Dimension reduction in time series

Abstract

We develop a new theory of dimension reduction in time series, which provides an initial phase when an adequate parsimoniously parameterized time series model is not yet available. In this thesis, we define a notion of Time Series Central Subspace and Time Series Central Mean Subspace, and estimate them using newly developed methods, when the lag of the series and minimum dimension are known. The estimators are shown to be strongly consistent. In addition, we also discuss estimation of the minimum dimension and the lag. The theory of dimension reduction in time series poses many challenges, but a variety of encouraging results presented through extensive simulations and real data analysis seem to suggest that our method has a great potential for providing a viable and meaningful alternative to traditional time series analysis. In fact, superior performance of our nonlinear or linear time series models for several real data sets serve as a testament that our methods are very useful in time series analysis. We believe that the ideas and methods presented here are of interest to time series analyst in fields such as Economics, Business, Climatology, among others. We hope that this work will stimulate a new way of analyzing time series data.