  We develop new higher-order asymptotic techniques for the Gaussian maximum
likelihood estimator in a spatial panel data model, with fixed effects,
time-varying covariates, and spatially correlated errors. Our saddlepoint
density and tail area approximation feature relative error of order
$O(1/(n(T-1)))$ with $n$ being the cross-sectional dimension and $T$ the
time-series dimension. The main theoretical tool is the tilted-Edgeworth
technique in a non-identically distributed setting. The density approximation
is always non-negative, does not need resampling, and is accurate in the tails.
Monte Carlo experiments on density approximation and testing in the presence of
nuisance parameters illustrate the good performance of our approximation over
first-order asymptotics and Edgeworth expansions. An empirical application to
the investment-saving relationship in OECD (Organisation for Economic
Co-operation and Development) countries shows disagreement between testing
results based on first-order asymptotics and saddlepoint techniques.
