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,ℓXk,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)(1,−1,1), (−1,1,1)(−1,1,1) and (−1,−1,−1)(−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 nn when the parameters are in the faces or on the edges of the tetrahedron, while on the vertices the rate is n3/2n3/2.