Application Of Quantile Regression Estimator In Generalized Autoregressive Conditional Heteroskedasticity Model

GARCH(1,1)model is used for analyzing time series.As we all know,time series in economy are difficult to analyze because they always involved with heavy-tailed residuals and volatility clustering.Common statistical software often used quasi-maximum likelihood estimator to estimate corresponding parameters in GARCH(1,1)model but this method can be effective when residuals are normal distributed.For time series that have heavy-tailed residuals,we need to find other method to estimate their parameters.This paper mainly discusses the application of quantile regression estimator in time series analysis.In order to use this method,first we need to transform the original data.Then the inspiration of using quantile regression in GARCH(1,1)model resembles using linear regression in the similar situation.However,there are many differences between two regression methods.When using quantile regression estimator,we don't need to minimize the sum of errors.Instead,we focus on the minimization of loss functions because loss function is closely related to the quantiles of the data.In order to minimize the loss function,we need to use a two-stage estimation method.In the first stage,we take advantage of the classical quasi-maximum likelihood estimator and replace an unknown parameter with an estimated parameter.In the second stage,we can use statistical software to calculate the exact quantile regression estimator.This paper also discusses the consistency and asymptotic properties of quantile regression estimator,after that two practical applications of this estimator are presented.One application is to predict the value at risk that is important in economy,another application is to hypothesis the correlation of different parameters.