## The Annals of Statistics

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

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

N. H. Chan and C. Z. Wei

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

**Keywords**

Nearly nonstationary autoregressive process least squares stochastic integral

#### Citation

Chan, N. H.; Wei, C. Z. Asymptotic Inference for Nearly Nonstationary AR(1) Processes. Ann. Statist. 15 (1987), no. 3, 1050--1063. doi:10.1214/aos/1176350492. https://projecteuclid.org/euclid.aos/1176350492