  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.
