Asymptotic inference for an unstable spatial AR model S ´ ANDOR BARAN a , GYULA PAP a and MARTIEN C. A. VAN ZUIJLEN b,* a Institute of Informatics, University of Debrecen, H-4010 Debrecen, P.O.Box 12, Hungary b Department of Mathematics, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen The Netherlands Abstract The spatial autoregressive process X k,ℓ = α(X k−1,ℓ + X k,ℓ−1 )+ ε k,ℓ , k,ℓ  0 is investigated. We consider the least squares estimator  α m,n of α based on the observations {X k,ℓ :1  k  m and 1  ℓ  n}. In the stable (i.e., asymptotically stationary) case when |α| < 1/2, asymptotic normality (mn) 1/2 (  α m,n − α) D −→ N (0,σ 2 α ) as m, n →∞ with m/n → constant > 0 can be derived from previous more general results due to Basu and Reinsel [16, 7, 12]. In the unstable case when |α| =1/2, we prove again asymptotic normality, but (in contrast to the doubly geometric spatial model) with a suprising rate of convergence, namely, (mn) 5/8 (  α m,n − α) D −→ N (0,σ 2 ) as m, n →∞ with m/n → constant > 0. Keywords: stable and unstable spatial autoregressive models, asymptotic information matrix, expansion in the Local Central Limit Theorem. 1 Introduction Consider the AR(1) time series model X k =  αX k−1 + ε k , k  1, 0, k =0. The least squares estimator  α n of α based on the observations {X k : k =1,...,n} is  α n = ∑ n k=1 X k−1 X k ∑ n k=1 X 2 k−1 . It is well known that in the stable (or, in other words, asymptotically stationary) case when |α| < 1, the sequence (  α n ) n1 is asymptotically normal (see [1, 2]), namely, n 1/2 (  α n − α) D −→ N (0, 1 − α 2 ). * Corresponding author. E-mail: zuijlen@sci.kun.nl This research has been supported by the Hungarian Scientific Research Fund under Grant No. OTKA– T032361/2000 and Grant No. OTKA-F032060/2000. 1