Open Access
February 2017 Identifying the number of factors from singular values of a large sample auto-covariance matrix
Zeng Li, Qinwen Wang, Jianfeng Yao
Ann. Statist. 45(1): 257-288 (February 2017). DOI: 10.1214/16-AOS1452


Identifying the number of factors in a high-dimensional factor model has attracted much attention in recent years and a general solution to the problem is still lacking. A promising ratio estimator based on singular values of lagged sample auto-covariance matrices has been recently proposed in the literature with a reasonably good performance under some specific assumption on the strength of the factors. Inspired by this ratio estimator and as a first main contribution, this paper proposes a complete theory of such sample singular values for both the factor part and the noise part under the large-dimensional scheme where the dimension and the sample size proportionally grow to infinity. In particular, we provide an exact description of the phase transition phenomenon that determines whether a factor is strong enough to be detected with the observed sample singular values. Based on these findings and as a second main contribution of the paper, we propose a new estimator of the number of factors which is strongly consistent for the detection of all significant factors (which are the only theoretically detectable ones). In particular, factors are assumed to have the minimum strength above the phase transition boundary which is of the order of a constant; they are thus not required to grow to infinity together with the dimension (as assumed in most of the existing papers on high-dimensional factor models). Empirical Monte-Carlo study as well as the analysis of stock returns data attest a very good performance of the proposed estimator. In all the tested cases, the new estimator largely outperforms the existing estimator using the same ratios of singular values.


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Zeng Li. Qinwen Wang. Jianfeng Yao. "Identifying the number of factors from singular values of a large sample auto-covariance matrix." Ann. Statist. 45 (1) 257 - 288, February 2017.


Received: 1 June 2015; Revised: 1 February 2016; Published: February 2017
First available in Project Euclid: 21 February 2017

zbMATH: 06710511
MathSciNet: MR3611492
Digital Object Identifier: 10.1214/16-AOS1452

Primary: 62H25 , 62M10
Secondary: 15B52

Keywords: High-dimensional factor model , high-dimensional time series , large sample auto-covariance matrices , number of factors , phase transition , random matrices , Spiked population model

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 1 • February 2017
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