Parametric Quantile Regression Using Mixture Distributions: Estimating Value at Risk and Expected Shortfall

Abstract

Classical quantile regression has an asymmetric absolute error loss function with no derivative at the origin, making parameter estimation difficult. By setting the residual to follow an asymmetric Laplace distribution, quantile regression becomes an equivalent parametric model. However, the asymmetric Laplace distribution does not fit the distribution of asset returns, which is characterized by sharp peaks and heavy tails. To better fit the residuals, we design some general flexible error distributions via mixture distributions whose quantile at a given level is zero. We then propose a novel parametric dynamic quantile regression model using these distributions. It has a smooth loss function, which allows us to apply the maximum likelihood approach to estimate parameters easily. We demonstrate the validity of our model through a risk management application. It has the unique advantage of providing the analytical formulae for value at risk and expected shortfall over other risk measure models. An empirical study of stock markets in 25 countries shows that our model outperforms benchmark models in backtesting out-of-sample forecasts of value at risk and expected shortfall.