The Theory Of Quantile Regression And Applications

Quantile regression is a basic tool for estimating conditional quantiles of a response variable Y given a vector of regressors X. It can be used to measure the effect of regressors not only in the center of a distribution, but also in the upper and lower tails. So it has much more advantages than the classical least square regression. The theory of quantile regression, Copula quantile regression, extremal quantiles and applications of quantile regression in many fields are discussed in this paper. The main achievements of this work are listed as follows:1. The basic extreme value theory is introduced, which is the basis of other chapters. We choose logistic distribution and use bivariate excess threshold model and bivariate point process model to measure the tail dependence of Shanghai and Shenzhen Stock market. The results show that the return rates of Shanghai and Shenzhen Stock markets have strong tail dependence and the two models are excellent for application.2. The linear trend of the annual maximum sea level at Fremantle Port, Western Australia, related with time and Southern Oscillation index during 1897-1989 is analyzed by linear conditional quantile regression model. And the result is compared with that of the classical least square regression. The results show that, under different quantiles, the linear trend of the annual maximum sea level related with time and Southern Oscillation Index is different, and quantile regression can provide much more information than the classical least square regression. So it is of great significance for prediction and prevention.3. The theory of Copula quantile regression is studied and the quantile curves of several common Copulas are obtained. The accuracy of quantile regression estimation is shown by simulation research. We choose clayton Copula and use Copula nonlinear conditional quantile regression model to measure the tail area risk dependence in Shanghai and Shenzhen stock markets. And then the result of this approach is compared with the tail dependence measure by extreme value method. The results show that Shanghai and Shenzhen stock markets have different risk dependence under different quantiles and extreme value theory method only focuses on the estimation of tail dependence.4. By studying the estimation method and asymptotic behaviors of extremal quantiles, we apply its behaviors to the research of VaR. The conditional quantile regression model of return rates of Shanghai stock market is established, which describes the trend of rates under extremal quantiles. Conditional VaR in very extreme quantiles is predicated by using extrapolation methods under the proper tail model. Comparison with the prediction of the ordinary quantile regression model is also given. The results show that the tendencies of the two predictions are similar and the value estimated by the extrapolation methods is relatively small.