Real-Time Multivariate Density Forecast Evaluation and Calibration: Monitorlng the Risk of High-Frequency Returns on Foreign Exchange

Abstract

We provide a framework for evaluating and improving multivariate density forecasts.Among other things, the multivariate framework lets us evaluate the adequacy of density forecastsinvolving cross-variable interactions, such as time-varying conditional correlations. We alsoprovide conditions under which a technique of density forecast "calibration" can be used toimprove deficient density forecasts. Finally, motivated by recent advances in financial riskmanagement, we provide a detailed application to multivariate high-frequency exchange ratedensity forecasts.