| Markowitz’s mean-variance portfolio model pioneered modern portfolio theory.By studying the optimal allocation of assets for rational investors under uncertain risk levels and expected returns,it can help investors and market supervisors’ decisions provide a scientific basis.However,with the deepening of financial time series theoretical research and the continuous development of financial empirical research,people gradually realize the drawbacks of the meanvariance portfolio framework.On the one hand,the normal return assumption and quadratic expected utility function of the mean-variance portfolios are difficult to achieve,which makes investors face serious welfare losses.Scott and Horvath(1980)pointed out that when the portfolio return distribution is asymmetric or thick-tailed,investors cannot use the quadratic utility function to describe the complete distribution form,and investors must consider the higher-order moments of the portfolio return(such as skewness and kurtosis)in order to avoid a larger welfare loss.On the other hand,the classical mean-variance portfolios usually assumes that the covariance matrix of the assets is constant,and this static portfolio may expose investors to severe wealth loss in the face of short-term large fluctuations in the market.Compared with static portfolios,a large number of theoretical and empirical studies have shown that dynamic portfolios constructed based on conditional expectations and conditional covariance matrices can significantly improve the out-of-sample performance of portfolios.In order to solve these two important defects of the classic mean-variance portfolios,firstly,the highorder moment characteristics of the asset need to be considered to better approximate the investor’s utility function,secondly,the multivariate dynamic structure of the asset needs to be considered to obtain a better out-of-sample performance.The dynamic high-order moment portfolio can effectively solve the existing drawbacks of the classical mean-variance portfolios,and it is also an inevitable trend of quantitative economic modeling of the portfolio.However,its application faces a lot of challenges.First,the application of dynamic higher-order moment portfolio faces the problem of "estimation difficulties".Since the dynamic higherorder moment portfolio inevitably needs to estimate the conditional coherent higher-order moments of the assets,"the curse of dimensionality" problem makes it difficult to estimate.Some scholars have proposed that highly structured modeling solves “the curse of dimensionality” to a certain extent,but the problem of "model misspecification" is equally serious.Therefore,how to effectively reduce the impact of "the curse of dimensionality" and at the same time take into account the problem of "model misspecification " has become the focus and difficulty of solving the "estimation difficulties".Secondly,the application of dynamic highorder moment portfolio faces the problem of "difficulty in optimization".The solution problem for higher-order moment portfolios is usually non-convex optimization,which is an NP-hard problem.At the same time,the optimization of the gradient function and the calculation of the Hesse matrix also face “the curse of dimensionality” problem,which is difficult to calculate or impossible to calculate when there are many assets,which makes the high-dimensional dynamic high-order moment portfolio impossible to achieve.Therefore,how to establish a set of optimization theoretical framework suitable for high-dimensional dynamic highorder moment portfolio has become the focus and difficulty of solving "optimization difficulties".This paper breaks through the bottlenecks of "estimation difficulties" and "optimization difficulties" faced by existing high-dimensional dynamic high-order moment portfolios through factor models.By systematically combing and summarizing the research results of the existing literature,aiming at the shortcomings and problems in the research,the high-dimensional high-order moment portfolio optimization theory and the dynamic higher-order co-moment estimation theory are improved and supplemented.The specific research contents of this paper are as follows:(1)This paper establishes a high-dimensional and high-order moment portfolio optimization theory based on factor models.By considering the properties of portfolio optimization when the higher-order co-moments have a factor structure,the mathematical expressions of the higher-order moment gradient function and Hesser matrix based on the factor model are obtained,and the optimization complexity of the high-order moment portfolio based on the factor model is obtained.Spend.Furthermore,based on the single-factor structure and the multifactor structure,the assumptions required for the optimal convexity of high-order moment portfolios are discussed,which provides an optimization theoretical basis for solving high-dimensional dynamic high-order moment portfolios.(2)This paper combines the single-factor pricing model and the time-varying semi-parametric distribution modeling,and proposes a high-dimensional timevarying higher-order co-moment modeling based on the single-factor semiparametric distribution,namely the single-factor time-varying semi-nonparametric(SF-TVSNP)Model.On the basis of assuming that the return on assets is generated by a single-factor model,a time-varying semi-parametric distribution structure is given to factors and heterogeneity components,so as to realize the estimation of the conditional coherent higher-order moments.On the one hand,the model effectively alleviates “the curse of dimensionality” problem of higher-order co-moment estimation through the single-factor model,and on the other hand,it reduces the possible "model misspecification" and increases the robustness of the estimation by introducing a time-varying semi-parameter distribution.By studying the SFTVSNP model setting,model estimation,conditional coherent higher-order moment estimation and time-varying higher-order moment test methods,a highdimensional time-varying coherent higher-order moment estimation framework based on single-factor semiparametric distribution is established.Finally,using the data of the constituent stocks of the CSI 300 Index,the SF-TVSNP model proposed in this paper is compared with other existing dynamic and static portfolio modeling methods in terms of economic value.