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 in a spatial panel model with individual and time fixed effects. Compared to the likelihood procedures currently available for this model, our adjusted QMLE does not require any conditions on the spatial weights matrix, and has better finite sample properties, particularly when the number of covariates is large. Saddlepoint confidence intervals for the spatial autoregressive parameter based on the adjusted QMLE are proposed. In simulation, they perform very well against other higher-order methods.