Related Theories And Applications Of High Dimensional Covariance Matrix

Over the past decades,as high-dimensional data has become more and more available in many fields,methods about estimating the second moments of high-dimensional data have received considerable attention.Substantive progress has been achieved in two complementary directions:cross-section and time series.From the view of cross-section,the biggest challenge is high dimensionality,and the corresponding methods mainly include sparseness,factor model,and shrinkage method.From the view of time-series,conditional heteroskedasticity is the most important factor to consider,and Generalized Autoregressive Conditional Heteroskedasticity(GARCH)model,which includes VEC,BEKK,DCC,and so on,is the most typical model.Development of theories from the two separate directions is rather rapid,while few papers combine these two streams together successfully.As a result,estimating high-dimensional covariance matrices in finance remains to be a challenge.In this context,we will systematically research the theory and application of estimating high-dimensional covariance matrices,first in separate directions,and then on how to combine them together effectively to fit the application in finance.In theory,this paper focus on three models:high-dimensional factor model,shrinkage method,and GARCH model which uses factor model or shrinkage method in the estimation.For high-dimensional factor model,we review the methods of determining the number of factors,as well as methods of estimating factor models.Most importantly,we introduce how to get the estimated covariance matrices by factor models using threshold functions.For shrinkage,three linear methods and the whole process to get the bona fide nonlinear estimator are introduced.On this basis,we investigate how to estimate high-dimensional GARCH model effectively using factor model or shrinkage method,which is the theoretical innovation of this paper.In application,using data from American stock market,we construct global minimum variance(GMV)portfolios,and compare Markowitz portfolios with Sorting portfolios based on 61 separate return predictive signals.In asset weighing,different methods are used to estimate the covariance matrices.The predictive results show that the Markowitz portfolios which use DCC-NL model to get the estimated covariance matrices yield the highest Sharpe Ratio.From both views of literature and application,we are the first to construct high-dimensional Markowitz portfolios using covariance matrices estimators from the DCC-NL model.The main conclusion of the study can be summarized as follows:First,we review alternative methods of estimating high-dimensional covariance matrices.Different from the general summary papers of factor models which only focus on how to determine the number of factors and how to estimate the models in different specifications,we pay more attention to the threshold assumptions on covariance matrix of the residuals,and how to get the final estimator of the high-dimensional covariance matrix.Besides,to our knowledge,we are the first to summarize the shrinkage methods,including three linear shrinkage methods and the nonlinear shrinkage method based on QuEST function.Second,by investigating advanced theories in GARCH model,we find that the essential problems in the Maximum Likelihood Estimation of GARCH models are the need to inverse high-dimensional matrices many times,and that there are too many parameters to be estimated.The first problem can be effectively solved by composite likelihood estimation proposed by Engle et al.(2008),while the popular solution to the second problem--using sample covariance to replace the MLE estimator of nuisance parameter--is biased in high dimension.Third,we clearly put forward the idea of using methods of estimating unconditional covariance matrices in the estimation of GARCH model,and introduce DCC-POET and DCC-NL,which respectively apply factor model and nonlinear shrinkage technique on the sample covariance matrix of the standardized residuals.Fourth,Our Monte Carlo simulations prove that nonlinear shrinkage technique improves the estimation of DCC and BEKK for various cross-section dimensions.Furthermore,the improvement increases with the ratio of cross-section dimension to time length,and the improvement from nonlinear shrinkage technique is more significant than that from linear shrinkage technique.Fifth,we collect 61 return predictive signals which have significant performance from authority literature and describe how their factor scores are computed.The 61 signals are classified into 6 groups:momentum,value-versus-growth,investment,profitability,intangible,and trading frictions.In practice,investors always construct portfolios based on several signals.Thus,this work not only embodies another innovation of this paper,but also has important implications for investors.Sixth,we construct GMV portfolios,and compare Markowitz portfolios with Sorting portfolios in American stock market,separately based on the 61 signals we collected.These applications demonstrate that the portfolio selection rule in predictive tests of cross-sectional anomalies should incorporate a suitable estimator of covariance matrix of stock returns,and DCC-NL and BEKK-NL estimators are the suitable estimators.To the best of our knowledge,this paper considers more signals than any other literature that research the performance of covariance matrix estimators by constructing portfolios.More importantly,this paper increases the maximum cross-section dimension considered from 200 to 1000.By doing so,we make our results more robust and meaningful.