The Annals of Mathematical Statistics

The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case II

John S. White

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Abstract

A standard linear regression model is \begin{equation*}\tag{1}x_t = \alpha y_t + u_t\quad (t = 1, 2, 3, \cdots T)\end{equation*} where $\alpha$ is an unknown parameter, the $y$'s are known parameters and the $u$'s are NID $(0, \sigma^2)$. The maximum likelihood estimators for $\alpha$ and $\sigma^2$ are \begin{equation*}\tag{2}\hat \alpha = \frac{\Sigma x_t y_t}{\Sigma y^2_t},\quad\hat \sigma^2 = \frac{\Sigma (x_t - \hat \alpha y_t)^2}{T}.\end{equation*} The statistic \begin{equation*}\tag{3}\frac{(\hat \alpha - \alpha)}{\hat \sigma} (\Sigma y^2_t)^{\frac{1}{2}} \big(\frac{T - 1}{T}\big)^{\frac{1}{2}}\end{equation*} then has a $t$ distribution with $T - 1$ d.f. and its limiting distribution is $N(0, 1)$. One approach to time-series analysis is to set $y_t = x_{t-1}, y_1 = x_0 = \text{a constant}$. The model (1) is then transformed into the stochastic difference equation. \begin{equation*}\tag{4}x_t = \alpha x_{t-1} + u_t.\quad (t = 1,2, \cdots, T)\end{equation*} The maximum likelihood estimators for $\alpha$ and $\sigma^2$ in (4) are \begin{equation*}\tag{5}\hat \alpha = \frac{\Sigma x_t x_{t-1}}{\Sigma x^2_{t-1}},\quad \hat \sigma^2 = \frac{\Sigma (x_t - \hat \alpha x_{t-1})^2}{T}\end{equation*} which are exactly the values one would obtain by substituting $y_t = x_{t-1}$ in (2). In this paper it is shown that the limiting distribution of \begin{equation*}\tag{6}W = \frac{(\hat \alpha - \alpha)}{\hat \sigma} (\Sigma x^2_{t-1})^{\frac{1}{2}},\end{equation*} which is the analogue of (3), has a limiting $N(0, 1)$ distribution, except perhaps when $|\alpha| = 1$. This result is well-known for $|\alpha| < 1$ and was proved by Mann and Wald [1] under much more general conditions. The feature of the proof presented here is that it also holds in the explosive case $(|\alpha| > 1).$

Article information

Source
Ann. Math. Statist., Volume 30, Number 3 (1959), 831-834.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706213

Digital Object Identifier
doi:10.1214/aoms/1177706213

Mathematical Reviews number (MathSciNet)
MR107348

Zentralblatt MATH identifier
0133.42403

JSTOR
links.jstor.org

Citation

White, John S. The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case II. Ann. Math. Statist. 30 (1959), no. 3, 831--834. doi:10.1214/aoms/1177706213. https://projecteuclid.org/euclid.aoms/1177706213


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