Spatial two-stage least squares
The endogeneity of the spatially lagged dependent variable can also be addressed by means of an instrumental variables or two-stage least squares (2SLS) approach (Anselin, 1980, 1988a, 1990; Kelejian and Robinson, 1993; Kelejian and Prucha, 1998). As demonstrated in Kelejian and Robinson (1993), the choice of an instrument for Wy follows from the conditional expectation in the reduced form (14.10),
E[y|X] = (I - pW)-1Xp = Xp + pWXp + p2W2Xp + ... . (14.28)
Apart from the exogenous variables X (which are always instruments), this includes their spatial lags as well, suggesting WX as a set of instruments.
Under a set of reasonable assumptions that are easily satisfied when the spatial weights are based on contiguity, the spatial two-stage least squares estimator achieves the consistency and asymptotic normality properties of the standard 2SLS (see, e. g. the theorems spelled out in Schmidt, 1976).25 A straightforward extension is the application of 3SLS to the spatial SUR model with a spatial lag (Anselin, 1988a, ch. 10).