41. Quantile Regression and Value at Risk

This paper studies quantile regression (QR) estimation of Value at Risk (VaR). VaRs estimated by the QR method display some nice properties. In this paper, different QR models in estimating VaRs are introduced. In particular, VaR estimations based on quantile regression of the QAR models, copula models, ARCH models, GARCH models, and the CaViaR models are systematically introduced. Comparing the proposed QR method with traditional methods based on distributional assumptions, the QR method has the important property in that it is robust to non-Gaussian distributions. Quantile estimation is only influenced by the local behavior of the conditional distribution of the response near the specified quantile. As a result, the estimates are not sensitive to outlier observations. Such a property is especially attractive in financial applications since many financial data like, say, portfolio returns (or log returns) are usually not normally distributed. To highlight the importance of the QR method in estimating VaR, we apply the QR techniques to estimate VaRs in International Equity Markets. Numerical evidence indicates that QR is a robust estimation method for VaR.