## The Annals of Statistics

- Ann. Statist.
- Volume 16, Number 1 (1988), 367-401.

### Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes

N. H. Chan and C. Z. Wei

#### Abstract

An autoregressive process $y_n = \beta_1y_{n-1} + \cdots + \beta_py_{n-p} + \varepsilon_n$ is said to be unstable if the characteristic polynomial $\phi(z) = 1 - \beta_1z - \cdots - \beta_pz^p$ has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of $(\beta_1, \cdots, \beta_p)$ is derived and characterized as a functional of stochastic integrals under a $2 + \delta$ moment assumption on $\varepsilon_n$. Up to the present, distributional results were available only with substantial restrictions on the possible roots which did not suggest the form of the distribution for the general case. To establish the limiting distribution, a result concerning the weak convergence of a sequence of random variables to a stochastic integral, which is of independent interest, is also developed.

#### Article information

**Source**

Ann. Statist., Volume 16, Number 1 (1988), 367-401.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350711

**Digital Object Identifier**

doi:10.1214/aos/1176350711

**Mathematical Reviews number (MathSciNet)**

MR924877

**Zentralblatt MATH identifier**

0666.62019

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62E20: Asymptotic distribution theory 60F17: Functional limit theorems; invariance principles

**Keywords**

Unstable autoregressive process least squares stochastic integral limiting distribution

#### Citation

Chan, N. H.; Wei, C. Z. Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes. Ann. Statist. 16 (1988), no. 1, 367--401. doi:10.1214/aos/1176350711. https://projecteuclid.org/euclid.aos/1176350711