-Gracia 2008; Dall’erba and Le Gallo 2008 for applications of this procedure). In addition, the case of endogenous variables and a spatial error process has been considered by Kelejian and Prucha (2004). Their paper generalizes the Kelejian and Prucha (1998) feasible generalized spatial two-stage least squares estimator to allow for additional endogenous variables on the right hand side when there is an explicit set of simultaneous equations. Kelejian and Prucha (2007) consider a general spatial regression model that allows for endogenous regressors, their spatial lags, as well as exogenous regressors, emphasizing that their model may, in particular, represent the ith equation of a simultaneous system of equations, but also mentioning its applicability to endogeneity in general. Fingleton and Le Gallo (2008a, b) develop the approach to consider endogeneity from various sources with either autoregressive or moving average error processes. However, there are certain specific aspects of spatial econometrics that lead to a somewhat different treatment of the endogeneity problem and its solution. In this chapter, we outline the problem in the spatial context, focusing on the relative impact of different sources of endogeneity. In particular, we focus on endogeneity and hence the inconsistency of the usual OLS estimators induced by omitting a significant variable that should be in the regression model but which is unmeasured and hence is present in the residual. We also consider simultaneity and errors-in-variables. The outline of the chapter is as follows. The next section describes the main sources of inconsistency considered in this chapter, namely omitted variables, simultaneity and measurement error. Also, we consider the particular case of omitted variables in a spatial context. Then, we perform the Monte-Carlo simulations aimed at analyzing the performance of a spatial Durbin model as a potential remedy for bias and inconsistency. The last section concludes.