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 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 <i>L</i>₁) 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.

本文针对高频离散观测过程的跳跃间线性相关性分析，构建了稳健推断方法。与经典线性回归不同，跳跃回归由固定时间区间内发生的少量跳跃，以及跳跃时刻附近过程的其余分量共同决定。该其余分量包含过程的连续鞅部分与观测噪声。通过提升采样频率，隐藏于观测价格中的此类分量的作用将渐近收缩。我们的推断程序对异常值具备稳健性，而在当前跳跃观测数量相对较少的场景下，异常值问题尤为关键。该稳健性通过在估计环节使用非光滑损失函数（如L₁）得以实现。与经典稳健方法不同，本文所提目标函数的极限仍保持非光滑特性。所提方法对观测过程中的测量误差同样具备稳健性，该特性通过对高频增量实施局部平滑得以实现。在针对金融数据的实证应用中，我们通过对比稳健型与普通最小二乘（OLS）类跳跃回归在金融市场动荡时期（如所谓的"flash crashes"）的表现，阐释了该稳健技术的应用价值。本文的补充材料可在线获取。