High frequency and high dimensional data modeling

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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.