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。在金融资产的随机动态行为建模中，资产收益率之间的协方差是关键因素之一，这在现代金融学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，协方差矩阵及其逆是关键的统计量。随着最近高频金融数据的可用，比如每分钟甚至每5秒记录一次观测， 固定时间范围内资产收益率的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的数据，从而能够捕捉到一些有趣的统计数据的每日变化，而这些数据是从每日或每周的数据中无法观察到的。另一方面，数据总是被市场微观结构噪声污染。如果没有适当的建模，这种微结构噪声可能会在很大程度上主导对综合变化的估计，从而扰乱其所有统计特性。与高频伴随的另一个现象是异步性，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)。两个资产的准确观测时间很少同时出现，这就造成了即使在日常数据频率较低的情况下，也很难对资产的协变性进行统计推断。超前/滞后关系是高频下的另一个重要问题。一些资产倾向于跟随其他资产的路径，只有很小的时间滞后。经验上观察到强不对称的互相关函数，特别是在期货/股票的情况下。为了获得稳定和准确的协方差估计，需要对这种关系进行仔细的建模。 由于上述困难，有关高频下的超前/滞后协变的文献有限。据我们所知，对整合的同期和领先/滞后协变的联合建模在文献中尚未见过。 在这项拟议的研究中，我们计划研究在存在超前/滞后关系的情况下，带有噪声和异步数据的高频协方差估计。综合同期和领先/滞后协变的联合建模将填补金融计量经济学领域的空白。本文的研究结果可以提供更稳定、更准确的综合协方差估计以及协方差矩阵估计，从而进一步为投资组合优化和风险管理提供坚实的支持。

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.