A CONTINUOUS-TIME APPROXIMATION TO THE UNSTABLE 1ST-ORDER AUTOREGRESSIVE PROCESS - THE CASE WITHOUT AN INTERCEPT

Consider the first-order autoregressive process y(t) = alpha-y(t-1) + e(t), y0 a fixed constant, e(t) approximately i.i.d. (0, sigma-2), and let alpha-triple-overdot 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 theta-triple-overdot be the continuous time maximum likelihood (conditional upon y0) estimator of theta based upon a single path of data of length N. We first show that the exact distribution of N(theta-triple-overdot - theta) is the same as the asymptotic distribution of T(alpha-triple-overdot - alpha) as the sampling interval converges to zero. This asymptotic distribution permits explicit consideration of the effect of the initial condition y0 upon the distribution of alpha-triple-overdot. We use this fact to provide an approximation to the finite sample distribution of alpha-triple-overdot for arbitrary fixed y0. The moment-generating function of N(theta-triple-overdot - theta) is derived and used to tabulate the distribution and probability density functions. We also consider the moment of theta-triple-overdot and the power function of test statistics associated with it. In each case, the adequacy of the approximation to the finite sample distribution of alpha-triple-overdot is assessed for values of alpha in the vicinity of one. The approximations are, in general, found to be excellent.