Abstract This article reviews recently proposed likelihood ratio tests of goodness-of-fit and independence of interval forecasts. It recasts them in the framework of Pearson chi-squared statistics, and considers their extension to density forecasts. The use of the familiar framework of contingency tables increases the accessibility of these methods to users, and allows the incorporation of two recent developments, namely a more informative decomposition of the chi-squared goodness-of-fit statistic and the calculation of exact small-sample distributions. The tests are applied to two series of density forecasts of inflation, namely the US Survey of Professional Forecasters and the Bank of England fan charts. This first evaluation of the fan chart forecasts finds that, whereas the current-quarter forecasts are well-calibrated, this is less true of the one-year-ahead forecasts. The fan charts fan out too quickly and the excessive concern with the upside risks was not justified over the period considered.