Exact Distribution of the Least Squares Estimator in a First- Order Autoregressive Model

This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. Only rarely the accuracy improves by including further term beyond the first term of this expansion in the approximation. As a matter of fact, often the accuracy of such an approximation with additional term(s) deteriorates. An application of the finding is illustrated with examples.