Quantile Regression in Risk Calibration

Abstract

Quantile regression studies the conditional quantile function Q Y |X (τ ) on X at level τ which satisfies F Y |X  Q Y |X (τ )  = τ , where F Y |X is the conditional CDF of Y given X, ∀τ ∈ (0, 1). Quantile regression allows for a closer inspection of the conditional distribution beyond the conditional moments. This technique is particularly useful in, for example, the Value-at-Risk (VaR) which the Basel accords (2011) require all banks to report, or the "quantile treatment effect" and "conditional stochastic dominance (CSD)" which are economic concepts in measuring the effectiveness of a government policy or a medical treatment.Given its value of applicability, to develop the technique of quantile regression is, however, more challenging than mean regression. It is necessary to be adept with general regression problems and M -estimators; additionally one needs to deal with non-smooth loss functions. In this dissertation, chapter 2 is devoted to empirical risk management during financial crises using quantile regression. Chapter 3 and 4 address the issue of high-dimensionality and the nonparametric technique of quantile regression.Chapter 2 applies nonparametric confidence bands for quantile functions to investigate the tail dependence of stock returns. It is shown that strong nonlinear correlation exists when stock prices drop, confirming the fact that in financial crises, firms are more dependent on each other than when the market is booming. This sheds light on the risk management of counterparty risk.In Chapter 3, motivated by applications in economics like quantile treatment effects, or conditional stochastic dominance, we focus on the construction of confidence corridors for nonparametric multivariate kernel quantile and expectile regression functions. Through an uniform kernel Bahadur representation for M -estimators, strong Gaussian approximation and asymptotic extreme value theory we derive the asymptotic confidence corridor for the nonparametric kernel conditional quantile/expectile functions. We find that the bands for quantile/expectile functions are wide when τ is close to 0 and 1 due to the variance of the estimator. The coverage ratios given by the asymptotic confidence corridors are meager. To deal with this issue, we propose a novel smoothing bootstrap which gives satisfactory coverage ratios while keeping the size of the confidence corridors in a reasonable range. Our method contributes to the differentiation between the "risk reduction CSD" and "potential enhancement CSD", which is not possible by using techniques based on previous research in CSD like Delgado and Escanciano (2013). This differentiation is crucial as the two types of CSD may induce different utility to the government and citizens. After applying our method to the data set from National Supported Work Demonstration, a temporary internship program offered to disadvantaged workers, it is found that this program tends to be "potential enhancement CSD" and it may not help foster the employment of less capable people as much...