We propose an $\ell_1$-penalized estimator for high-dimensional models of Expected Shortfall (ES). The estimator is obtained as the solution to a least-squares problem for an auxiliary dependent variable, which is defined as a transformation of the dependent variable and a pre-estimated tail quantile. Leveraging a sparsity condition, we derive a nonasymptotic bound on the prediction and estimator errors of the ES estimator, accounting for the estimation error in the dependent variable, and provide conditions under which the estimator is consistent. Our estimator is applicable to heavy-tailed time-series data and we find that the amount of parameters in the model may grow with the sample size at a rate that depends on the dependence and heavy-tailedness in the data. In an empirical application, we consider the systemic risk measure CoES and consider a set of regressors that consists of nonlinear transformations of a set of state variables. We find that the nonlinear model outperforms an unpenalized and untransformed benchmark considerably.

翻译：本文提出了一种针对高维预期亏损模型(Expected Shortfall, ES)的ℓ₁惩罚估计量。该估计量通过求解辅助因变量的最小二乘问题得到，其中辅助因变量定义为原始因变量的变换形式与预估计尾部分位数的组合。基于稀疏性条件，我们推导了ES估计量预测误差和估计误差的非渐近界，同时考虑了因变量估计误差的影响，并给出了估计量一致性成立的条件。该估计量适用于厚尾时间序列数据，且研究发现模型中参数数量可随样本量增长，其增长速率取决于数据的依赖性和厚尾特征。在实证应用中，我们考虑系统性风险测度CoES，并采用一组由状态变量非线性变换构成的回归因子。结果表明，非线性模型在性能上显著优于未经惩罚和未经变换的基准模型。