Estimation of latent factors for high-dimensional time series

Abstract

This paper deals with the dimension reduction of high-dimensional time series based on common factors. In particular we allow the dimension of time series p to be as large as, or even larger than, the sample size n. The estimation of the factor loading matrix and the factor process itself is carried out via an eigenanalysis of a p £ p non-negative de¯nite matrix. We show that when all the factors are strong in the sense that the norm of each column in the factor loading matrix is of the order p1=2, the estimator of the factor loading matrix is weakly consistent in L2-norm with the convergence rate independent of p. This result exhibits clearly that the `curse' is canceled out by the `blessing' of dimensionality. We also establish the asymptotic properties of the estimation when factors are not strong. The proposed method together with their asymptotic properties are further illustrated in a simulation study. An application to an implied volatility data set, together with a trading strategy derived from the ¯tted factor model, is also reported.