Parameter estimation in a spatial unilateral unit root autoregressive model

Spatial unilateral autoregressive model X k , ℓ = α X k − 1 , ℓ + β X k , ℓ − 1 + γ X k − 1 , ℓ − 1 + ε k , 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 n 3 / 2 .