跳跃回归再探
Jump regression revisited
- Christian-Albrechts-Universität zu Kiel（基尔大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文针对跳跃过程提出一种更贴近经典回归的设定，基于高频观测数据估计回归系数，并给出两种渐近框架下的相合性与中心极限定理。
中文摘要 AI 辅助
在过去的几十年里，随机过程的回归分析一直是一个重要的研究课题。通常，将依赖过程 $Y$ 对解释变量 $Z$ 进行回归，会得到形如 $dY_t = \beta_t dZ_t + dX_t$ 的积分关系，其中 $X$ 是某个残差过程，且 $\beta$ 通常是常数或分段常数。在跳跃过程的情形下，这种关系在跳跃层面上本质上归结为 $\Delta Y_t = \beta \Delta Z_t + \Delta X_t$。这里残差过程 $X$ 与 $Z$ 不会同时跳跃，因此 $Y$ 的任何跳跃要么与 $Z$ 严格成比例，要么纯粹是特质性的。在本文中，我们讨论一个相关但更现实且更接近经典回归的设定，即 $\Delta Y_t = (\beta + \eta_t) \Delta Z_t$，其中 $(\eta_t)_{t \ge 0}$ 是独立同分布的。我们基于 $(Y,Z)$ 的高频观测提出 $\beta$ 的估计量，并在两种不同的渐近机制下讨论其相合性及相关的中心极限定理。
英文摘要
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 $dY_t = β_t dZ_t + dX_t$ with some residual process $X$, and often with constant or piecewise constant $β$. In the case of jump processes such a relationship essentially boils down to $ΔY_t = βΔZ_t + ΔX_t$ 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 $ΔY_t = (β+ η_t) ΔZ_t$ for i.i.d. $(η_t)_{t \ge 0}$. We propose an estimator for $β$ based on high-frequency observations of $(Y,Z)$ and discuss consistency and associated central limit theorems in two different asymptotic regimes.
