The Adjustment of Prediction Intervals to Account for Errors in Parameter Estimation

Standard approximate 1-a prediction intervals (PIs) need to be adjusted to take account of the error in estimating the parameters. This adjustment may be aimed at setting the (unconditional) probability that the PI includes the value being predicted equal to 1-a. Alternatively, this adjustment may be aimed at setting the probability that the PI includes the value being predicted equal to 1-a, conditional on an appropriate statistic T. For an autoregressive process of order p, it has been suggested that T consist of the last p observations. We provide a new criterion by which both forms of adjustment can be compared on an equal footing. This new criterion of performance is the closeness of the coverage probability, conditional on all of the data, of the adjusted PI and 1-a. In this paper, we measure this closeness by the mean square of the difference between this conditional coverage probability and 1-a. We illustrate the application of this new criterion to a Gaussian zero-mean autoregressive process of order 1-a and one-step-ahead prediction. For this example, this comparison shows that the adjustment which is aimed at setting the coverage probability equal to 1-a conditional on the last observation is the better of the two adjustments