Modified QML estimation of spatial autoregressive models with unknown heteroskedasticity and nonnormality

• We introduce a heteroskedasticity robust QML-type estimator for SAR model.
• We present heteroskedasticity robust estimators of the standard errors.
• Monte Carlo results show that the proposed estimator outperforms the GMM estimators.
• Extension of the proposed methods is given to the more general SARAR(1, 1) model.
• Regular QML estimator can be consistent but is outperformed by the proposed one.

In the presence of heteroskedasticity, Lin and Lee (2010) show that the quasi-maximum likelihood (QML) estimator of the spatial autoregressive (SAR) model can be inconsistent as a ‘necessary’ condition for consistency can be violated, and thus propose robust GMM estimators for the model. In this paper, we first show that this condition may hold in certain situations and when it does the regular QML estimator can still be consistent. In cases where this condition is violated, we propose a simple modified QML estimation method robust against unknown heteroskedasticity. In both cases, asymptotic distributions of the estimators are derived, and methods for estimating robust variances are given, leading to robust inferences for the model. Extensive Monte Carlo results show that the modified QML estimator outperforms the GMM and QML estimators even when the latter is consistent. The proposed methods are then extended to the more general SARAR models.