AISM 52, 71-83

## 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}

^{1}Department of Statistics, North
Carolina State University, Raleigh, NC 27695-8203, U.S.A.

^{2}Department 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} = *alpha*
*Z*_{s-1,t}+*beta*
*Z*_{s,t-1} - *alpha beta*
*Z*_{s-1,t-1}+
*epsilon*_{st}.
Moreover, for the sequence
*alpha*_{n} = *e*
^{c/n},
*beta*_{n} =
*e*^{d/n} of local Pitman-type
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.

**Source**
( TeX , DVI )