Adjusted QMLE for the spatial autoregressive parameter

One simple, and often very effective, way to attenuate the impact of nuisance parameters on maximum likelihood estimation of a parameter of interest is to recenter the profile score for that parameter. We apply this general principle to the quasi-maximum likelihood estimator (QMLE) of the autoregressive parameter $\lambda$ in a spatial autoregression. The resulting estimator for $\lambda$ has better finite sample properties compared to the QMLE for $\lambda$, especially in the presence of a large number of covariates. It can also solve the incidental parameter problem that arises, for example, in social interaction models with network fixed effects, or in spatial panel models with individual or time fixed effects. However, spatial autoregressions present specific challenges for this type of adjustment, because recentering the profile score may cause the adjusted estimate to be outside the usual parameter space for $\lambda$. Conditions for this to happen are given, and implications are discussed. For inference, we propose confidence intervals based on a Lugannani--Rice approximation to the distribution of the adjusted QMLE of $\lambda$. Based on our simulations, the coverage properties of these intervals are excellent even in models with a large number of covariates.