High frequency and high dimensional data modeling

金融时间序列数据的统计建模一直是一个非常活跃的研究领域。在对金融资产的随机动态行为进行建模时，资产收益之间的协方差是一个关键因素，它在现代金融中起着至关重要的作用。例如，在投资组合优化和风险管理中，协方差矩阵及其逆矩阵是关键统计量。随着最近高频金融数据的可用性，例如每分钟甚至每5秒记录的观察， 固定时间范围内资产收益率的综合协方差估计问题引起了研究者的极大关注。但高频是一把双刃剑。它为统计学家提供了大量可用数据，允许捕捉一些有趣的统计数据的每日变化，这些数据无法从每日或每周数据中观察到。另一方面，数据总是被市场微观结构噪声所污染。如果没有适当地建模，这种微结构噪声可能会在很大程度上主导积分变化的估计，因此破坏其所有的统计特性。另一个伴随出现的高频现象是资产相关性，它类似于变量中测量误差的衰减效应，使资产相关性的估计偏向于零（Epps效应）。两个资产的精确观测时间很少是同时的，这导致即使是低频日数据也难以统计推断资产协变。超前/滞后关系是另一个高频重要问题。一些资产倾向于跟随其他资产的路径，但时间滞后很小。强烈的非对称交叉相关函数的经验观察，特别是在未来/股票的情况下。这种关系需要仔细建模，以获得稳定和准确的协变估计。 由于上述困难，在高频下的超前/滞后协变的文献是有限的。据我们所知，在文献中还没有看到综合同期和领先/滞后协变的联合建模。 在这项研究中，我们计划调查的高频协方差估计与噪声和异步数据中存在的领先/滞后关系。综合同期协变和超前/滞后协变的联合建模将填补金融计量经济学领域的一个空白。本文的研究结果可以提供更稳定、更准确的协方差估计以及协方差矩阵估计，为投资组合优化和风险管理提供有力的支持。

The statistical modeling of financial time series data has been a very active research field. Among the key elements in modeling the stochastic dynamic behavior of financial assets is the covariance between the asset returns, which plays a crucial role in modern finance. In portfolio optimization and risk management, for instance, the covariance matrix and its inverse are key statistics. With the recent availability of high frequency financial data, say observations recorded every minute or even every 5 seconds, the estimation of integrated covariance of asset returns over a fixed time horizon attracted tremendous attention from researchers'. But high frequency is a double-edged sword. It provides large amount of available data to statisticians allowing to capture the daily variation of some interesting statistics that are unobservable from daily or weekly data. On the other hand, the data are always contaminated with market micro-structure noise. If not appropriately modeled, this micro-structure noise could very much dominate the estimation of the integrated variation and hence disrupts all its statistical properties. Another accompanying phenomenon with the high frequency is the asynchronicity, which, similar to the attenuation effect of measurement error in variables, biases the estimation of correlation of assets towards zero ( Epps effect). The exact observation times of two assets are rarely simultaneous, which causes difficulties in statistical inference of assets covariation even with low frequency daily data. Lead/lag relationship is another important issue at high frequency. Some assets tend to follow the path of others with a small time lag. Strongly asymmetric cross correlation functions are empirically observed, especially in the future/stock case. This relationship needs to be carefully modeled in order to obtain stable and accurate covariation estimation. Due to the above mentioned difficulties, the literature about the lead/lag covariation at high frequency is limited. Joint modeling of integrated contemporaneous and lead/lag covariation is not yet seen in literature, to our best knowledge. In this proposed research, we plan to investigate the high-frequency covariance estimation with noisy and asynchronous data in the presence of lead/lag relationships. The joint modeling of integrated contemporaneous and lead/lag covariation will fill a gap in the area of financial econometrics. The outcome of this research could provide more stable and accurate estimation of integrated covariance as well as covariance matrix estimation, which can further provide solid support to portfolio optimization and risk management.