High-dimensional Covariance Matrix Estimation And The Application On Portfolio Selection

As an indispensable statistical parameter in multivariate statistical analysis,covariance matrix plays an important role in classical multivariate statistical analysis,so how to get an accurate covariance matrix estimation is a basic problem in multivariate statistical analysis.On the other hand,its application is also very extensive.Especially in the field of financial asset investment,financial investment return and risk are always the first consideration.With the covariance matrix’s description of correlation,the amount of fluctuation is measured by covariance matrix,which becomes a powerful tool for financial assets investment.Portfolio model and onther quantitive investment method also improve the covariance matrix role position in the field of financial investment.The era of big data has quietly arrived.In this context,the current statistical analysis work is faced with higher and larger variable dimensions and data size than before.Covariance matrix estimation are faced with a very serious challenge.When the variable dimension or the ratio of the variable dimension to the sample size increases significantly,the traditional covariance matrix construction method based on sample data will become untrustworthy,and a series of problems such as the increase of the number of parameters to be estimated,the accumulation of estimation errors,and the inability to invert the matrix singularity will bring new problems to the statistical theory research.In addition,in the application field of high-dimensional covariance matrix,related problems will follow.For example,as one of the input variables,the high-dimensional covariance matrix may have singularity,so the construction of high-dimensional portfolio model cannot be carried out smoothly.In addition,the covariance matrix elements significantly increase and the estimation error accumulates,so it is difficult to guarantee the good performance of the high-dimensional portfolio.Many scholars have studied the estimation of covariance matrix under the background of high dimension statistics.Based on different research purposes,the study of high-dimensional covariance matrix estimation can be roughly summarized into two research contexts,one is the static estimation of high-dimensional covariance matrix,and the other is the dynamic prediction of high-dimensional covariance matrix.Compared with the traditional sample covariance matrix estimation,the existing research method of high-dimensional covariance matrix has made great progress,which not only ensures that the estimator has positive definite characteristics,but also reduces the estimated parameters and improves the estimation accuracy through the sparse method.However,the hypothesis of data distribution more or less affects the actual application of the estimation method,and the shrinkage estimation used is not perfect.Some time series methods need to split matrix,which also destroys the structure of the matrix.To sum up,based on the static estimation and dynamic prediction analysis within the framework of the high-dimensional covariance matrix,this paper has researched in three aspects of the study,following the overall compliance statistics basic research paradigm of academic papers,to improve the model and solve the problem as the research objective,firstly introduced relevant methods,pointing out its existing problems and shortcomings,and put forward a new method as innovation to compensate,and tried to give convergence rate of model or the the upper bound of error prediction,guaranteed the feasibility of the model in theory,by conducting numerical simulation,showed the proposed method is better,finally in the empirical analysis of financial assets portfolio,made the practical application of the new method.The specific research content is as follows:Chapter 1 introduces the background and significance of this paper.Covariance matrix estimation is the basic problem in multivariate statistical analysis.It is facing great challenges in the high-dimensional context of big data era at present,and its application is worth further discussion.Therefore,the topic selection of this paper has theoretical significance and application value.Chapter 2 is a literature review on the theoretical research of high dimensional covariance matrix estimation and its application in the field of portfolio.Existing literature research perspectives could be summed up in two categories: static estimation and dynamic modeling,carding literature under each kind of perspective,additionally to summarize its application in the field of portfolio,the last at the end of the chapter to evaluate the existing research methods,pointing out that the existence of the problem,in order to draw out the paper research content.Chapter 3 is the theoretical preliminary knowledge.The construction method of the traditional sample covariance matrix is described,and the serious defect of its estimation failure in the high-dimensional data background is demonstrated by numerical simulation.This paper briefly introduces the mean-variance portfolio theory to pave the way for the subsequent construction of the minimum variance portfolio model.Chapter 4 is the first research content of this paper.A new robust estimation of high-dimensional covariance matrix is established,so called central-regularized robust estimation.On the basis of sub-sample sampling,the central regularization is used to improve the mean-median covariance matrix estimator,so as to obtain a positive definite sparse high-dimensional covariance matrix robust estimation.Chapter 5 is the second research content of this paper,which improves the shrinkage estimation method in the process of dynamic modeling of high-dimensional covariance matrix,and introduce VAR-EN model.On the basis of LASSO estimation,L2 norm punishment is applied to the penalty term in this paper to form the elastic net algorithm,which makes the variable selection function of the model more perfect and the sparse parameter matrix obtained more reasonable.Chapter 6,a new dynamic modeling method of high-dimensional covariance matrix is proposed.In this paper,which is called predictive factor model for matrix-valued data based on Cholesky decomposition.Cholesky decomposition and vector autoregressive method are innovatively utilized to solve the applicability of the factor model for high-dimensional matrix-valued data and endow the model with prediction function.Chapter 7 is the last chapter,the content is summary and prospect.This paper summarizes the research contents and conclusions of this paper,pointing out the shortcomings of this research,looking forward to the future development direction of high-dimensional covariance matrix research.The research content of this paper conforms to the basic paradigm of statistical academic research.The three research contents are the three innovation points of this paper,focusing on the improvement of the research method of high-dimensional covariance matrix.The specific innovation points are as follows:1.The central-regularized robust estimation method is proposed to solve the problem that the mean-median robust estimation cannot obtain the positive definite sparse matrix.Original mean-median estimate for high-dimensional covariance matrix of robust estimation method is simple and easy to calculate,but couldn’t guarantee on model set estimation matrix of positive definite feature,and high-dimensional covariance matrix estimation is not sparse,this article proposed the central-regularized robust estimation to improve,resultly we can obtain a robust high-dimensional sparse positive-definited covariance matrix.2.VAR-EN model is proposed to solve the problems that variable selection and unreasonable sparsity of estimated coefficient matrix in var-lasso method is incomplete.The VAR-LASSO method takes the realized covariance matrix as the modeling sample,sparsely estimates the parameter matrix through the LASSO contraction.However,limited by the defects of LASSO method itself,the model’s function of variable selection is not complete,and the prediction accuracy needs to be improved.In this paper,the LASSO estimation is improved to form the elastic net algorithm,and the VAR-EN model is proposed to improve it.3.The predictive matrix-valued factor model based on Cholesky decomposition was put forward,which solved the applicability of the original matrix value factor model for dynamic modeling of high-dimensional covariance matrix.Factor analysis for matrix-valued time series data was introduced in recent years,matrix dimensions can be effectively reduced through factor analysis,the ideology of dimension reduction is instructive for high-dimensional data analysis,however,the original model does not impose structure on modeling matrix set,and will not be directly applied to the research process,factor analysis itself does not have also predicts a function,which makes the model application value discount greatly.The predictive matrix-valued factor model proposed in this paper,based on Cholesky decomposition,not only solves the applicability of the original model,but also has the predictive function.The research content of this paper takes methodology research as the starting point and empirical application as the foothold,and focuses on pointing out the research defects of traditional methods,making up for and improving them,and demonstrating the application effects through empirical analysis.In general,the study of high-dimensional covariance matrix estimation in this paper still follows the ideas of dimensionality reduction and sparsity,and its specific research contents grasp the forefront of statistical research methods in the era of big data.There are novel perspectives and ideas in topic selection,and the three innovations have certain academic value and research significance.