The Annals of Statistics

Estimation for Autoregressive Processes with Unit Roots

David P. Hasza and Wayne A. Fuller

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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

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

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62J05: Linear regression

Time series autoregression nonstationary differencing


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.

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