Modeling Time-Varying Higher Order Moments Using GAS Model And High-Frequency Data

The autoregressive conditional heteroscedasticity(ARCH)model of Engle(1982)pioneered the time-varying volatility model.In his paper,breaking the assumption that the traditional econometric model has one pried static prediction variance,and based on the assumption that the conditional variance has zero mean and is sequence uncorrelated and unconditional variance is static,the dynamic characteristics of time-varying variance structure are investigated.Subsequently,Bollerslev(1986)expanded the ARCH process.In his paper,the lag of the conditional variance is introduced into the updating process of the current conditional variance,and the generalized autoregressive conditional heteroscedasticity(GARCH)model is proposed.Currently,the idea of the ARCH and GARCH model has been widely used to study the volatility of financial time series by domestic and foreign researchers,which provides a scientific basis for investment participants and market supervisors to make decisions.However,with the development of the financial market,the stylized facts that the distribution of return on assets has the characteristics of high-order moments such as peak,fat-tail and asymmetric have been confirmed by many research.This also leads to the limitations of GARCH-type models in studying the typical characteristics of financial markets.Furthermore,in the financial investment decision,the researchers also find that the return on assets not only has the above typical characteristics,but also these characteristics may be dynamic,which leads to the discussion on the dynamic modeling of time-varying high-order moments.When conducting the dynamic modeling of high-order moments of financial asset return,it is necessary to conduct the dynamic modeling of both time-varying second-order moments and high-order moments.Currently,there are relatively many researches on the time-varying modeling of the second moment of financial asset return.The mainstream approach is to adopt GARCH-type model or extend GARCH-type model to multiple cases,focusing on the study of the volatility of financial assets and the conditional dependent structure between assets.However,there are few literatures on modeling the multivariate dynamic high-order co-moments.The problem lies in the difficulty of parameter estimation of multivariate dynamic high-order co-moments model,which easily induces the ”curse of dimension”.In order to solve this problem,current research on dynamic modeling of timevarying high-order moments is based on the idea of autoregressive conditional density(ARCD)model and adopts GARCH-type model for dynamic setting.Few scholars adopt the modeling method of generalized autoregressive score(GAS)model.Even though those who use GAS model framework,they concentrated on the modeling the volatility.However even though there are some researches on the construction of time-varying high-order moment models by using the GAS models,most of them focus on the application.With the advent of high-frequency data in the financial market,investors urgently need to dig deep into the rich information behind the high-frequency financial data.However,on the one hand,most of the traditional volatility models and high-order moment models use daily return to extract current information.And this information is precisely used to explain the fluctuations of the next period,which makes the volatility model based on daily data(low-frequency)not only unable to quickly capture the change of volatility,but also the rich market information of high-frequency data has not been fully explored.And only using daily data to take the statistically inference of the time-varying volatility will lose the effectiveness of the model.On the other hand,the previous time-varying high-order moment modeling is based on the classical GARCH-type model for dynamic structure setting,this modeling idea only considers the conditional moment information of asset return,and does not make full use of the density structure information of the distribution of financial asset return.Although this method can reflect the dynamic structure characteristics of high-order moment,but in the face of the huge data information in the financial market,it is impossible to fully mine the sample data information.This,in turn,affects the estimation effect and prediction accuracy of the model.Based on the probability function score function and under the observation-driven model framework,Creal,Koopman and Lucas(2011)and Creal,Koopman and Lucas(2013)proposed a generalized autoregressive score(GAS)model.In fact,one of the characteristics and advantages of the model framework is based on score function,and the model structure makes full use of the density function structure of data distribution,rather than just considering conditional moment information,which is also the biggest difference between this model and other observation-driven models.And the proposal of GAS model provides one of the idea for making full use of the complete information of data distribution.Based on the generalized autoregressive score(GAS)model and autoregressive conditional density(ARCD)model,this paper combines the highfrequency data of the financial market,and strives to break through the bottleneck of the time-varying high-order moment modeling.By systematically summarizing the existing research,we can find the shortcomings and