Asymptotic inference for near unit roots in spatial autoregression

Asymptotic inference for estimators of (αn, βn) in the spatial autoregressive model Zij(n) = αnβnZi Zi-1, j(n) + βnZi, j-1(n) - αn βnZi-1, j-1(n) + εij is obtained when αn and βn are near unit roots. When αn and βn are reparameterized by αn = ec/n and βn = ed/n, it is shown that if the one-step Gauss-Newton estimator of λ1αn + λ2 βn 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