A misspecification test for the higher order co-moments of the factor model

ABSTRACT

The traditional estimation of higher order co-moments of non-normal random variables by the sample analog of the expectation faces a curse of dimensionality, as the number of parameters increases steeply when the dimension increases. Imposing a factor structure on the process solves this problem; however, it leads to the challenging task of selecting an appropriate factor model. This paper contributes by proposing a test that exploits the following feature: when the factor model is correctly specified, the higher order co-moments of the unexplained return variation are sparse. It recommends a general to specific approach for selecting the factor model by choosing the most parsimonious specification for which the sparsity assumption is satisfied. This approach uses a Wald or Gumbel test statistic for testing the joint statistical significance of the co-moments that are zero when the factor model is correctly specified. The asymptotic distribution of the test is derived. An extensive simulation study confirms the good finite sample properties of the approach. This paper illustrates the practical usefulness of factor selection on daily returns of random subsets of S&P 100 constituents.

KEYWORDS:

• Factor models
• higher order co-moments
• sparsity
• curse of dimensionality
• diagnostic test

AMS SUBJECT CLASSIFICATIONS:

• 62F03
• 91B25

Acknowledgments

We are grateful to two the anonymous referees and to the editor whose valuable suggestions and remarks allowed us to significantly improve the original paper. We also thank Dries Cornilly and Giang Nguyen for their constructive comments and assistance.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The 30 tickers of the stocks are selected in terms of the largest total factor exposure from S&P 100 components. They are as follows: AAPL, AIG, AMZN, BAC, BIIB, BK, C, CELG, CL, COF, CSCO, DUK, GILD, GS, INTC, JPM, MET, MO, MRK, MS, ORCL, OXY, PCLN, PEP, QCOM, SPG, TXN, USB, WFC, and XOM.

Additional information

Funding

Wanbo Lu's research is sponsored by the National Science Foundation of China (71771187, 71101118), the Program for New Century Excellent Talents in University (NCET–13–0961), and the Fundamental Research Funds for the Central Universities (JBK170970) of China. Dong Yang's research is supported in part by China Scholarship Council, the Fundamental Research Funds for the Central Universities (JBK1807047, JBK1805004) of China, and the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics.