Predictor Selection for Positive Autoregressive Processes
Let observations <italic>y</italic><sub>1</sub>, …, <italic>y<sub>n</sub></italic> be generated from a first-order autoregressive (AR) model with positive errors. In both the stationary and unit root cases, we derive moment bounds and limiting distributions of an extreme value estimator, <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_836974_ilm0001.gif"/></inline-formula>, of the AR coefficient. These results enable us to provide asymptotic expressions for the mean squared error (MSE) of <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_836974_ilm0002.gif"/></inline-formula> and the mean squared prediction error (MSPE) of the corresponding predictor, <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_836974_ilm0003.gif"/></inline-formula>, of <italic>y</italic><sub><italic>n</italic> + 1</sub>. Based on these expressions, we compare the relative performance of <inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_836974_ilm0004.gif"/></inline-formula> (<inline-formula><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="uasa_a_836974_ilm0005.gif"/></inline-formula>) and the least-squares predictor (estimator) from the MSPE (MSE) point of view. Our comparison reveals that the better predictor (estimator) is determined not only by whether a unit root exists, but also by the behavior of the underlying error distribution near the origin, and hence is difficult to identify in practice. To circumvent this difficulty, we suggest choosing the predictor (estimator) with the smaller accumulated prediction error and show that the predictor (estimator) chosen in this way is asymptotically equivalent to the better one. Both real and simulated datasets are used to illustrate the proposed method. Supplementary materials for this article are available online.