(3)Combined with multi-factor pricing model,nonlinear time-varying factor load modeling and time-varying semi-parametric distribution modeling,this paper proposes a high-dimensional time-varying higher-order co-moment modeling based on variable coefficient multi-factor structure,namely varying-coefficient multifactor time-varying semi-nonparametric(VC-MF-TVSNP)model.Considering that the single factor setting of the SF-TVSNP model has time-varying effects of omitted variables and factor loadings,in order to further reduce the impact of "model misspecification" on the estimation of dynamic coherent higher-order moments,it is assumed that the return on assets is generated by a multi-factor model.On the basis of the nonlinear time-varying structure of factor loading and intercept term is given,the semi-parametric multivariate time-varying higher-order moment structure is given to the factor,and the time-varying semi-parametric distribution structure is given to the heterogeneity component,so as to realize the construction of dynamic cooperative higher-order moment.mold.By studying the model setting,model estimation,conditional coherent higher-order moment estimation and model checking methods of VC-MF-TVSNP,a high-dimensional time-varying coherent higher-order moment estimation framework based on variable coefficient and multi-factor structure is established.Finally,using the French industry fund data,the statistical significance and economic value of the VC-MF-TVSNP model proposed in this paper and the SF-TVSNP model,as well as other dynamic and static high-order moment portfolios and mean-variance portfolios,are systematically and comprehensively compared.The research innovations of this paper are mainly reflected in the following three aspects:(1)Based on the factor model,this paper solves the "optimization difficulties" problem caused by "non-convex optimization" and "curse of dimensionality" in portfolio optimization of higher-order moments.The existing higher-order moment portfolio optimization theory and higher-order co-moment estimation are separated.Most optimization algorithms are based on the naive estimation of higher-order comoments,and “the curse of dimensionality” of naive estimation makes it perform well in practical applications.Considering that high-order moment portfolio optimization and higher-order co-moment estimation are an inseparable whole,this paper proposes a set of optimization theories suitable for high-dimensional highorder moment portfolios based on factor models.The main conclusions of the theory are applicable to all satisfies the assumed factor model.By studying the algorithmic complexity and convexity of the high-order moment portfolio optimization based on the factor model,the excellent properties of the high-order moment portfolio optimization based on the factor model under mild conditions are proved.Higher order moment portfolios provide a theoretical basis for optimization.(2)This paper solves the problem of "estimation difficulties" caused by "curse of dimensionality" and "model misspecification" in dynamic higher-order comoment estimation based on factor model combined with semi-parametric method.Due to the high complexity and nonlinearity of higher-order co-moment estimation,most of the existing structural modeling methods focus on parametric static modeling,and the dynamic structure estimation of asset higher-order co-moments is still in its infancy.In this paper,a single-factor time-varying semi-nonparametric(SF-TVSNP)model is constructed by combining the single-factor model with the time-varying semi-parametric distribution,and further considering the single-factor structural omitted variable problem and the time-varying effect of factor loading,the SF-TVSNP model is extended to varying-coefficient single-factor time-varying semi-nonparametric(VC-MF-TVSNP)model.On the one hand,the VC-MFTVSNP model alleviates “the curse of dimensionality” problem to a certain extent based on the multi-factor structure;on the other hand,based on the semi-parametric method,it can well describe the time-varying high-order moment characteristics and nonlinear time-varying characteristics of asset returns.The factor loading feature reduces the risk of "model misspecification".By studying the setting,model estimation,conditional higher-order co-moment estimation and model testing methods of the two models,the theoretical literature on high-dimensional dynamic higher-order co-moment estimation has been enriched.(3)This paper enriches the application research on high-dimensional dynamic high-order moment portfolios.On the one hand,this paper conducts a highdimensional dynamic high-order moment investment analysis based on the constituent stock data of China’s CSI 300 index and the American French industrial fund data,and also analyzes the time-varying high-order moment characteristics and factor structure time-varying characteristics of the stock markets of the two countries.The empirical research results show that: First,both the Chinese market factor and the American Fama-French three factors have significant time-varying high-order moment characteristics,and the time-varying high-order moment characteristics of the heterogeneity component of French industrial funds are common.It shows that it is necessary to model the time-varying coordinative higher-order moments of assets;secondly,through the consistency test of the factor loading function form of the VC-MF-TVSNP model,the results show that the factor loading of the Fama-French three-factor pricing model is There are significant nonlinear time-varying characteristics,and assuming the factor loadings to be constant will face serious model misspecification problems;The investment portfolio of TVSNP and VC-MF-TVSNP model has better out-of-sample performance,all risk indicators are better than existing methods,and can generate higher and more stable additional economic value,while VC-MF-TVSNP model The out-of-sample performance is better than the SF-TVSNP model;fourth,the portfolio based on the SF-TVSNP and VC-MF-TVSNP models has passed the robustness test,and the estimated sample length,portfolio objective function,risk aversion coefficient and other user settings The parameters are not sensitive and have high investment robustness.On the other hand,this paper develops the R package “Semi DMFMC”,which implements the SF-TVSNP model,the VC-MFTVSNP model and other related nested models.By setting the relevant user parameters,this package can complete the relevant model estimation,model The application of test and high-order moment portfolio provides a good application basis for the promotion of high-dimensional dynamic high-order moment portfolio. |
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