On the variances of a spatial unit root model

TL;DR

This paper investigates the asymptotic behavior of variances in a spatial autoregressive model with unit root parameters, identifying different scaling limits depending on the location within the stability domain.

Contribution

It provides a detailed analysis of the variance limits of a spatial unit root model at boundary points, edges, and faces of the stability domain.

Findings

Variance scales as n^{-1/4} inside the domain

Variance scales as n^{-1/2} on the edges

Variance scales as n^{-1} at the vertices

Abstract

The asymptotic properties of the variances of the spatial autoregressive model $X_{k,\ell}=\alpha X_{k-1,\ell}+\beta X_{k,\ell-1}+\gamma X_{k-1,\ell-1}+\epsilon_{k,\ell}$ are investigated in the unit root case, that is when the parameters are on the boundary of domain of stability that forms a tetrahedron in $[-1,1]^3$. The limit of the variance of $n^{-\varrho}X_{[ns],[nt]}$ is determined, where on the interior of the faces of the domain of stability $\varrho=1/4$, on the edges $\varrho =1/2$, while on the vertices $\varrho =1$.