Consistency and Limit Distributions of Estimators of Parameters in Explosive Stochastic Difference Equations

Abstract

Let $\{X_t, t \geq 1\}$ be a stochastic process which satisfies the following set of assumptions: ASSUMPTION 1: For every $t, X_t$ satisfies \begin{equation*}\tag{1}X_t = \alpha_1X_{t - 1} + \alpha_2X_{t - 2} + \cdots + \alpha_kX_{t - k} + u_t,\end{equation*} where $\alpha_1, \cdots, \alpha_k$ are $k$ finite real numbers (unknown parameters) and $u_t, t$ positive, are independent, identically distributed random variables with mean zero and a finite positive variance $\sigma^2$. ASSUMPTION 2: The distribution of $u_t$ is continuous. (Actually $\mathrm{Pr}\{u_t = 0\} = 0$ suffices.) ASSUMPTION 3: The roots $m_1, \cdots, m_k$ of the characteristic equation \begin{equation*}\tag{2}m^k - \alpha_1m^{k - 1} - \alpha_2m^{k - 2} - \cdots - \alpha_k = 0,\end{equation*} of (1), are distinct. ASSUMPTION 4: There is a unique root $\rho$ of (2) such that $|\rho| > 1$, and $|\rho| > \max_{j = 2, \cdots, k} |m_j|$. Here $\rho$ is identified with $m_1$ for convenience. Since complex roots enter in pairs, it follows from this assumption that $\rho$ is real. Note that there can be $m_j, j > 1$, such that $|m_j| > 1$. ASSUMPTION 5: For $t$ non-positive, $u_t = 0$. If Assumption 4 holds, the process $\{X_t, t \geqq 1\}$ is said to be (strongly) explosive, and the corresponding difference equation (1) is called an explosive (linear homogeneous) stochastic difference equation; this is the subject of the present paper. Under the assumptions above, it follows (cf., C. Jordan [5], p. 564, Mann and Wald [8], p. 178, and also the footnote on p. 22 of [10]) that $X_t = \sum^t_{r = 1}\sum^k_{q = 1} \lambda_qm^{t - r}_qu_r,$ $t$ positive, and that $\lambda_q$ satisfy the relations \begin{equation*}\tag{3}\delta_{1t} = \sum^k_{q = 1} \lambda_qm^{t - 1}_q,\quad t = 1, 0, -1, \cdots, - (k - 2),\end{equation*} where $\delta_{1t} = 1$ if $t = 1$ and 0 otherwise. (Note that $\sum^k_{q = 1}\lambda_q = 1$.) For convenience, define the random variables \begin{equation*}\tag{4}X_{i,t} = \sum^t_{r = 1} m^{t - r}_iu_r,\quad i = 1, 2, \cdots, k, (m_1 = \rho),\end{equation*} so that $X_{i,t} = 0$ for $t$ non-positive. Thus one may write $X_t$ as follows: \begin{equation*}\tag{5}X_t = \lambda_1X_{1,t} + \lambda_2X_{2,t} + \cdots + \lambda_kX_{k,t}.\end{equation*} The first part of this paper is devoted to finding a consistent estimator of $\rho$ and its limit distribution. Consequently, in Section 3 some lemmas will be proved for use in the consistency proof (Theorem I). Similarly, in Section 5, some lemmas leading to the proof of the limit distribution of the estimator (Theorem II) will be given. In the second part, the consistency of the Least Squares (L.S.) or Maximum Likelihood (M.L.) estimators of the "structural parameters" $\alpha_i$ of (1) will be considered (Theorem III). The procedure becomes much more involved because the direct application of the usual limit theorems is not possible, since the process under consideration is explosive. It is noteworthy that Lemmas 9, 10, 14-16, and Theorem I are rather general, in that they hold under the only global Assumptions 1-5 above, and the further requirement $|m_j| < 1, j = 2, \cdots, k$, so essential for the rest of the analysis of this paper, is unnecessary for them. The corresponding problem, in the case $|\rho| < 1$, has been completely solved by Mann and Wald [8]. If $k = 1$ in (1), the results of this paper reduce to those obtained by Rubin [13], White [14], and T. W. Anderson [1]. The vector case has also been treated by Anderson in [1], but a comparison of the results in this case with those of the present paper shows that they do not imply each other except in the first order. In the latter case, however, both reduce to Rubin's [13] result. The available results on stochastic difference equations are summarized in a table at the end of the paper. Some of the details and computations omitted in this paper may be found in [10]. In the following section, some known lemmas related to stochastic convergence are collected and stated in a convenient form, as they will be constantly referred to in both parts of the paper. (For proofs, see [2], [3], [4], [6] and [9].)