Sankhya: The Indian Journal of Statistics

1996, Volume 58, Series B, Pt. 3, pp. 389--395

ASYMPTOTIC DISTRIBUTION OF PERIODOGRAM ORDINATE FOR A NEARLY NONSTATIONARY AR(1) PROCESS

By

B. B. BHATTACHARYYA, North Carolina State University, Raleigh

and

G. D. RICHARDSON, University of Central Florida, Orlando

SUMMARY.  A first-order autoregressive model $Y_{t}-\mu = \alpha_n(Y_{t-1}-\mu)+\epsilon_t$, $1 \leq t \leq n$, is called nearly nonstationary when the sequence {$\alpha_n$} converges to one. Reparameterizing $\alpha_n=e^c/n$, the limiting distribution of the normalized periodogram ordinate of a time series obeying this model is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c. This determines the limiting distribution of a unit root test in the frequency domain proposed by Akdi (1995) under Pitman-type alternatives of the form ec/n

AMS (1991) subject classification.  62M10, 62F12.

Key words and phrases. Nearly nonstationary, first-order autoregressive process, periodogram ordinate.

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