Parameter estimation in a spatial unilateral unit root autoregressive model

Spatial unilateral autoregressive model Xk,ℓ=αXk−1,ℓ+βXk,ℓ−1+γXk−1,ℓ−1+εk,ℓ 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 n3/2.