Asymptotic inference for near unit roots in spatial autoregression. (English) Zbl 0890.62018

Bhattacharyya, B. B.; Richardson, G. D.; Franklin, L. A.

Ann. Stat. 25, No. 4, 1709-1724 (1997).

Summary: Asymptotic inference for estimators of \((\alpha_n, \beta_n)\) in the spatial autoregressive model \[ Z_{ij} (n)= \alpha_nZ_{i-1,j} (n)+ \beta_nZ_{i,j-1} (n)-\alpha_n\beta_n Z_{i-1,j-1} (n)+\varepsilon_{ij} \] is obtained when \(\alpha_n\) and \(\beta_n\) are near unit roots. When \(\alpha_n\) and \(\beta_n\) are reparameterized by \(\alpha_n= e^{c/n}\) and \(\beta_n= e^{d/n}\), it is shown that if the “one-step Gauss-Newton estimator” of \(\lambda_1 \alpha_n+ \lambda_2 \beta_n\) 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.

MSC:

62F12	Asymptotic properties of parametric estimators
62M30	Inference from spatial processes
60F17	Functional limit theorems; invariance principles
60G44	Martingales with continuous parameter

Keywords:

spatial autoregressive process; near unit roots; Gauss-Newton estimation; central limit theory; maximal inequality; two-parameter martingale

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