On Same-Realization Prediction in the Multivariate Long Memory Process with the VAR Procedure

This paper proposes an easy-to-implement approach to forecasting the multivariate long memory process on same realization and further examines its usefulness on forecasting multivariate volatility series. This procedure bases on the extension of the analysis of Lewis and Reinsel (1985) to the multivariate fractionally integrated model, that is, the vector autoregressive (VAR (k)) model to approximate the multivariate long memory system. Under suitable assumptions on the long memory parameter d and lag length k, the consistency of the multivariate least squares (LS) coefficient estimator and that of the residual covariance matrix estimator Σ̂ _k are derived. In addition, the one-step ahead prediction error generated by the VAR(k)-approximation model is shown to converge in probability to its population counterpart, even though the exact orders of the multivariate long memory process are unknown and the long memory parameter d varies across each series of the multivariate long memory model. Moreover, insights from our theoretical analysis are confirmed by a set of Monte Carlo experiments, which are consistent with the findings of Lewis and Reinsel (1985) for the short memory process. An empirical application to the multivariate realized and option implied volatility series illustrates the usefulness of our forecasting procedure, when compared to the current volatility forecasting methods