Bayesian ex Post Evaluation of Recursive Multi-Step-Ahead Density Prediction

Abstract

This research is focused on a formal Bayesian method of recursive multi-step-ahead density prediction and its ex post evaluation. Our approach remains within the framework of the standard (classical or orthodox) Bayesian paradigm based on the Bayes factor and on the use of the likelihood-based update. We propose a new decomposition of the predictive Bayes factor into the product of partial Bayes factors, for both a finite number of consecutive k-step-ahead forecasts (where $k>1$) and the recursive updates of the posterior odds ratios based on updated data sets. The first factor in the decomposition is related to the relative k-step-ahead forecasting ability of models, while the second one measures the updating effect.

To illustrate the usefulness of the proposed measures, we apply the new decomposed predictive Bayes factors to compare the forecasting ability of models when the true data generating process (DGP) is known, using simulated data sets. Taking into account the effect of updating, the posterior odds ratios leads to the conclusion that the best model coincides with the true DGP. However, the highest k-step-ahead forecasting ability (considered alone) can be achieved by some other, less adequate models. Next, we investigate the predictive ability of different Vector Error Correction (VEC) models with conditional heteroscedasticity, combining three macroeconomic variables: unemployment, inflation and interest rates, separately for the US and Polish economies. The results show that the inference about the models’ predictive performance depends on the forecast horizon as well as on taking into account the updating effect.