Evaluation and combination of conditional quantile forecasts

We propose an encompassing test for comparing conditional quantile forecasts in an out-of-sample framework. Our test provides a basis for forecast combination when encompassing is rejected. Its central features are (1) use of the "tick" loss function, (2) a conditional approach to out-of-sample evaluation, and (3) derivation in an environment with asymptotically nonvanishing estimation uncertainty. Our approach is valid under general conditions; the forecasts can be based on nested or nonnested models and can be obtained by general estimation procedures. We illustrate the test properties in a Monte Carlo experiment and apply it to evaluate and compare four popular value-at-risk models.