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。金融资产的随机动态行为建模的关键因素之一是资产收益率之间的协方差，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，协方差矩阵及其逆是关键的统计量。随着最近高频金融数据的出现，例如每分钟甚至每5秒记录一次的观察结果，固定时间范围内资产回报的综合协方差估计引起了研究人员的极大关注。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，使他们能够捕捉一些有趣的统计数据的日常变化，这些统计数据从每日或每周的数据中无法观察到。另一方面，这些数据总是受到市场微观结构噪声的污染。如果没有适当地建模，这种微观结构噪声可能会在很大程度上主导对综合变化的估计，从而破坏其所有统计特性。伴随着高频率的另一个现象是异步性，它类似于变量测量误差的衰减效应，使资产的相关性估计偏向于零（Epps效应）。两种资产的精确观测次数很少同时发生，这使得即使在低频日数据下也难以进行资产共变的统计推断。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，并具有较小的时间滞后。强烈不对称的互相关函数是经验观察，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共变估计。由于上述困难，有关高频超前/滞后共变的文献有限。据我们所知，在文献中尚未见过综合同期和超前/滞后共变的联合建模。在本研究中，我们计划研究在存在领先/滞后关系的情况下，具有噪声和异步数据的高频协方差估计。综合同期和领先/滞后共变的联合建模将填补金融计量经济学领域的空白。研究结果可提供更稳定、准确的综合协方差估计和协方差矩阵估计，为投资组合优化和风险管理提供有力支持。

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.