Efficient estimation of forecast uncertainty based on recent forecast errors

Multi-step-ahead forecasts of forecast uncertainty in practice are often based on the horizon-specific sample means of recent squared forecast errors, where the number of available past forecast errors decreases one-to-one with the forecast horizon. In this paper, the efficiency gains from the joint estimation of forecast uncertainty for all horizons in such samples are investigated. Considering optimal forecasts, the efficiency gains can be substantial if the sample is not too large. If forecast uncertainty is estimated by seemingly unrelated regressions, the covariance matrix of the squared forecast errors does not have to be estimated, but simply needs to have a certain structure. In Monte Carlo studies it is found that seemingly unrelated regressions mostly yield estimates which are more efficient than the sample means even if the forecasts are not optimal. Seemingly unrelated regressions are used to address questions concerning the inflation forecast uncertainty of the Bank of England.