Asymptotically efficient root estimators for spatial autoregressive models with spatial autoregressive disturbances

This paper considers closed-form root estimators for spatial autoregressive models with spatial autoregressive disturbances (SARAR model). We first derive a simple consistent closed-form estimator. Then we construct feasible moment conditions that are quadratic in the spatial lag and spatial error dependence parameters separately, which generate root estimators with closed forms. We consider both the cases with homoskedastic and unknown heteroskedastic disturbances. In the homoskedastic case, the root estimator can be asymptotically as efficient as the quasi-maximum likelihood estimator (QMLE); in the heteroskedastic case, it can be asymptotically as efficient as a method of moments estimator (MME) that sets adjusted quasi-maximum likelihood scores to zero, where the adjusted scores have zero means at the true parameters. The root estimators and their associated standard errors can avoid the computation of any matrix determinants or inverses, so they are computationally simple without iterations, especially valuable for big data where the sample size is large.