Estimation And Application Of Joint Value-at-Risk And Expected Shortfall Models

Value-at-risk(VaR)and expected shortfall(ES)characterize the distribution from the perspective of the quantile of the distribution and the average value below the quantile,which are different from mean,variance,and bias.With the deepening of the research on the distribution characteristics of dependent variables in various disciplines,VaR and ES models have received extensive attention in the fields of finance,economics and biology.Since ES is defined as the average value exceeding VaR,it is not reasonable to model only ES without VaR.In the past five years,scholars have made a series of contributions to joint VaR and ES models.Focusing on the joint modeling of VaR and ES,this dissertation develops joint VaR and ES models for different practical situations,gives the corresponding parameter estimators,and shows the large sample properties of the estimators.First,when modeling the VaR and ES of real data,there are many theoretically supported explanatory factors and statistical models that are candidates for researchers.Traditional model selection methods can only screen out a single model and corresponding explanatory factors.However,the result of model selection may be degraded due to the changing external environment.Aiming at this problem,this dissertation proposes the Jackknife model averaging framework of joint VaR and ES regression,combining the advantages of all potential candidate models,and obtains the corresponding VaR and ES model averaging estimators through the weighted average of the Jackknife weight vectors.Under the premise that the number of candidate models and the dimensionality of explanatory variables are allowed to approach infinity with increasing sample size,the consistency and asymptotic normality of the parameter estimators of each candidate model are proven.After that,it is proven that the convergence speed of the parameter estimators is uniform with respect to different candidate models.Then,we prove that the model averaging estimator of VaR and ES is asymptotically optimal in the sense of minimizing the out-of-sample final prediction error.In addition,through simulation studies and empirical analysis,we confirm that the model averaging estimators of the VaR and ES perform well in both the heteroscedastic and homoscedastic cases.Censored data are widely studied in disciplines such as life sciences,economics,and social sciences.Censored VaR regression,that is,censored quantile regression,has always been an enduring topic in the field of statistics.However,unlike VaR,which only depends on a certain quantile,ES is related to the entire tail of the distribution,which leads to difficulties in estimating ES in the case of censoring.For this reason,this dissertation proposes the regression framework of VaR and ES for censored data,and proposes two different algorithms for estimating the coefficients of VaR and ES models.Based on the statistical theories of Kaplan-Meier estimators and Kaplan-Meier integrals,the two algorithms construct the loss function required for parameter estimation in the censored situation by generating quasi complete data and the redistribution-of-mass methodology.Afterwards,under some regularity conditions,the consistency of the parameter estimators obtained by the two algorithms is proven.The finite-sample performance of the two parameter estimators given in this dissertation is verified by simulation studies involving the covariates that are correlated with error terms or censored variables.Finally,the traditional method of estimating the VaR and ES of time series data based on parametric model needs to model the mean-variance form of the whole distribution first.For example,an ARMA-GARCH model is first constructed,and then the VaR and ES of the time series are calculated based on the model form.However,this method essentially estimates parameters for the purpose of more accurate mean-variance modeling.Therefore,this dissertation proposes VaR and ES estimation methods for a class of location-scale models including the ARMA-GARCH model.Our method estimates the parameters of the location-scale model with the aim of making the VaR and ES estimates more accurate.Moreover,the consistency and asymptotic normality of the parameter estimators are proven in this dissertation.Then,an extensive simulation study and empirical analysis of the ARMA(1,1)-GARCH(1,1)time series data was conducted.The results of simulation studies and empirical analysis through a Diebold-Mariano type test statistic confirm that our method estimates VaR and ES more accurately than the traditional quasi-maximum likelihood estimation method.