TESTING FOR UNIT ROOTS IN A NEARLY NONSTATIONARY SPATIAL AUTOREGRESSIVE PROCESS

B. B. BHATTACHARYYA^{1}, X. LI^{2}, M. PENSKY^{2} AND G. D. RICHARDSON^{2}

1Department of Statistics, North Carolina State University, Raleigh, NC 27695-8203, U.S.A. 2Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, U.S.A.

(Received May 20, 1996; revised June 30, 1998)

Abstract. The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model Z_{st} = αZ_{s−1,t} +βZ_{s,t−1} − αβZ_{s−1,t−1} + ϵ_{st}. Moreover, for the sequence α_{n} = e^{c/n}, β_{n} = e^{d/n} of local Pitmantype alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.

Key words and phrases: First-order autoregressive process, unit roots, nearly nonstationary, periodogram ordinate, local Pitman-type alternatives, Ornstein-Uhlenbeck process.