# Econometrica

## Journal Of The Econometric Society

### An International Society for the Advancement of Economic Theory in its Relation to Statistics and Mathematics

Edited by: Guido W. Imbens • Print ISSN: 0012-9682 • Online ISSN: 1468-0262

Econometrica: Jan, 1991, Volume 59, Issue 1

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

Pierre Perron

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.