# Spatial Regression Models

2.1 Spatial lag and spatial error models

In the standard linear regression model, spatial dependence can be incorporated in two distinct ways: as an additional regressor in the form of a spatially lagged dependent variable (Wy), or in the error structure (Е[є;є;] Ф 0). The former is referred to as a spatial lag model and is appropriate when the focus of interest is the assessment of the existence and strength of spatial interaction. This is inter­preted as substantive spatial dependence in the sense of being directly related to a spatial model (e. g. a model that incorporates spatial interaction, yardstick com­petition, etc.). Spatial dependence in the regression disturbance term, or a spatial error model is referred to as nuisance dependence. This is appropriate when the concern is with correcting for the potentially biasing influence of the spatial autocorrelation, due to the use of spatial data (irrespective of whether the model of interest is spatial or not).

Formally, a spatial lag model, or a mixed regressive, spatial autoregressive model is expressed as

y = pWy + Xp + e, (14.9)

where p is a spatial autoregressive coefficient, e is a vector of error terms, and the other notation is as before.16 Unlike what holds for the time series counterpart of this model, the spatial lag term Wy is correlated with the disturbances, even when the latter are iid. This can be seen from the reduced form of (14.9),

y = (I – pW)-1Xp + (I – pW)-1E, (14.10)

in which each inverse can be expanded into an infinite series, including both the explanatory variables and the error terms at all locations (the spatial multiplier). Consequently, the spatial lag term must be treated as an endogenous variable and proper estimation methods must account for this endogeneity (OLS will be biased and inconsistent due to the simultaneity bias).

A spatial error model is a special case of a regression with a non-spherical error term, in which the off-diagonal elements of the covariance matrix express the structure of spatial dependence. Consequently, OLS remains unbiased, but it is no longer efficient and the classical estimators for standard errors will be biased. The spatial structure can be specified in a number of different ways, and (except for the non-parametric approaches) results in a error variance-covariance matrix of the form E[ee’] = Q(0),

where 0 is a vector of parameters, such as the coefficients in an SAR error

process.