Forecast Evaluations for Multiple Time Series: A Generalized Theil Decomposition

The mean square error (MSE) compares point forecasts or a location parameter of the forecasting distribution with actual observations by the quadratic loss criterion. This paper shows how the Theil decomposition of the MSE error into a bias, variance and noise component which was proposed for univariate time series can be used to evaluate and compare multiple time series forecasts. Thus, for multivariate time series the ordinary and the alternative Theil decomposition is applied to decompose the MSE matrix. As an alternative we propose the average predictive ordinate criterion (APOC) which evaluates the ordinates of the predictive distribution for comparing forecasts of volatile time series. The multivariate Theil decomposition for the MSE and APOC criterion is used to compare and evaluate 3-dimensional VAR-GARCH-M time series forecasts for stock indices and exchange rates.