Model-Free Prediction of Multivariate Time Series

This paper extends a model-free prediction framework from univariate to multivariate time series. We show that, under a mild uniformly bounded first-moment condition, a multivariate time series admits a VARMA-type representation and an associated infinite-past linear solution. These results are obtained without assuming that the observed data are generated by a parametric VAR, VARMA, or other specifically postulated stochastic model. Motivated by this representation, we develop a practical forecasting method based on truncating the infinite-past solution and estimating the resulting coefficient matrices by multivariate least squares. The proposed procedure is simple to implement and is applicable to a broad class of stationary or nonstationary, linear or nonlinear multivariate time series when the required moment, identification, and empirical design conditions are plausible. We also describe recursive multi-step forecasting and give multivariate analogues of the main theoretical accuracy statements. The representation results require only a uniformly bounded first moment, whereas the forecast-accuracy results are stated under an explicit selected-solution identification assumption and additional empirical design conditions needed for multivariate least-squares estimation. Monte Carlo results, a real-data application, and additional sensitivity checks show that, in the lag-based implementation studied here, the method is empirically comparable to standard linear and regularized forecasting approaches, while not uniformly dominating them, and retains the conceptual advantage of a model-free formulation.