Empirical Likelihood Inference For Smooth Quantile Regression GARCH Models

The empirical likelihood estimation method is a commonly used method in nonparametric statistics,and a large number of scholars have used this method in different models,and the conditional heteroscedasticity structure is usually identified as capturing volatility in financial time series.Although generalized autoregressive conditional heteroscedasticity processes have proven to be very successful in modeling financial data,it is generally agreed that it is useful to consider a broader class of processes that are able to more flexibly represent the asymmetry and tail behavior of conditional benefit distributions.The quantile regression estimates the GARCH model discussed in this paper,while the quantile regression estimation for the GARCH model is highly nonlinear.In this paper,an empirical likelihood method is considered to estimate the parameters of quantile regression GARCH models,and an empirical likelihood ratio statistic is constructed.In order to realize the problem of non-differentiability of parameters and high-order refinement,this paper uses the kernel function instead of the empirical likelihood estimation equation of the schematic function,showing that the empirical likelihood estimator of the smooth quantile regression GARCH model is asymptotically equivalent to the standard quantile regression estimation GARCH model estimator,which theoretically proves that the smoothed empirical likelihood ratio statistic obeys the chi-square distribution,and the empirical likelihood method is good in the quantile regression GARCH model through numerical simulation.