Digital correlation matrix in multivariate statistics and its application for component selection and dynamic correlation modeling

Many financial models require the estimation of correlation coefficients. From portfolio optimization to option valuation, understanding the interactions among financial securities is the bottom line of asset pricing and risk valuation, and the correlation matrix is the key measure to explain how multiple financial assets are related to each other. As a large number of assets have been introduced in the market, research on large dimensional correlation matrices has become particularly important. Accurate estimation of large correlation matrices is challenging due to the complication of high dimensional statistical systems.; We aim to analyze correlations of large dimensional processes in the framework of principal component analysis and random matrix theory. Our study focuses on both static and time varying correlation models. In Chapter 2, we discuss factorization of large sample correlation matrices. Principal Component Analysis and Random Matrix Theory are popular methods in correlation analysis, and the critical issue is the methodology of component selection. We propose criteria for component selection using the statistical relationship between spectrums of digital correlations and Gaussian correlations. Comparing the statistics of spectrums of digital correlations and Gaussian linear correlations results in the zero correlation test, which is used to select the optimal number of factors. Another topic explored in this research is modeling the dynamics of time-varying correlations, which is addressed in Chapter 3. In regards to dynamic correlations, we propose several time-varying models capturing the fluctuation of digital correlations. Digital correlations are independent of individual volatilities, and show the genuine picture of correlation dynamics. The proposed models include both digital GARCH and factor digital GARCH, which are variations of the classical multivariate GARCH model.