  Newton-step approximations to pseudo maximum likelihood estimates of spatial
autoregressive models with a large number of parameters are examined, in the
sense that the parameter space grows slowly as a function of sample size. These
have the same asymptotic efficiency properties as maximum likelihood under
Gaussianity but are of closed form. Hence they are computationally simple and
free from compactness assumptions, thereby avoiding two notorious pitfalls of
implicitly defined estimates of large spatial autoregressions. For an initial
least squares estimate, the Newton step can also lead to weaker regularity
conditions for a central limit theorem than those extant in the literature. A
simulation study demonstrates excellent finite sample gains from Newton
iterations, especially in large multiparameter models for which grid search is
costly. A small empirical illustration shows improvements in estimation
precision with real data.
