Properties of estimators of the parameters of autoregressive time series

Pantula, Sastry
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Assuming that the errors of an autoregressive process form a sequence of martingale differences, the limiting distribution of the least squares estimator is derived. The limiting distribution of the least squares estimator is normal if the roots of the characteristic equation are less than unity in absolute value. It is shown that the limiting distribution for the unit root case obtained under the assumption of independent errors holds for martingale errors;For samples of the size encountered in practice, the least squares estimators are biased. Using large sample theory, approximations to the bias in the least squares estimators of the parameters of a stationary autoregressive process due to estimation of the mean are derived. The bias expression is used to develop modifications of the least squares estimator. The modification is extended to include the case when exactly one of the roots of the characteristic equation is equal to one;Estimation of the parameters of an autoregressive process with a mean that is a function of time is considered. Approximate expressions for the bias of the least squares estimators that is due to estimating the mean function are derived. For the special case when the mean function is a polynomial in time, a reparametrization that isolates the bias is proposed. Using the approximate expressions, a method of modifying the least squares estimators is proposed. Methods are suggested for the seasonal autoregressive processes;Two Monte Carlo studies examining the small sample properties of various estimators of the parameters of second-order autoregressive processes are considered. A second-order autoregressive process with constant mean, and a second-order autoregressive process with mean function linear in time are considered. Generally speaking the modified estimators performed better than the least squares estimator.