Estimation for Latent Factor Models for High-Dimensional Time Series

TL;DR

This paper introduces a method for estimating latent factor models in high-dimensional time series, allowing for cases where the number of variables exceeds the sample size, and demonstrates the estimator's consistency and asymptotic properties.

Contribution

It develops eigenanalysis-based estimators for factor loadings in high-dimensional time series, including cases with weak factors, and establishes their asymptotic properties.

Findings

Estimator is weakly consistent when all factors are strong.

Convergence rates are independent of the dimension p.

A two-step procedure is effective for weak factors.

Abstract

This paper deals with the dimension reduction for 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 for the factor loading matrix and the factor process itself is carried out via an eigenanalysis for a $p\times p$ non-negative definite 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 $p^{1/2}$, the estimator for the factor loading matrix, as well as the resulting estimator for the precision matrix of the original $p$-variant time series, are weakly consistent in $L_2$-norm with the convergence rates independent of $p$. This result exhibits clearly that the `curse' is canceled out by the `blessings' in dimensionality. We also establish the asymptotic properties of the estimation when not all factors are strong. For the latter case, a two-step estimation procedure is preferred accordingly to the asymptotic theory. The proposed methods together with their asymptotic properties are further illustrated in a simulation study. An application to a real data set is also reported.