The Estimations Of High-Dimensional Conditional Covariance Matrices With Three Factor-GARCH Models

Nowadays,Advances in science and information technology make high-dimensional time series and space-time data widely accessible in many scientific fields,including e-conometrics,environmentology,engineering and statistics.How to make inferences about high-dimensional conditional covariance matrix plays an important role in these fields,e-specially in risk management,portfolio allocation,asset pricing,risk-adjusted return and hedging and so on.Although theoretically,the common multi-variate GARCH models can be directly used in multi-variate time series,and make inference for conditional co-variance matrix.However,the common multi-variate GARCH models can't be used in moderate dimensional time series,let alone ultra/high-dimensional ones.The main prob-lems are curse of dimension,hard to identify the factor or be positive-definite.Hence,it is a great challenge to combine the multi-variate GARCH models with the present dimension/parametrization-reduction technologies for high-dimensional conditional co-variance.The proposed thesis focuses on the above issue,with assumptions that common factor is known and unknown.In Chapter 1,we first introduce the developments of multivariate GARCH model-s,the developments of factor model and its applications in the inferences of covariance matrix,two important dimension/parametrization-reduction methods,and the present es-timating methods of high-dimensional conditional covariance matrix.The proposed thesis focuses on the estimations of high-dimensional conditional co-variance matrix by combining multi-variate GARCH model with factor model.Some famous or new methods are used,which are the thresholding estimation for sparse ma-trix,quasi-maximum likelihood estimation for GARCH model,eigenanalysis method for identifying factor loading matrix in latent factor model,and least square estimation for es-timating factor loading matrix in strict and approximate factor model.The main contents and conclusions of the proposed thesis include the following three aspects:(1)For?,we establish a model by combining the ideas of a strict factor model and a symmetric GARCH model to describe the dynamics of a high-dimensional con-ditional covariance matrix.In this model,the common factor is observable and under CCC-GARCH structure,and the idiosyncratic components is conditional uncorrelated.Hence,the conditional covariance matrix of the proposed time series is combine with a nonnegative definite matrix and a diagonal matrix.To obtain the estimation of the ma-trix,quasi maximum likelihood estimation(QMLE)and least square estimation(LSE)methods are used to estimate the parameters in the model,and the plug-in method is in-troduced to obtain the estimation of conditional covariance matrix.Asymptotic properties are established for the proposed method,and simulation studies are given to demonstrate its performance.A financial application is presented to support the methodology.(2)For?,our thesis proposes a method for modelling volatilities(conditional covariance matrices)of high dimensional dynamic data.We combine the ideas of approx-imate factor models for dimension reduction and multivariate GARCH models to estab-lish a model to describe the dynamics of high dimensional volatilities.In the proposed model,the common factor is assumed to be known and under CCC-GARCH structure.Hence,the conditional covariance matrix is combined with a nonnegative definite matrix and a sparse matrix.To estimate the proposed matrix,thresholding technique is applied to the estimation of the error covariance matrix,and quasi maximum likelihood estimation(QMLE)method is used to estimate the parameters of the common factor conditional co-variance matrix.Asymptotic theories are developed for the proposed estimation.Monte Carlo simulation studies and real data examples are presented to support the methodology.(3)For high-dimensional time series,we introduce a model for the high-dimensional conditional covariance matrix by combining a latent factor model and multivariate GARCH model.In the proposed model,the common factor is unknown or unobservable which is much common but important in applications,and common factor and idiosyn-cratic components are half correlated,e.g.for an integer k>0,Cov(x_t,e_t-k|(?)_t-k-1)(?)0but Cov(x_t,e_t-k|(?)_t-k-1)=0.We use an eigenanalysis method to identify the factor loading and common factor,and give the corresponding matrices'estimations.And we establish the asymptotic properties for the proposed estimation withis fixed and??.In conclusions,the results show that,there are several advantages of the proposed models.Our researches perfect the theory of estimation for conditional covariance matrix by combining factor model and multivariate GARCH model and broaden the application scope of the proposed methods.So the research of this thesis have certain theoretical and application value.