High-dimensional Dynamic Covariance Matrix Estimation And Its Application

With the rapid increase in data storage capacity and arithmetic power,statistical analysis of high-dimensional data has become a hot research topic.The covariance matrix serves as a favourable tool to provide a good insight into the structural features within a data set.Almost all areas of multivariate analysis and many application problems involve the use of covariance matrices,such as mean-variance optimal portfolio selection,risk management and asset pricing.Therefore,the estimation of high-dimensional covariance matrices is an interesting and important research topic.Much of the current literature is devoted to the study of high-dimensional static covariance matrices.However,it is not sufficient to study static covariance matrices alone,for example in portfolio allocation,where the use of dynamic covariance matrices would be more relevant.To introduce a dynamic structure for covariance matrices,one cannot simply assume each entry of a covariance matrix is a function of time because this would not serve very well in prediction.In order to address the difficulty of estimating dynamic high-dimensional covariance matrices,this paper embeds a dynamic single-index varying-coefficient model based on auto-regressive idea equipped with the factor model,thereby constructing a dynamic model of individual risky asset return,namely the Adaptive Dynamic Single-Index Coefficient Model(ADSICM),and thus obtaining a dynamic covariance matrix among the assets to be allocated.For this dynamic covariance matrix,the presence of a factor structure avoids the estimation of too many unknown parameters and functions.In this paper,the main consideration is to estimate the index function and coefficient function within it using the estimation method of B splines,using a local constant kernel estimation method to estimate the factor covariance matrix.To demonstrate the validity of this estimation procedure,numerical simulations are carried out in this paper.The simulation results show that the proposed estimation procedure performs well when the sample size is limited.Finally,the proposed high-dimensional dynamic covariance matrix is applied to the portfolio allocations of the 16 sector indices of the SZSE and the 30 industrial indices of the US,respectively to explore the significance of the proposed dynamic structure in real-world applications.The empirical results show that portfolio allocations based on the dynamic high-dimensional covariance matrix proposed in this paper are high-quality,with higher average annual returns than both portfolio allocations based on the sample covariance matrix and those based on the compressed estimation method,over the period from 2013 to 2022.Also,to illustrate the validity of the proposed approach to complex asset allocation,the paper also compares the average annual returns of a portfolio based on naive diversification and those based on the Dow Jones Industrial Index.