Factor modeling for high-dimensional time series: Inference for the number of factors

TL;DR

This paper proposes a dimension-reduction based method for estimating the number of factors in high-dimensional stationary time series, demonstrating its effectiveness and the 'blessing of dimensionality' through theoretical analysis and empirical tests.

Contribution

It introduces a new eigenanalysis-based estimator for the number of factors that remains effective even when both sample size and dimension grow large, and explores its properties under different asymptotic regimes.

Findings

Estimator performs well with large dimensions and sample sizes.

The method benefits from the 'blessing of dimensionality' in estimation accuracy.

Numerical results confirm the theoretical advantages and robustness of the approach.

Abstract

This paper deals with the factor modeling for high-dimensional time series based on a dimension-reduction viewpoint. Under stationary settings, the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix, and is therefore applicable when the dimension of time series is on the order of a few thousands. Asymptotic properties of the proposed method are investigated under two settings: (i) the sample size goes to infinity while the dimension of time series is fixed; and (ii) both the sample size and the dimension of time series go to infinity together. In particular, our estimators for zero-eigenvalues enjoy faster convergence (or slower divergence) rates, hence making the estimation for the number of factors easier. In particular, when the sample size and the dimension of time series go to infinity together, the estimators for the eigenvalues are no longer consistent. However, our estimator for the number of the factors, which is based on the ratios of the estimated eigenvalues, still works fine. Furthermore, this estimation shows the so-called "blessing of dimensionality" property in the sense that the performance of the estimation may improve when the dimension of time series increases. A two-step procedure is investigated when the factors are of different degrees of strength. Numerical illustration with both simulated and real data is also reported.