## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 1 (2017), 257-288.

### Identifying the number of factors from singular values of a large sample auto-covariance matrix

Zeng Li, Qinwen Wang, and Jianfeng Yao

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 1 (2017), 257-288.

**Dates**

Received: June 2015

Revised: February 2016

First available in Project Euclid: 21 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1487667623

**Digital Object Identifier**

doi:10.1214/16-AOS1452

**Mathematical Reviews number (MathSciNet)**

MR3611492

**Zentralblatt MATH identifier**

06710511

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62H25: Factor analysis and principal components; correspondence analysis

Secondary: 15B52: Random matrices

**Keywords**

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

#### Citation

Li, Zeng; Wang, Qinwen; Yao, Jianfeng. Identifying the number of factors from singular values of a large sample auto-covariance matrix. Ann. Statist. 45 (2017), no. 1, 257--288. doi:10.1214/16-AOS1452. https://projecteuclid.org/euclid.aos/1487667623

#### Supplemental materials

- Supplement to “Identifying the number of factors from singular values of a large sample auto-covariance matrix”. A supplementary file [Li, Wang and Yao (2016)] collects several technical proofs used in the paper.Digital Object Identifier: doi:10.1214/16-AOS1452SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.