Rank Tests at Jump Events

Abstract

We propose a test for the rank of a cross-section of processes at a set of jump events. The jump events are either specific known times or are random and associated with jumps of some process. The test is formed from discretely sampled data on a fixed time interval with asymptotically shrinking mesh. In the first step, we form nonparametric estimates of the jump events via thresholding techniques. We then compute the eigenvalues of the outer product of the cross-section of increments at the identified jump events. The test for rank r is based on the asymptotic behavior of the sum of the squared eigenvalues excluding the largest r. A simple resampling method is proposed for feasible testing. The test is applied to financial data spanning the period 2007–2015 at the times of stock market jumps. We find support for a one-factor model of both industry portfolio and Dow 30 stock returns at market jump times. This stands in contrast with earlier evidence for higher-dimensional factor structure of stock returns during “normal” (nonjump) times. We identify the latent factor driving the stocks and portfolios as the size of the market jump.

KEY WORDS:

• Factor model
• High-frequency data
• Jumps
• Rank test
• Semimartingale

ACKNOWLEDGMENTS

We would like to thank the editor, the associate editor, and anonymous referees for their thoughtful comments and suggestions which led to significant improvements. We would also like to thank Markus Pelger (our discussant) and seminar participants at various conferences for helpful discussions and suggestions. Li’s and Todorov’s research have been partially supported by NSF grants SES-1326819 and SES-1530748, respectively.