Asymptotic inference for near unit roots in spatial autoregression

Asymptotic inference for estimators of (αₙ, βₙ) in the spatial autoregressive model Zᵢⱼ(n) = αₙ Zi-1, j(n) + βₙ Zi, j-1(n) - αₙ βₙ Zi-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 λ₁ αₙ + 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.