problems existing in the research.And from the various aspects such as theoretical research and empirical analysis,the dynamic modeling of time-varying highorder moments is systematically studied,which make the dynamic modeling of time-varying high-order moments to be improved and supplemented.The main research content of this paper is divided into the following three parts:(1)In this paper,a univariate time-varying high-frequency high-order moment model,that is,the realized GAS-ARCD model,is constructed by combining high-frequency data,generalized autoregressive score model and autoregressive conditional density model.By assuming that the return follows the skew student’s t distribution and the realized variance measure follows a flexible distribution(Chi-square and F distribution),and based on the ideas of GAS model and ARCD model,this paper sets the joint dynamic score structure of return and realized variance measure.And according to the combination of SKST-Chisq and SKST-F distribution,the specific form of the score function is derived in detail.And in this paper the maximum likelihood estimation method is given,as well as the consistency of the parameters and the conditions of the asymptotic properties will to be proved.In conclusion,the univariate high-frequency high-order moment model is systematically modeled and studied.At the same time,through simulation research and empirical analysis,the dynamic model proposed in this paper are systematically and comprehensively compared with the corresponding static model.(2)In this paper,combined with high-frequency data,the multivariate time-varying high-frequency high-order moment model,that is,the realized MGAS-SKST-MF model,is constructed based on the generalized autoregressive score model.Based on the univariate time-varying high-order moment dynamic model in the first part,this paper extends it to the multivariate case,by assuming that the return follows the multivariate skew student t distribution and the realized covariance matrix measure follows the Matrix F distribution with fat-tailed characteristics.The joint dynamic score structure of the return vector and the realized covariance measure matrix is set,and the specific form of the score matrix of the dynamic structure of the time-varying covariance matrix is derived in detail according to the SKST-Matrix-F distribution combination.Meanwhile,the consistency of the parameters and the conditions of the asymptotic properties are to be proved,and the realized MGAS-SKST-MF model is constructed.Through simulation research and empirical analysis,the realized MGAS-SKST-MF model proposed in this paper is systematically and comprehensively compared with the current classical volatility model.(3)In this paper,combined with high-frequency data,generalized autoregressive score model and autoregressive conditional density model,a generalized multivariate time-varying high-frequency high-order moment model is proposed,that is,the realized MGAS-ARCD model.Based on the univariate time-varying higher-order moment dynamic model in Part I and the model in the multivariate case in Part II,this paper expands it to the general case.On the basis of assuming that the return on financial assets follows the multivariate biased student’s t distribution with time-varying covariance structure and time-varying shape parameters,and that the realized covariance measure matrix follows a flexible matrix distribution(Wishart distribution and Matrix F distribution),the joint dynamic score structure of the return on assets vector and the realized covariance measure matrix are set.And according to the combination of SKST-Wishart and SKST-Matrix-F distributions,the specific forms of the score matrix and the score function of the time-varying covariance matrix,time-varying skewness parameter and time-varying kurtosis parameter are derived in detail.Meanwhile,the consistency of the model parameters and the conditions of the asymptotic properties are given,and the realized MGAS-ARCD-SKST-W and realized MGAS-ARCD-SKST-MF models are constructed.This model not only nests the realized MGAS-SKST-MF model proposed in the second part,but also nests the current more classical volatility model,which has a generalized model form.Through simulation research and empirical analysis,the realized MGAS-ARCD model proposed in this paper is systematically and comprehensively compared with the current classical volatility model.The research innovation of this paper is reflected in the following three aspects:(1)Based on the idea of generalized autoregressive score(GAS)model and under the background of high-frequency data,this paper uses the modeling method of autoregressive conditional density(ARCD)model,and systematically and comprehensively studies the modeling problem of time-varying high-frequency high-order moments from the theoretical research and empirical analysis.Due to the current research on the dynamic modeling of time-varying high-order moments,most of the classical GARCH-type models are used for dynamic setting,and few literature adopts the modeling framework of GAS models,even if there is research which use on the GAS model,it is concentrated in volatility modeling research.However,even if there is research on the construction of time-varying high-order moment models based on GAS models,it is mostly focused on the application.Combining the idea of GAS model with the modeling framework of ARCD model,this paper