Robust Jump Regressions∗

We develop robust inference methods for studying linear dependence between the jumps of discretely observed processes at high frequency. Unlike classical linear re-gressions, 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 impor-tance in the current setting of relatively small number of jump observations. This is achieved by using non-smooth loss functions (like L1) in the estimation. Unlike clas-sical robust methods, the limit of the objective function here remains non-smooth. 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 empiri-cal application to financial data we illustrate the usefulness of the robust techniques by contrasting the behavior of robust and OLS-type jump regressions in periods in-cluding disruptions of the financial markets such as so called “flash crashes.