Forecast evaluation tests and negative long-run variance estimates in small samples

This paper shows that the long-run variance can frequently be negative when computing standard Diebold-Mariano-type tests for equal forecast accuracy and forecast encompassing if one is dealing with multi-step-ahead predictions in small, but empirically relevant, sample sizes. We therefore consider a number of alternative approaches for dealing with this problem, including direct inference in the problem cases and the use of long-run variance estimators that guarantee positivity. The finite sample size and power of the different approaches are evaluated using extensive Monte Carlo simulation exercises. Overall, for multi-step-ahead forecasts, we find that the test recently proposed by Coroneo and Iacone (2016), which is based on a weighted periodogram long-run variance estimator, offers the best finite sample size and power performance.