## The Annals of Statistics

- Ann. Statist.
- Volume 40, Number 2 (2012), 694-726.

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

#### 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.

#### Article information

**Source**

Ann. Statist. Volume 40, Number 2 (2012), 694-726.

**Dates**

First available in Project Euclid: 17 May 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1337268209

**Digital Object Identifier**

doi:10.1214/12-AOS970

**Mathematical Reviews number (MathSciNet)**

MR2933663

**Zentralblatt MATH identifier**

1273.62214

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Secondary: 60G99: None of the above, but in this section

**Keywords**

Autocovariance matrices blessing of dimensionality eigenanalysis fast convergence rates multivariate time series ratio-based estimator strength of factors white noise

#### Citation

Lam, Clifford; Yao, Qiwei. Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist. 40 (2012), no. 2, 694--726. doi:10.1214/12-AOS970. https://projecteuclid.org/euclid.aos/1337268209

#### Supplemental materials

- Supplementary material: Detailed proofs of Theorems 3 and 4. The document contains detailed proofs of Theorem 3 and 4 in the paper.Digital Object Identifier: doi:10.1214/12-AOS970SUPP