Multivariate density forecast evaluation and calibration in financial risk management: High-frequency returns on foreign exchange

We provide a framework for evaluating and improving multivariate density forecasts. Among other things, the multivariate framework lets us evaluate the adequacy of density forecasts involving cross-variable interactions, such as time-varying conditional correlations. We also provide conditions under which a technique of density forecast "calibration" can be used to improve deficient density forecasts, and we show how the calibration method can be used to generate good density forecasts from econometric models, even when the conditional density is unknown. Finally, motivated by recent advances in financial risk management, we provide a detailed application to multivariate high-frequency exchange rate density forecasts.