Robust Jump Regressions

ABSTRACT

We develop robust inference methods for studying linear dependence between the jumps of discretely observed processes at high frequency. Unlike classical linear regressions, jump regressions are determined by a small number of jumps occurring over a fixed time interval and the rest of the components of the processes around the jump times. The latter are the continuous martingale parts of the processes as well as observation noise. By sampling more frequently the role of these components, which are hidden in the observed price, shrinks asymptotically. The robustness of our inference procedure is with respect to outliers, which are of particular importance in the current setting of relatively small number of jump observations. This is achieved by using nonsmooth loss functions (like L₁) in the estimation. Unlike classical robust methods, the limit of the objective function here remains nonsmooth. The proposed method is also robust to measurement error in the observed processes, which is achieved by locally smoothing the high-frequency increments. In an empirical application to financial data, we illustrate the usefulness of the robust techniques by contrasting the behavior of robust and ordinary least regression (OLS)-type jump regressions in periods including disruptions of the financial markets such as so-called “flash crashes.” Supplementary materials for this article are available online.

KEYWORDS:

• High-frequency data
• Jumps
• Microstructure noise
• Robust regression
• Semimartingale

Supplementary Materials

The supplementary materials consist of two appendices. Appendix A contains the proofs of the results in the main text. Appendix B provides an example that illustrates the adverse consequence of mistakenly ignoring measurement errors in the jump regression setting.

Acknowledgment

The authors thank Yuan Xue, who provided excellent research assistance.

Funding

Li’s and Todorov’s research have been partially supported by NSF grants SES-1326819 and SES-0957330 and SES-1530748, respectively.