A CLT for second difference estimators with an application to volatility and intensity
A CLT for second difference estimators with an application to volatility and intensity
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DOI：
10.1214/22-aos2176
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发表时间：
2020-11
期刊：
影响因子：
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通讯作者：
E. A. Stoltenberg;P. Mykland;Lan Zhang
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文献类型：
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作者：
E. A. Stoltenberg;P. Mykland;Lan Zhang
本文介绍了一个或多个连续时间半鞅点参数过程的二次协变估计的一般方法。该估计器适用于各种现货参数过程，也可用于估计随机波动率模型的杠杆效应。我们介绍的估计量是基于所观察到的过程的第二个差异的平方增量的总和，并且计算差异的间隔是滚动和重叠的。后一个特征使我们能够充分利用数据，并且从充分性考虑，应该优于仅基于观察窗口的一个分区的估计。本文的主要结果是这样的三角形阵列滚动二次变化的中心极限定理。我们强调了这一定理的广泛适用性，展示了它如何可能被应用到一个新的杠杆效应估计。然而，本研究的主要动机是，连续时间半鞅被观察到的离散时间可能取决于可观察过程的特征，而不是其水平，例如其（不可观察的）现货波动率过程。作为我们的估计的主要应用程序，因此，我们展示了它如何可以用来估计现货波动率过程和强度过程的观测时间之间的二次协变，当这两个都采取了半鞅。通过模拟实验研究了该估计量的有限样本性质，并将其应用于苹果股票的实证分析。我们对苹果股票的分析表明，对数价格过程的现货波动率过程与该股票交易的时间（因此观察到）之间存在相当强的相关性。
In this paper we introduce a general method for estimating the quadratic covariation of one or more spot parameters processes associated with continuous time semimartingales. This estimator is applicable to a wide range of spot parameter processes, and may also be used to estimate the leverage effect of stochastic volatility models. The estimator we introduce is based on sums of squared increments of second differences of the observed process, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are only based on one partition of the observational window. The main result of the paper is a central limit theorem for such triangular array rolling quadratic variations. We highlight the wide applicability of this theorem by showcasing how it might be applied to a novel leverage effect estimator. The principal motivation for the present study, however, is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its (non-observable) spot-volatility process. As the main application of our estimator, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded (hence observed).