Improved Frequentist Prediction Intervals for Autoregressive Models by Simulation

Abstract

It is well known that the so called plug-in prediction intervals for autoregressive processes, with Gaussian disturbances, are too narrow, i.e. the coverage probabilities fall below the nominal ones. However, simulation experiments show that the formulas borrowed from the ordinary linear regression theory yield one-step prediction intervals, which have coverage probabilities very close to what is claimed. From a Bayesian point of view the resulting intervals are posterior predictive intervals when uniform priors are assumed for both autoregressive coefficients and logarithm of the disturbance variance. This finding opens the path how to treat multi-step prediction intervals which are obtained easily by simulation either directly from the posterior distribution or using importance sampling. A notable improvement is gained in frequentist coverage probabilities. An application of the method to forecasting the annual gross domestic product growth in the United Kingdom and Spain is given for the period 2002–2011 using the estimation period 1962–2001.