Outer-product-of-gradients tests for spatial autoregressive models

For Lagrangian multiplier (LM) tests of restrictions on parameters in spatial autoregressive (SAR) models with (SARAR models) or without SAR disturbances, their outer-product-of-gradient (OPG) variants can be simple and robust to unknown heteroskedasticity. However, for certain tests, asymptotic distributions of test statistics might depend on the constrained maximum likelihood or quasi maximum likelihood (QML) estimators, so their OPG variants would not be valid. To overcome such a hurdle, we propose to use C(α)-type score vectors to obtain valid OPG variants. Such OPG tests can be systematically constructed for SARAR models with homoskedastic and heteroskedastic disturbances, which might not be normally distributed. They also have the advantage that any n-consistent estimator can be used in place of a restricted QML estimate. In particular, OPG tests based on generalized method of moments (GMM) estimates are computationally simple and powerful compared to LM tests. Corresponding OPG tests based on C(α)-type gradient vectors in the GMM framework are also investigated.