Fx. Diebold et al., Multivariate density forecast evaluation and calibration in financial riskmanagement: High-frequency returns on foreign exchange, REV ECON ST, 81(4), 1999, pp. 661-673
We provide a framework for evaluating and improving multivariate density fo
recasts. Among other things, the multivariate framework lets us evaluate th
e 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 multivar
iate high-frequency exchange rate density forecasts.