Estimation Of High-dimensional Covariance Matrices And Their Application To Investment Portfolios

With the improvement of data availability,many financial institutions or individuals are faced with an increasing number of dimensions of asset data.When the dimension is too high or even exceeds the sample data size,the influence of "Curse of dimensions " and the impact of noise makes the traditional multivariate GARCH model no longer applicable,which cause further research on covariance matrix estimation by scholars.This paper first introduces the traditional multivariate GARCH model,and explains the limitation of the traditional multivariate GACRH model on the estimation of sample covariance matrix when the dimension is too high,and then introduces the factor model,thresholding,shrinkage and other high-dimensional covariance matrix estimation method.In order to solve the problem of "Curse of dimension",this paper is applied to the common multivariate GARCH model—DCC-GARCH model by the improved Cholesky decomposition.When the traditional Cholesky decomposition method is applied to the estimation of high-dimensional covariance matrices,it is often assumed that the order of assets is given.In fact,this assumption is not reasonable,which affect the estimation effect of covariance matrix.In order to solve this problem,the improved Cholesky decomposition method is considered to be applied to the estimation of high-dimensional covariance matrix.The substitution matrix is introduced into the Cholesky decomposition process with the covariance matrix to rank the asset vectors randomly,so as to obtain the ensemble estimate under multiple permutations of the asset covariance matrix.Montecarlo method is used to simulate stock return,and then some criteria are used to compare the estimation effect of different models.Then,the CSI 300 index constituent stocks were taken as the research object to compare the prediction effect of covariance matrix for the empirical analysis,as well as the portfolio returns and volatility constructed by the predicted covariance matrix,and the following conclusions were drawn:(1)With the increase of asset dimensions,the "curse of dimension" and the influence of noise faced by the estimation covariance matrix are more and more serious,which makes the estimation and prediction effect of traditional multivariate GARCH model,such as DCC model,worse and worse,and the efficiency of its portfolio is also lower and lower.This model is no longer suitable for estimating andpredicting the covariance matrix of high-dimensional assets,and needs to be improved.The improved multivariate GARCH model described in this paper is more effective in estimating the covariance,and the constructed portfolio has greater returns and less risk.(2)The DCC-IMCF model obtained by applying the improved Cholesky decomposition to the DCC model obviously optimizes the estimation and prediction efficiency of the multivariate GARCH model.Compared with the modified DCC(DCC-MCF)model,the estimated and predicted covariance matrix of DCC-IMCF is closer to the real covariance matrix,and its application effect is better in the investment portfolio,resulting in higher returns and less portfolio risks.