On unit roots for spatial autoregressive models

Abstract

In this paper we consider the unit root problem for one rather simple autoregressive model Y t,s =aY t−1,s + bY t,s−1 +ε t,s on a two-dimensional lattice. We show that the growth of variance of Y t,s is essentially different from corresponding growth in the unit root case for AR(1) or AR(2) time series models. We also show that the dimension of the lattice plays an important role: the growth of variance of autoregressive field on a d-dimensional lattice is different for d = 2, 3 and d 4.

On the variances of a spatial unit root model*

“…This model is stable in case |α| + |β| < 1 and unstable if |α| + |β| = 1 (Basu and Reinsel, 1993). Paulauskas (2007) determined the exact asymptotic behaviour of the variances of the process, while Baran et al (2007) proved the asymptotic normality of the LSE of the parameters both in stable and unstable cases.…”

“…Observe that as for m, n ∈ N we have F (−n, −m; −n − m; 1) = m+n n −1 and F (−n, −m; −n − m; 0) = 1, moving average representations of the doubly geometric model of Martin (1979) and of the spatial models studied by Paulauskas (2007) and Baran et al (2004Baran et al ( , 2007, respectively, are special forms of (1.5).…”

“…In Baran et al (2004) the asymptotic normality of the LSE of the unknown parameter α is proved both in stable and unstable cases. The case p 1 = p 2 = 1, α 1,0 =: α, α 0,1 =: β and α 1,1 = 0 was studied by Paulauskas (2007) and Baran et al (2007). This model is stable in case |α| + |β| < 1 and unstable if |α| + |β| = 1 (Basu and Reinsel, 1993).…”

Parameter estimation in a spatial unilateral unit root autoregressive model

“…Observe that as for m, n ∈ N we have F (−n, −m; −n − m; 1) = m+n n −1 and F (−n, −m; −n − m; 0) = 1, moving average representations of the doubly geometric model of Martin (1979) and of the spatial models studied by Paulauskas (2007) and Baran et al (2004Baran et al ( , 2007, respectively, are special forms of (1.6). Now, put ε k,ℓ := (−1) k+ℓ ε k,ℓ for k, ℓ ∈ N. Then { ε k,ℓ : k, ℓ ∈ N} are independent random variables with E ε k,ℓ = 0, Var ε k,ℓ = 1 and sup{E ε 8…”

Efficiency of the OLS estimator in the vicinity of a spatial unit root

“…Spatial unit roots have also been analyzed in the case of SAR processes on regular ACCEPTED MANUSCRIPT lattices; see, e.g., Bhattacharyya et al (1997), Baran et al (2004) andPaulauskas (2007).…”