Evaluating, Comparing and Combining Density Forecasts Using the KLIC with an Application to the Bank of England and NIESR ‘Fan’ Charts of Inflation*

TLDR

The Kullback–Leibler information criterion (KLIC) is attractive because it is equivalent to the Berkowitz likelihood ratio test and parallels methods used for comparing and combining point forecasts. This paper proposes and analyzes KLIC as a unified statistical tool for evaluating, comparing, and combining density forecasts. The authors develop KLIC, demonstrate its equivalence to the Berkowitz test, and apply it to two UK inflation fan charts from the Bank of England and NIESR. Monte Carlo experiments reveal properties of combined density forecasts, and the application to the Bank of England and NIESR fan charts illustrates KLIC’s practical utility.

Abstract

Abstract This paper proposes and analyses the Kullback–Leibler information criterion (KLIC) as a unified statistical tool to evaluate, compare and combine density forecasts. Use of the KLIC is particularly attractive, as well as operationally convenient, given its equivalence with the widely used Berkowitz likelihood ratio test for the evaluation of individual density forecasts that exploits the probability integral transforms. Parallels with the comparison and combination of point forecasts are made. This and related Monte Carlo experiments help draw out properties of combined density forecasts. We illustrate the uses of the KLIC in an application to two widely used published density forecasts for UK inflation, namely the Bank of England and NIESR ‘fan’ charts.