Parameter estimation in a spatial unit root autoregressive model

TL;DR

This paper analyzes the asymptotic behavior of least squares estimators in a spatial autoregressive model at the boundary of stationarity, revealing different convergence rates and normality in various boundary cases.

Contribution

It characterizes the limiting distribution and convergence rates of estimators in a spatial unit root autoregressive model at the boundary of stability.

Findings

Normal limiting distribution of estimators on faces and edges.

Rate of convergence is n on faces and edges.

Rate of convergence is n^{3/2} at vertices.

Abstract

Spatial unilateral autoregressive model $X_{k,\ell}=\alpha X_{k-1,\ell}+\beta X_{k,\ell-1}+\gamma X_{k-1,\ell-1}+\epsilon_{k,\ell}$ is investigated in the unit root case, that is when the parameters are on the boundary of the domain of stability that forms a tetrahedron with vertices $(1,1,-1), \ (1,-1,1),\ (-1,1,1)$ and $(-1,-1,-1)$. It is shown that the limiting distribution of the least squares estimator of the parameters is normal and the rate of convergence is $n$ when the parameters are in the faces or on the edges of the tetrahedron, while on the vertices the rate is $n^{3/2}$.