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This paper considers the generalized method of moments (GMM) estimation of a spatial autoregressive (SAR) model with SAR disturbances, where we allow for endogenous regressors in addition to a spatial lag of the dependent variable. We do not assume any reduced form of the endogenous regressors, thus we allow for spatial dependence and heterogeneity in endogenous regressors, and allow for nonlinear relations between endogenous regressors and their instruments. Innovations in the model can be homoscedastic or heteroskedastic with unknown forms. We prove that GMM estimators with linear and quadratic moments are consistent and asymptotically normal. In the homoscedastic case, we derive the best linear and quadratic moments that can generate an optimal GMM estimator with the minimum asymptotic variance.

GMM estimation of a spatial autoregressive model with autoregressive disturbances and endogenous regressors