Forecasting Multiple Time Series With One-Sided Dynamic Principal Components

Abstract

We define one-sided dynamic principal components (ODPC) for time series as linear combinations of the present and past values of the series that minimize the reconstruction mean squared error. Usually dynamic principal components have been defined as functions of past and future values of the series and therefore they are not appropriate for forecasting purposes. On the contrary, it is shown that the ODPC introduced in this article can be successfully used for forecasting high-dimensional multiple time series. An alternating least-squares algorithm to compute the proposed ODPC is presented. We prove that for stationary and ergodic time series the estimated values converge to their population analogs. We also prove that asymptotically, when both the number of series and the sample size go to infinity, if the data follow a dynamic factor model, the reconstruction obtained with ODPC converges in mean square to the common part of the factor model. The results of a simulation study show that the forecasts obtained with ODPC compare favorably with those obtained using other forecasting methods based on dynamic factor models. Supplementary materials for this article are available online.

Keywords:

• Dimensionality reduction
• Dynamic factor models
• High-dimensional time series

Supplementary Material

The supplemental material available online contains the proofs of all the theoretical results stated in the article.

Acknowledgments

The authors would like to thank Stefano Soccorsi for help with the programs to run the FHLZ procedure. They also acknowledge the help provided by five anonymous referees whose useful comments and suggestions led to important improvements in the paper in both presentation and content.

Additional information

Funding

D.P. has been supported by Grant ECO2015-66593-P of MINECO/FEDER/UE. E.S. was partially funded by a CONICET Ph.D fellowship and by grant PIP 112-201101-00339 from CONICET. Part of this work was conducted while E.S. was a Postdoctoral Research Fellow at the Department of Statistics, and at the Department of Computer Science at the University of British Columbia.