The Research On Modelling Mixed-Frequency Multi-Factor Higher Order Co-Moments

Markowitz’s mean-variance model lays the foundation for modern investment theory.It has got rid of the history of analyzing the porfolios by experience alone,and has a profound impact on the theoretical and empirical research in the financial field.However,a large number of studies have shown that there is a serious welfare loss in a portfolio constructed based on the mean-variance model under the condition that the investor preference is non-quadratic or the asset return rate is non-normal.Therefore,in the face of the objective existence of high-order moment features such as skewness,leptokurtic and thick tails,investors need to consider the impact of higher-order moments for optimizing portfolio.At present,the problem of portfolio optimization based on higher-order moments has been widely concerned by scholars at home and abroad.In general,there are two methods to analyze higher-order moments.The direct method is used by optimizing the function of each moment of the portfolio yield as an objective function to form mean-variance-skewness or mean-variance-skewnesskurtosis portfolio optimization problem.The indirect method is used by maximizing the expected utility function through high-order Taylor expansion on the expected utility.In both above methods,they are inevitably needed to estimate the higherorder moment matrix.When the number of assets is large,the co-skewness and cokurtosis will face the “curse of dimensionality” when we consider the higher-order moment portfolio optimization.In order to solve the sampling error faced by higherorder matrix estimation,a more direct method is to increase the observable sample.However,considering that the Chinese financial market started late,the available low-frequency observation data is insufficient,so in practice there is a reality difficulties.Another method is to apply a structured constraint on each moment matrix by assuming that the return is subject to a specific data generation process.This method greatly reduces the number of parameters to be estimated at the cost of increasing the model misspecification risk.It significantly reduces the estimation error in each moment element and is used as the main means in higher moment estimation.Further,in the selection and setting of the factor model,most scholars simply use a single idex model or use principal component analysis to obtain factors.These two methods either do not theoretically study the real data generation process of the return,or ignore the information implied in the higher moments,so that a more accurate estimation of the higher-order moment cannot be made.Based on the previous researches of previous scholars,based on the mixed frequency factor model method,the use of higher frequency data makes the model contain more historical information and the ability to increase the number of factors to improve the ability to explain return.Attempts to solve the "curse of dimensionality" problem faced by higher-order moment estimation,and further improve the performance of high-order moment portfolio.Specifically,the main research contents of this paper are as follows:First,this paper proposes a method for identifying the optimal number of factors based on the high-order moment modeling of the mixed frequency multifactor model.Under the assumption that the return generated by the mixed frequency multi-factor model,the disturbance term should have independent and identical distribution characteristics,and then the moments obtained by the disturbance term should have significant sparsity.Based on this assumption,the asymptotic distribution of the higher-order moment real distribution and the estimated higher-order moments obtained by using the sample estimation,which ensures the high-order moment matrix sparsity test which can be constructed by the disturbance term.Then we can use it to judge the adequacy of the factor model,and to identify the optimal number of factors.In the construction of specific statistics,this paper proposes the parametric Wald test and the non-parametric Gumbel test respectively.By using a large number of Monte Carlo simulations,the number of optimal factors of the two methods under different dimensions is tested under finite samples.Then we can see the size and power of the two tests.Second,in order to improve the ability of factor model to explain the return,and thus improve the performance of high-order moment portfolio,this paper proposes to use the mixed frequency version of the Fama-French multi-factor model to estimate the each moment.In the selection of the mixed frequency model,the mixed data sampling(MIDAS)model proposed by Ghysels(2004)is used.The model assumes that the actual observable data sampling frequency used as the explanatory variable is not lower than the observed data frequency corresponding to the explanatory variable.Therefore,the information contained in the higher frequency factor is used to improve the ability to interpret return under the condition that the observation sample is limited,thereby reducing the estimation error of the higher order moment matrix.In terms of MIDAS model setting,it is divided into unconstrained mixed data sampling(U-MIDAS)model and restricted mixed data sampling(R-MIDAS)model according to whether the parameters to be restricted,because the frequency difference and the factor variable,the large number of parameters to be estimated in the U-MIDAS model,so this paper will focus on the R-MIDAS model and apply it to the higher-order moment portfolio modeling.Thirdly,using the data of listed companies in China’s A-share market as an example,the high-order moment estimation method established by the mixed frequency multi-factor model is compared with other existing higher-order moment estimation methods from the aspects of statistical significance and economic value..Under the assumption that investors have the same relative risk aversion coefficient,the CRRA type utility function and Taylor expansion to the fourth moment are used,we obtain the different higher moment estimation results by using the differential evolution algorithm.In order to judge the usefulness of mixed frequency multifactor high-order moment modeling from a statistical point of view,this paper uses monetary utility gain(MUG)as a measure,then we randomly select a basket of stocks from the A-share market to build an optimal portfolio and repeat its method.Then we can get the summary by performing 100 times to obtain the MUG description statistics of each method.In addition,considering the relative risk aversion,the sample length actually used in estimating the higher moments and the influence of survivor biased on the higher moment portfolio are verified by a large number of robustness tests.In order to verify the practical value of the mixed frequency multi-factor high-order moment modeling from the economic value,a modified expected shortfall(mES)considering the high-order moment characteristics of the return on assets is used,we randomly draw a basket of stocks from the A-share market,the economic value of the mixed frequency multifactor model and other higher order moment estimation methods are verified.The main innovations of this paper are mainly reflected in the following three aspects:(1)This paper proposes an estimation method that can effectively reduce the sampling error for the "dimension disaster" problem in high-order moment estimation.(2)In this paper,a method for identifying the optimal number of factors based on the perturbation term is proposed for how to select the optimal number of factors when using multi-factor model for high-order moment modeling.(3)This paper introduces the higher-order moment matrix estimated by the mixing model into the higher-order moment portfolio through the fourth-order Taylor series expansion based on the expected utility function,thus constructing a higher-order moment portfolio strategy,and other structures.The comprehensive and meticulous comparison of the modeling methods found that the model has very good economic value.