On the variances of a spatial unit root model

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 $\varrho =1\mathrm{/}4$, on the edges $\varrho =1\mathrm{/}2$, while on the vertices $\varrho =1$.