## The Annals of Statistics

### Asymptotic inference for near unit roots in spatial autoregression

#### Abstract

Asymptotic inference for estimators of $(\alpha_n, \beta_n)$ in the spatial autoregressive model $Z_{ij}(n) = \alpha_n Z_{i-1, j}(n) + \beta_n Z_{i, j-1}(n) - \alpha_n \beta_n Z_{i-1, j-1}(n) + \varepsilon_{ij}$ 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.

#### Article information

Source
Ann. Statist., Volume 25, Number 4 (1997), 1709-1724.

Dates
First available in Project Euclid: 9 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1031594738

Digital Object Identifier
doi:10.1214/aos/1031594738

Mathematical Reviews number (MathSciNet)
MR1463571

Zentralblatt MATH identifier
0890.62018

#### Citation

Bhattacharyya, B. B.; Richardson, G. D.; Franklin, L. A. Asymptotic inference for near unit roots in spatial autoregression. Ann. Statist. 25 (1997), no. 4, 1709--1724. doi:10.1214/aos/1031594738. https://projecteuclid.org/euclid.aos/1031594738

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