Asymptotic inference for near unit roots in spatial autoregression

Asymptotic inference for estimators of (α n , β n ) in the spatial autoregressive model z ij (n) = α n z i-1 , j (n) + β n Z i,j-1 (n) - α n β n Z i-1 , j-1 (n) + e ij obtained when α n and β n are near unit roots. When α n and β n are reparameterized by α n = e c/n and β n = e d/n , it is shown that if the one-step Gauss-Newton estimator of λ 1 α n + λ 2 β n is properly normalized and embedded in the function space D([0,1] 2 ), the limiting distribution is a Gaussian process. The key idea in the proof relies on a maximal inequality for a two-parameter martingale which may be of independent interest. A simulation study illustrates the speed of convergence and goodness-of-fit of these estimators for various sample sizes.