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

Abstract

Several authors have studied the discrete stochastic process $(x_t)$ in which the $x$'s are related by the stochastic difference equation \\begin{equation*}\\tag{1.1}x_t = \\alpha x_{t - 1} + u_t, \\quad t = 1,2, \\cdots, T,\\end{equation*} where the $u$'s are unobservable disturbances, independent and identically distributed with mean zero and variance $\\sigma^2$, and $\\alpha$ is an unknown parameter. The statistical problem is to find some appropriate function of the $x$'s as an estimator for $\\alpha$ and examine its properties. We may rewrite (1.1) as \\begin{equation*}\\tag{1.2}x_t = u_t + \\alpha u_{t - 1} + \\cdots + \\alpha^{t - 1}u_1 + \\alpha^tx_0.\\end{equation*} From (1.2) we see that the distribution of the successive $x$'s is not uniquely determined by that of the $u$'s alone. The distribution of $x_0$ must also be specified. Three distributions which have been proposed for $x_0$ are the following: (A) $x_0$ = a constant (with probability one), (B) $x_0$ is normally distributed with mean zero and variance $\\sigma^2/(1 - \\alpha^2)$, (C) $x_0 = x_T$. Distribution (B) is perhaps the most appealing from a physical point of view, since if $x_0$ has this distribution and if the $u$'s are normally distributed, then the process is stationary (e.g., see Koopmans [4]). However, there are several analytic difficulties which arise in the statistical treatment of this process. Distribution (C), the so-called circular distribution, has been proposed as an approximation to (B) and is much easier to analyze (e.g., see Dixon [2]). Distribution (A) has been studied extensively by Mann and Wald [5]. An interesting feature of distribution (A) is that $\\alpha$ may assume any finite value, while for distributions (B) and (C) $\\alpha$ must be between $-1$ and 1. From (1.2) we see that a process satisfying (1.1) and (A) has \\begin{equation*}\\tag{1.3}\\operatorname{var}(x_t) = \\sigma^2(1 + \\alpha^2 + \\cdots + \\alpha^{2(t - 1)})\\end{equation*} If $|\\alpha| \\geqq 1, \\lim_{t = \\infty} \\operatorname{var}(x_t) = \\infty$ and the process is said to be "explosive." Mann and Wald [5] considered only the case $|\\alpha| < 1$. They showed that the least squares estimator for $\\alpha$ is the serial correlation coefficient \\begin{equation*}\\tag{1.4}\\hat\\alpha = \\frac{\\sum x_t x_{t - 1}}{\\sum x^2_{t - 1}}\\end{equation*} and that (for $|\\alpha| < 1$) this estimator is asymptotically normally distributed with mean $\\alpha$ and variance $(1 - \\alpha^2)/T$. Rubin [6] showed that the estimator $\\hat\\alpha$ is consistent (i.e., $\\operatorname{plim} \\hat\\alpha = \\alpha$) for all $\\alpha$. In this paper the asymptotic distribution of $\\hat\\alpha$ will be studied under the assumption that the $u$'s are normally distributed. For $|\\alpha| > 1$, it is shown that the asymptotic distribution of $\\alpha$ is the Cauchy distribution. For $|\\alpha| = 1$, a moment generating function is found, the inversion of which will yield the asymptotic distribution.