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

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 $(|\alpha| > 1).$