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September, 1979 Estimation for Autoregressive Processes with Unit Roots
David P. Hasza, Wayne A. Fuller
Ann. Statist. 7(5): 1106-1120 (September, 1979). DOI: 10.1214/aos/1176344793

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.

Citation

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David P. Hasza. Wayne A. Fuller. "Estimation for Autoregressive Processes with Unit Roots." Ann. Statist. 7 (5) 1106 - 1120, September, 1979. https://doi.org/10.1214/aos/1176344793

Information

Published: September, 1979
First available in Project Euclid: 12 April 2007

zbMATH: 0419.62068
MathSciNet: MR536511
Digital Object Identifier: 10.1214/aos/1176344793

Subjects:
Primary: 62M10
Secondary: 62J05

Rights: Copyright © 1979 Institute of Mathematical Statistics

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Vol.7 • No. 5 • September, 1979
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