On determination of the number of factors in an approximate factor model

Jinshan Liu; Jiazhu Pan; Qiang Xia; Li Xiao

doi:10.1515/snde-2020-0055

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On determination of the number of factors in an approximate factor model

Jinshan Liu , Jiazhu Pan , Qiang Xia and Li Xiao

Published/Copyright: October 3, 2022

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 27 Issue 3

Abstract

This paper proposes a ridge-type method for determining the number of factors in an approximate factor model. The new estimator of factor number is obtained by maximizing both the ratio of two adjacent eigenvalues and the cumulative contribution rate of the factors which represents the explanatory power of the common factors for response variables. Our estimator is proved to be as asymptotically consistent as those in (Ahn, S., and A. Horenstein. 2013. “Eigenvalue Ratio Test for the Number of Factors.” Econometrica 81: 1203–27). But Monte Carlo simulation experiments show our method has better correct selection rates in finite sample cases. A real data example is given for illustration.

Keywords: approximate factor model; cumulative contribution rate; eigenvalue ratio; number of factors; ridge-type method

JEL Classification: C1; C3

Corresponding author: Qiang Xia, College of Mathematics and Informatics, South China Agricultural University, Guangzhou 510642, China, E-mail: xiaqiang@scau.edu.cn

Funding source: the National Natural Science Foundation of China

Award Identifier / Grant number: 12171161, 91746102

Funding source: Natural Science Foundation of Guangdong Province

Award Identifier / Grant number: 2022A1515011754

Funding source: East China Normal University

Award Identifier / Grant number: Unassigned

Funding source: Ministry of Education in China Project of Humanities and Social Sciences

Award Identifier / Grant number: 17YJA910002

Acknowledgments

We thank the Editor and the Referee(s) for their insightful comments and suggestions that help us significantly to improve our manuscript.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This research was supported by the National Natural Science Foundation of China (No. 12171161, 91746102), the Natural Science Foundation of Guangdong Province of China (No. 2022A1515011754), and Ministry of Education in China Project of Humanities and Social Sciences (No. 17YJA910002), the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science (East China Normal University), Ministry of Education.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

The proofs of our theorems will need the following lemmas.

Lemma 1

Under Assumptions A–D, for j = 1, …, r,

μ ̃ N T , j = μ N T , j + O p ( N − 1 / 2 ) + O p ( m − 1 ) .

Lemma 2

Under Assumptions A, C and D, for j = 1, …, [d^cm] − 2r, where d^c ∈ (0, 1], we have that

c ̲ + o p ( 1 ) ≤ m μ ̃ N T , r + j ≤ c ̄ + o p ( 1 ) ,

where c ̲ = c 2 2 y * * ( 1 − b y * ) 2 , y * * = lim m → ∞ ( N / M ) and c ̄ = c 1 2 ( 1 + y ) 2 .

Lemma 3

Under Assumptions A–D, V(r + 1) ≍ O_p(1).

For the proofs of Lemma 1 to Lemma 3, see Lemma A.11, Lemma A.9 and Lemma A.12 in Ahn and Horenstein (2013), respectively.

Proof of Theorem 1

It follows from Lemma A.11 and Lemma A.9 of Ahn and Horenstein (2013) that μ ̃ N T , j = μ N T , j + O p ( N − 1 / 2 ) + O p ( m − 1 ) ≍ O p ( 1 ) , for j = 1, …, r and μ ̃ N T , j ≍ O p ( 1 / m ) , for j = r + 1, …, [d^cm] − r. By Lemma A.12 of Ahn and Horenstein (2013), V(r + 1) ≍ O_p(1). Therefore, for j = 1, …, r − 1,

μ ̃ N T , j μ ̃ N T , j + 1 1 v + V ( j ) ≤ μ ̃ N T , j μ ̃ N T , j + 1 1 v + V ( r − 1 ) = μ ̃ N T , j μ ̃ N T , j + 1 1 v + μ ̃ N T , r + μ ̃ N T , r + 1 + V ( r + 1 ) = O P ( 1 ) O P ( 1 ) = O P ( 1 ) .

For j = r,

μ ̃ N T , r μ ̃ N T , r + 1 1 v + V ( r ) = μ ̃ N T , r μ ̃ N T , r + 1 1 v + μ ̃ N T , r + 1 + V ( r + 1 ) ≍ O P ( m ) O P ( 1 ) ≍ O P ( m ) .

For j = r + 1, …, k_max,

μ ̃ N T , j μ ̃ N T , j + 1 1 v + V ( j ) < μ ̃ N T , j μ ̃ N T , j + 1 1 v = O P ( 1 ) .

Thus, the consistency of the estimator r ̂ EC defined by (2.7) is verified.

Proof of Theorem 2

For j = 1, …, r − 1,

μ ̃ N T , j μ ̃ N T , j + 1 V ( j ) V ( j − 1 ) ≤ μ ̃ N T , j μ ̃ N T , j + 1 = O P ( 1 ) .

For j = r,

μ ̃ N T , r μ ̃ N T , r + 1 V ( r ) V ( r − 1 ) = μ ̃ N T , r μ ̃ N T , r + 1 V ( r ) μ ̃ N T , r + V ( r ) ≍ O P ( m ) O P ( 1 ) ≍ O P ( m ) .

For j = r + 1, …, k_max, k_max ≤ [d^cm] − r,

μ ̃ N T , j μ ̃ N T , j + 1 V ( j ) V ( j − 1 ) ≤ μ ̃ N T , j μ ̃ N T , j + 1 = O P ( 1 ) .

Therefore, the consistency of r ̂ CR defined by (2.8) is proved.

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Received: 2020-05-12

Revised: 2022-05-31

Accepted: 2022-06-03

Published Online: 2022-10-03

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Articles in the same Issue

https://doi.org/10.1515/snde-2020-0055

Keywords for this article

approximate factor model; cumulative contribution rate; eigenvalue ratio; number of factors; ridge-type method