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Abstract:
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Quantiles and expected shortfalls (ES) are commonly used risk measures in financial risk management. The two measurements are corrected while have distinguish features. In this project we consider joint modeling of conditional quantiles and ES. While the regression coefficients can be estimated jointly by minimizing strictly consistent loss functions, the computation is challenging especially when the dimension of parameters is large since the loss functions are neither differentiable nor convex. To reduce the computational effort, we propose a two-step estimation procedure by estimating the quantile regression parameters first using linear programming. We show that the two-step estimator has the same asymptotic properties as the joint estimator, but the former is numerically more efficient. We further develop score-type inference method for hypothesis testing and confidence interval construction. Compared to the Wald-type method, the score method is robust against heterogeneity and is less sensitive to the choice of bandwidth involved in estimating unknown nuisance parameters. We demonstrate the advantages of the proposed methods over existing approaches through numerical studies.
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