## The Annals of Statistics

### Asymptotic Inference for Nearly Nonstationary AR(1) Processes

#### Abstract

A first-order autoregressive process, $Y_t = \beta Y_{t - 1} + \epsilon_t$, is said to be nearly nonstationary when $\beta$ is close to one. The limiting distribution of the least-squares estimate $b_n$ for $\beta$ is studied when $Y_t$ is nearly nonstationary. By reparameterizing $\beta$ to be $1 - \gamma/n, \gamma$ being a fixed constant, it is shown that the limiting distribution of $\tau_n = (\sum^n_{t = 1}Y^2_{t - 1})^{1/2}(b_n - \beta)$ converges to $\mathscr{L}(\gamma)$ which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of $\tau_n$ proposed by Ahtola and Tiao (1984).

#### Article information

Source
Ann. Statist., Volume 15, Number 3 (1987), 1050-1063.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176350492

Digital Object Identifier
doi:10.1214/aos/1176350492

Mathematical Reviews number (MathSciNet)
MR902245

Zentralblatt MATH identifier
0638.62082

JSTOR