Omitted Variable Biases of OLS and Spatial Lag Models

Numerous authors have suggested that omitted variables affect spatial regression methods less than ordinary least-squares (OLS; Dubin 1988; Brasington and Hite 2005, Cressie 1993). To explore these conjectures, we derive an expression for OLS omitted variable bias in a univariate model with spatial dependence in the disturbances and explanatory variables. There are a number of motivations for making this set of assumptions regarding the disturbances and explanatory variables. First, in spatial regression models each observation represents a region or point located in space, for example, census tracts, counties or individual houses. Sample data used as explanatory variables in these models typically consists of socioeconomic, census and other characteristics of the regional or point locations associated with each observation. Therefore, spatial dependence in the explanatory variables seems likely, motivating our choice of this assumption. Note, the literature rarely examines the spatial character of the explanatory variables, but this can affect the relative performance of OLS as shown below. Second, application of OLS models to regional data samples frequently leads to spatial dependence in the regression disturbances, providing a justification for this assumption. Finally, there are a host of latent unobservable and frequently unmeasurable influences that are likely to impact spatial regression relationships. For example, factors such as location and other types of amenities, highway accessibility, school quality or neighborhood prestige may exert an influence on the dependent variable in hedonic house price models. It is unlikely that explanatory variables are readily available to capture all of these types of latent influences. This type of reasoning motivates our focus on the impact of omitted explanatory variables. Since the omitted and included explanatory variables are both likely to exhibit spatial dependence based on the same spatial connectivity structure, it seems likely that omitted and included variables will exhibit non-zero covariance. The expression we derive for OLS bias in these circumstances shows that positive dependence in the disturbances and explanatory variables when omitted variables are correlated with included explanatory variables magnifies the magnitude of conventional least-squares omitted variables bias.