proposes a complete set of time-varying high-order moment dynamic modeling methods from various aspects,such as model setting,score function and score matrix derivation,model estimation method,asymptotic property analysis,model test,simulation research,and empirical analysis.On the one hand,by constructing a univariate time-varying high-frequency high-order moment model based on GAS model,the dynamic characteristics of individual stocks in the financial market can be studied.On the other hand,by constructing a generalized multivariate time-varying high-frequency high-order moment model based on GAS model,the paper not only can get a generalized model form,that is,this model nests the popular multivariate volatility model and GAS model,but also systematically studies the dynamic characteristics and market behavior of multi-asset and multi-market moments in the financial market,enriching the theoretical literature of time-varying high-order moment dynamic modeling research.(2)By constructing and adopting a distribution structure that can more reasonably reflect the data form,this paper provides a model basis for systematic modeling.On the one hand,based on previous research,this paper constructs a univariate skew student t distribution with time-varying variance and time-varying distribution parameters(skewness parameters and kurtosis parameters),and a multivariate skew student t distribution with time-varying covariance matrix and time-varying distribution parameters(skewness parameters and kurtosis parameters).This kind of distribution can not only reflect the peak,fat-tail and asymmetric characteristics of the sample data,but also reflect the dynamic nature of these characteristics,which are particularly obvious in the financial market.On the other hand,with the availability and extensiveness of high-frequency data in the financial market,in order to not only quickly capture the change of volatility without losing the effectiveness of the model,this paper incorporates the dynamic modeling of time-varying high-order moments into the background of high-frequency data.Meanwhile,for the realized variance measure,a flexible distribution form,that is,a Chisquare distribution and a F distribution are adopted,and for the realized covariance matrix measure,a more flexible matrix distribution form,namely the Wishart distribution and the Matrix F distribution,are also adopted.This distribution can effectively reflect the fat-tailed characteristics of the realized variance measure and the realized covariance matrix measure.Through modeling research,it is found that through the different parameters settings in the above distribution combinations,the time-varying high-order moment dynamic model of this paper can nest the current popular volatility model and GAS model,which makes the time-varying high-order moment dynamic model not only reflect the distribution structure of the data more reasonably,but also make the model have generalized form.This provides a model distribution basis for systematic modeling in this paper.(3)This paper enriches the theoretical and application research of timevarying high-order moment dynamic modeling.On the one hand,in the process of time-varying high-order moment dynamic modeling,this paper uses the GAS model method to set the score-driven dynamic structure of time-varying variance,time-varying covariance matrix,and time-varying distribution parameters(skewness parameters and kurtosis parameters),and derives their respective score functions and score matrices through matrix derivation.At the same time,this paper proves the consistency and asymptotic normality of model parameters,and especially extends the proof that the model asymptotic meets the conditions in the presence of multiple time-varying parameters.This not only generalizes the theoretical results to the situation with the multiple time-varying parameters,but also generalizes the results of the univariate GAS model to the multivariate situation,which provides some theoretical references for the subsequent research of similar modeling methods.On the other hand,based on the high-frequency data of the China Shanghai Stock Exchange 50 constituents and the Dow Jones Industrial Average in the United States,this paper conducts dynamic modeling of time-varying high-order moments,and analyzes the time-varying high-order moment characteristics of the two stock markets.The empirical results show that,first,whether it is a constituent stock of the Shanghai Stock Exchange in China or a constituent stock of the Dow Jones Industrial Average in the United States,the return not only has the characteristics of volatility aggregation,but also has significant time-varying high-order moment characteristics.Second,compared with the static highorder moment model,the univariate time-varying high-frequency high-order moment dynamic model based on the GAS model has better performance not only in-sample but also out-of-sample.Third,the study found that compared with the current more classical volatility model and GAS model,the generalized time-varying high-frequency high-order moment dynamic model based on the GAS model also has better in-sample and out-of-sample performance.These research conclusions not only provide some scientific decision-making references for financial market investors and supervisors,but also take some references for the promotion of time-varying high-order moment dynamic modeling research.