Factor and Idiosyncratic Empirical Processes

ABSTRACT

The distributions of the common and idiosyncratic components for an individual variable are important in forecasting and applications. However, they are not identified with low-dimensional observations. Using the recently developed theory for large dimensional approximate factor model for large panel data, the common and idiosyncratic components can be estimated consistently. Based on the estimated common and idiosyncratic components, we construct the empirical processes for estimation of the distribution functions of the common and idiosyncratic components. We prove that the two empirical processes are oracle efficient when T = o(p) where p and T are the dimension and sample size, respectively. This demonstrates that the factor and idiosyncratic empirical processes behave as well as the empirical processes pretending that the common and idiosyncratic components for an individual variable are directly observable. Based on this oracle property, we construct simultaneous confidence bands (SCBs) for the distributions of the common and idiosyncratic components. For the first-order consistency of the estimated distribution functions, $\sqrt{T}=o\left(p\right)$ suffices. Extensive simulation studies check that the estimated bands have good coverage frequencies. Our real data analysis shows that the common-component distribution has a structural change during the crisis in 2008, while the idiosyncratic-component distribution does not change much. Supplementary materials for this article are available online.

KEYWORDS:

• Empirical process
• Factor model
• Principal component analysis
• Simultaneous confidence band

Supplementary Materials

The online supplementary materials contain Assumptions 1 and 2, additional technical proofs, codes for real data analysis and simulation studies, as well as the real data sets.

Acknowledgments

The helpful comments from the editor, the associate editor, and anonymous referees are gratefully acknowledged. Any communication please contact [email protected].

Additional information

Funding

This work is supported in part by National Natural Science Foundation of China awards NSFC 11801272, Natural Science Foundation of Jiangsu BK20180820.

Kong’s work is supported by NSFC 11201080, 11571250, HSSYF of Chinese Ministry of Education (12YJC910003). The authors from the Nanjing Audit University acknowledge the support of the PAPD of Jiangsu Higher Education Institutions. Wang’s work is supported by Natural Science Foundation of the Higher Education Institutions of Jiangsu Province 17KJB110005. Xu’s work is supported by NSFC 61640220 and Jiangsu NSF 16KJA520002. Wang's work is supported by Natural Science Foundation of the Higher Education Institutions of Jiangsu Province 17KJB110005 and NSFC 11771240.