Long-Run Forecasting in Multicointegrated Systems

In this paper long-run forecasting of multicointegrating variables is investigated. Multicointegration typically occurs in dynamic systems involving both stock and flow variables whereby a common feature in the form of shared stochastic trends is present across different levels of multiple time series. Hence, the effect of imposing this ”common feature” restriction on out-of-sample evaluation and forecasting accuracy of such variables is of interest. In particular, we compare the long-run forecasting performance of the multicointegrated variables between a model that correctly imposes the ”common feature” restrictions and a (univariate) model that omits these multicointegrating restrictions completely. We employ different loss functions based on a range of mean square forecast error criteria, and the results indicate that different loss functions result in different ranking of models with respect to their infinite horizon forecasting performance. We consider loss functions using a standard trace mean square forecast error criterion (penalizing the forecast errors of flow variables only), and a loss function evaluating forecast errors of changes in both stock and flow variables. The latter loss function is based on the triangular representation of cointegrated systems and was initially suggested by Christoffersen and Diebold (1998). It penalizes deviations from long-run relations among the flow variables through cointegrating restrictions. We present a new loss function which further penalizes deviations in the long run relationship between the levels of stock and flow variables. It is derived from the triangular representation of multicointegrated systems. Using this criterion, system forecasts from a model incorporating multicointegration restrictions dominate forecasts from univariate models. The paper highlights the importance of carefully selecting loss functions in forecast evaluation of models involving stock and flow variables.