Limiting Distributions of Least Squares Estimates of Unstable Autoregressive Processes

Abstract

An autoregressive process $y_n = \beta_1y_{n-1} + \cdots + \beta_py_{n-p} + \varepsilon_n$ is said to be unstable if the characteristic polynomial $\phi(z) = 1 - \beta_1z - \cdots - \beta_pz^p$ has all roots on or outside the unit circle. The limiting distribution of the least squares estimate of $(\beta_1, \cdots, \beta_p)$ is derived and characterized as a functional of stochastic integrals under a $2 + \delta$ moment assumption on $\varepsilon_n$. Up to the present, distributional results were available only with substantial restrictions on the possible roots which did not suggest the form of the distribution for the general case. To establish the limiting distribution, a result concerning the weak convergence of a sequence of random variables to a stochastic integral, which is of independent interest, is also developed.