Estimating autoRegressive processes with positive white noise

Autoregressive processes with positive white noise have become increasingly popular, particularly in financial econometrics (e.g. volatility processes), and are used not only for stationary dynamics but also in the presence of unit roots or explosive behaviour. We show that, for a broad class of distributions of the positive error term, the maximum likelihood estimator of the autoregressive parameters does not depend on the specific distribution assumed. In the first-order case, we analyse a simplified estimator that has been shown to be strongly consistent under general conditions. This strong consistency result remains valid for any non-negative value of the autoregressive parameter. We also study an analogous estimator for second-order autoregressive processes. Furthermore, in the non-stationary case, specifically with unit root or explosive dynamics, we introduce an even simpler consistent estimator based on the most recent observations. A key result of our Monte Carlo simulations is that, when the autoregressive parameter is greater than or equal to one, this simplified estimator performs almost as well as the more efficient one, even in small samples. Interestingly, the two estimators nearly coincide when the sequence of growth rates is decreasing, which is often approximately the case. For researchers concerned about the lack of smoothing in the simplified estimator, geometric averaging provides alternative estimators that remain consistent and computationally convenient.