Study On ALS Estimation For VaR Forecasts Based On MRS-GARCH Model

Compared with quantile based VaR(Value-at-Risk),expectile based VaR is more sensitive to the variable distribution's tail while they have similar definition.In fact,there is a one-toone mapping relationship between quantile and expectile,so it's feasible to use expectile regression models to estimate and forecast conditional VaR which is essentially a quantile of observations.Since the volatility of market is time-varying with heteroscedasticity and in different states for instance bull and bear market,the performance will be different.In that case,I consider using a Markov chain to describe the stochastic environment and GARCH(Generalized Autoregressive Conditional Heteroskedasticity)type models to approximate the heteroscedasticity,which is called MS-GARCH(Markov Regime Switch-Generalized Autoregressive Conditional Heteroskedasticity)model.Some statistical properties like consistency and asymptotic normality for both linear and nonlinear expectile regression models have been investigated during historical studies,so this paper will focus on the performance of nonlinear expectile regression model which is deduced from MS-GARCH model.Through reformulating the original mean-variance model,it's easy to get a location-scale model which is necessary in the parameter estimation of expectile regression model.As for the parameter estimation method for expectile regression,the ALS(Asymmetric Least Squares)method is used in a large range.Take the unknown regime process into consideration,I combine the filtered probability by Hamilton(1989)with ALS loss function using a full probability formula to calculate the final objective function of optimization.After the parameter estimation,one-day ahead VaR forecasts can be deduced through the conditional volatility calculation.Besides theoretical innovation,a simulation study and real data analysis are carried out in order to illustrate and compare different models with different estimation methods where rolling window is used to forecast one-day ahead VaR recursively.As for the measurement of performance,I choose both unconditional coverage backtest from Kupiec(1995)and conditional coverage backtest namely CC test from Christoffersen(1998)and DQ test from Engle and Manganelli(2004)to see which model is the best one with optimal estimator.At the same time,some index like biases and RMSE are used in simulation study to measure the accuracy of different types of models as well.