Prediction Inference for Time Series

In this paper we briefly review Bayesian and frequentist prediction inference for time series, and then advocate the use of guaranteed-content prediction intervals. These intervals are such that their content (or coverage) is guaranteed with a given high probability. They, thus, are more relevant for the observed time series at hand than classical prediction intervals, whose content is guaranteed merely on average over hypothetical repetitions of the prediction process. Guaranteed-content intervals should, therefore, conciliate Bayesians and frequentists when no prior belief on the parameterization is available. This type of prediction inference has, however, been ignored in the time series context because of a lack of results. This gap is filled by deriving asymptotic results for a general family of autoregressive models, thereby extending existing results in non-linear regression. The actual construction of guaranteed-content prediction intervals directly follows from this theory. Simulated and real data are used to illustrate the practical difference between classical and guaranteed-content prediction intervals for ARCH models.