Title:

Factor modeling for high dimensional time series

Chapter 1: Identifying the finite dimensionality of curve time series The curve time series framework provides a convenient vehicle to model some types of nonstationary time series in a stationary framework. We propose a new method to identify the finite dimensionality of curve time series based on the autocorrelation between different curves. Based upon the duality relation between row and column subspaces of a data matrix, we show that the practical implementation of our methodology reduces to the eigenanalysis of a real matrix. Furthermore, the determination of the dimensionality is equivalent to indentifying the number of nonzero eigenvalues of this same matrix. For this purpose we propose a simple bootstrap test. Asymptotic properties of our methodology are investigated. The proposed methodology is illustrated with some simulation studies as well as an application to IBM intraday return densities. Chapter 2: Methodology and convergence rates for factor modeling of multiple time series An important task in modeling multiple time series is to obtain some form of dimension reduction. We tackle this problem using a factor model where the estimation of the factor loading space is constructed via eigenanalysis of a matrix which is a simple function of the sample autocovariance matrices. The number of factors is then equal to the number of "nonzero" eigenvalues of this matrix. We use the term "nonzero" loosely because in practice it is unlikely that there will be any eigenvalues which are exactly zero. However, our theoretical results suggest that the sample eigenvalues whose population counterparts are zero are "superconsistent" (i.e. they converge to zero at a n rate) whereas the sample eigenvalues whose population counterparts are nonzero converge at an ordinary parametric rate of rootn. Here n denotes the sample size. This striking result is supported by simulation evidence and consequences for inference are discussed. In addition, we study the properties of the factor loading space under very general conditions (including possible nonstationarity) and a simple white noise test for empirically determining the number of nonzero eigenvalues is proposed and theoretically justified. We also provide an example of a heuristic threshold based estimator for the number of factors and prove that it yields a consistent estimator provided that the threshold is chosen to be of an appropriate order. Finally we conclude with an analysis of some implied volatility datasets.
