The Annals of Statistics

Estimation for Autoregressive Processes with Unit Roots

David P. Hasza and Wayne A. Fuller

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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


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