Evaluating quantile-bounded and expectile-bounded interval forecasts

In many different contexts, decision making is improved by the availability of probabilistic predictions. The accuracy of probabilistic forecasting methods can be compared using scoring functions, and insight provided by calibration tests. These tests evaluate the consistency of predictions with the observations. Our main agenda in this paper is interval forecasts and their evaluation. Such forecasts are usually bounded by two quantile forecasts. However, a limitation of quantiles is that they convey no information regarding the size of potential exceedances. By contrast, the location of an expectile is dictated by the whole distribution. This prompts us to propose expectile-bounded intervals. We provide interpretation, a consistent scoring function and a calibration test. Before doing this, we reflect on the evaluation of forecasts of quantile bounded intervals and expectiles, and suggest extensions of previously proposed calibration tests in order to guard against strategic forecasting. We illustrate ideas using day-ahead electricity price forecasting