Jump regression revisited

Abstract

Regression analysis for stochastic processes has been an important topic over the last decades. Typically, regressing a dependent process $Y$ on an explanatory $Z$ results in an integral relationship of the form $d{Y}_t={\beta }_{t}d{Z}_t+d{X}_t$ with some residual process $X$, and often with constant or piecewise constant $\beta$. In the case of jump processes such a relationship essentially boils down to $\Delta Y_t=\beta \Delta Z_t+\Delta X_t$ on the level of the jumps. Here the residual process $X$ is such that it never jumps together with $Z$, and so any jump in $Y$ is either exactly proportional to $Z$ or purely idiosyncratic. In this paper we discuss a related setup which appears more realistic and is closer to classical regression, namely $\Delta Y_t=(\beta +{\eta }_t)\Delta Z_t$ for i.i.d. $({\eta }_t{)}_{t\ge 0}$. We propose an estimator for $\beta$ based on high-frequency observations of $(Y,Z)$ and discuss consistency and associated central limit theorems in two different asymptotic regimes.