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A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case Without an Intercept

Pierre Perron

Abstract

Consider the first-order autoregressive process $y_t = \alpha y_{t-1} + e_t, y_0$ a fixed constant, $e_t \sim \text{i.i.d.} (0, \sigma^2)$, and let $\hat{alpha}$ be the least-squares estimator of $\alpha$ based on a sample of size $(T + 1)$ sampled at frequency h. Consider also the continuous time Ornstein-Uhlenbeck process $dy_t = \theta y_t dt + \sigma dw_t$ where $w_t$ is a Wiener process and let $\hat{\theta}$ be the continuous time maximum likelihood (conditional upon $y_0$) estimator of $\theta$ based upon a single path of data of length $N$. We first show that the exact distribution of $N(\hat{\theta} - \theta)$ is the same as the asymptotic distribution of T(\hat{\alpha} - \alpha)$ as the sampling interval converges to zero. This asymptotic distribution permits explicit consideration of the effect of the initial condition $y_0$ upon the distribution of $\alpha$. We use this fact to provide an approximation to the finite sample distribution of $\hat{\alpha}$ got arbitrary fixed $y_0$. The moment-generating function of $N(\hat{\theta} - \theta)$ is derived and used to tabulate the distribution and probability density functions. We also consider the moment of $\theta$ and the power function of test statistics associated with it. In each case, the adequacy of the approximation to the finite sample distribution of $\hat{\alpha}$ is assessed for values of $\alpha$ in the vicinity of one. The approximations are, in general, found to be excellent.