## The Annals of Statistics

- Ann. Statist.
- Volume 7, Number 5 (1979), 1106-1120.

### Estimation for Autoregressive Processes with Unit Roots

David P. Hasza and Wayne A. Fuller

#### Abstract

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \sum^p_{j = 1}\eta_jY_{t - j} + e_t$ for $t = 1, 2, \cdots$, where the $e_t$ are independent identically distributed $(0, \sigma^2)$ random variables and the initial conditions $(Y_{-p + 1}, Y_{-p + 2}, \cdots, Y_0)$ are fixed constants. It is assumed the true, but unknown, roots $m_1, m_2, \cdots, m_p$ of $m^p - \sum^p_{j = 1}\eta_jm^{p - j} = 0$ satisfy $m_1 = m_2 = 1$ and $|m_j| < 1$ for $j = 3, 4, \cdots, p$. Let $\hat{\mathbf{\eta}}$ denote the least squares estimator of $\mathbf{\eta} = (\eta_1, \eta_2, \cdots, \eta_p)'$ obtained by the least squares regression of $Y_t$ on $Y_{t - 1}, Y_{t - 2}, \cdots, Y_{t - p}$ for $t = 1, 2, \cdots, n$. The asymptotic distributions of $\hat{\mathbf{\eta}}$ and of a test statistic designed to test the hypothesis that $m_1 = m_2 = 1$ are characterized. Analogous distributional results are obtained for models containing time trend and intercept terms. Estimated percentiles for these distributions are obtained by the Monte Carlo method.

#### Article information

**Source**

Ann. Statist., Volume 7, Number 5 (1979), 1106-1120.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344793

**Digital Object Identifier**

doi:10.1214/aos/1176344793

**Mathematical Reviews number (MathSciNet)**

MR536511

**Zentralblatt MATH identifier**

0419.62068

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62J05: Linear regression

**Keywords**

Time series autoregression nonstationary differencing

#### Citation

Hasza, David P.; Fuller, Wayne A. Estimation for Autoregressive Processes with Unit Roots. Ann. Statist. 7 (1979), no. 5, 1106--1120. doi:10.1214/aos/1176344793. https://projecteuclid.org/euclid.aos/1176344793