Factor Modeling for High-Dimensional Interval-Valued Data

Yan Guo; Guchu Zou; Jianhong Wu

doi:10.1515/snde-2024-0019

Article

Factor Modeling for High-Dimensional Interval-Valued Data

Yan Guo , Guchu Zou and Jianhong Wu

Published/Copyright: February 4, 2025

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From the journal Studies in Nonlinear Dynamics & Econometrics

Abstract

The paper considers an approximate factor model for interval-valued panel data with both large numbers of cross-section units and time series observations. A ratio-type estimator is proposed for the number of interval-valued factors in the approximate factor model. A variant of the estimator is also suggested, which is robust to the case with dominant factors. Under certain conditions, the estimators can be proved to be consistent. Moreover, the estimators of interval-valued factors and the pooled loadings can be obtained by the principal component analysis method for point-valued data. Monte Carlo simulation studies show that the proposed estimators have the desired finite sample properties.

Keywords: approximate factor models; factor analysis; interval-valued data; principal components; the number of factors

JEL Classification: C01; C13

Corresponding author: Jianhong Wu, College of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China; and Lab for Educational Big Data and Policymaking, Ministry of Education, Shanghai 200234, China, E-mail: wujianhong@shnu.edu.cn

Yan Guo and Guchu Zou contributed equally to this work.

Acknowledgments

We are deeply grateful to Professor Jeremy Piger and two anonymous referees for valuable comments that led to substantial improvement of this paper.

Research funding: This research is supported in part by the National Nature Science Foundation of China (Grant No. 72173086).

Appendix: Technical Details

Proof of Theorem 1.

Under Assumptions 1–3, it follows from Lemma A.11 of Ahn and Horenstein (2013) that, for any k ≤ r, μ ̃ N T , k L = O p ( 1 ) and μ ̃ N T , k R = O p ( 1 ) . Thus, we have ω μ ̃ N T , k L = O p ( 1 ) and ( 1 − ω ) μ ̃ N T , k R = O p ( 1 ) , ω ∈ [ 0,1 ] , k = 1,2 , … , r . Accordingly, for any weight 0 ≤ ω ≤ 1,

ω μ ̃ N T , k L + ( 1 − ω ) μ ̃ N T , k R ω μ ̃ N T , k + 1 L + ( 1 − ω ) μ ̃ N T , k + 1 R = O p ( 1 ) , k = 1,2 , … , r − 1 .

Also, it follows from Lemma A.9 of Ahn and Horenstein (2013) that, for any r + 1 ≤ k ≤ [d^cm] − r, μ ̃ N T , k L = O p ( 1 m ) and μ ̃ N T , k R = O p ( 1 m ) . Then, we have ω μ ̃ N T , k L = O p ( 1 m ) and ( 1 − ω ) μ ̃ N T , k R = O p ( 1 m ) , for any r + 1 ≤ k ≤ [d^cm] − r and ω ∈ [0, 1]. Accordingly,

ω μ ̃ N T , k L + ( 1 − ω ) μ ̃ N T , k R ω μ ̃ N T , k + 1 L + ( 1 − ω ) μ ̃ N T , k + 1 R = O p ( 1 ) , k = r + 1 , r + 2 , … , r max .

However, when k = r,

ω μ ̃ N T , r L + ( 1 − ω ) μ ̃ N T , r R ω μ ̃ N T , r + 1 L + ( 1 − ω ) μ ̃ N T , r + 1 R = O p ( m ) → ∞ .

It then follows that

lim m → ∞ Pr ( r ̂ ω = r ) = 1 .

Proof of Theorem 2.

Let μ ̃ N T , k * be ω μ ̃ N T , k L + ( 1 − ω ) μ ̃ N T , k R . It follows from the proof of Theorem 1 that, under Assumptions 1–2, μ ̃ N T , k * = O p ( 1 ) , k ≤ r and μ ̃ N T , k * = O p ( 1 m ) , k ≥ r + 1 . Therefore, for k ≤ r,

2 Φ μ ̃ N T , k * − 1 = ∫ 0 μ ̃ N T , k * 2 2 π e − t 2 2 d t ≥ 2 2 π e − μ ̃ N T , k * 2 2 μ ̃ N T , k * = O p ( 1 ) .

For r + 1 ≤ k ≤ [d^cm] − r, we have 2 Φ μ ̃ N T , k * − 1 = O p ( 1 m ) , because

2 Φ μ ̃ N T , k * − 1 = ∫ 0 μ ̃ N T , k * 2 2 π e − t 2 2 d t ≥ 2 2 π e − μ ̃ N T , k * 2 2 μ ̃ N T , k * = O p 1 m , 2 Φ μ ̃ N T , k * − 1 = ∫ 0 μ ̃ N T , k * 2 2 π e − t 2 2 d t ≤ 2 2 π μ ̃ N T , k * = O p 1 m .

Thus, we have

2 Φ μ ̃ N T , k * − 1 2 Φ μ ̃ N T , k + 1 * − 1 = O p ( 1 ) , k = 1 , … , r − 1 , r + 1 , … , r max .

However, when k = r,

2 Φ μ ̃ N T , r * − 1 2 Φ μ ̃ N T , r + 1 * − 1 = O p ( m ) → ∞ .

It then follows that

lim m → ∞ Pr ( r ̃ ω = r ) = 1 .

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2024-0019).

Received: 2024-03-13

Accepted: 2024-12-19

Published Online: 2025-02-04

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Supplementary Material Details

https://doi.org/10.1515/snde-2024-0019

Keywords for this article

approximate factor models; factor analysis; interval-valued data; principal components; the number of factors