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：我们针对期望短缺（ES）的高维模型提出了一种$\ell_1$惩罚估计量。该估计量通过求解一个辅助因变量的最小二乘问题获得，此辅助因变量被定义为原始因变量与预估计尾部分位数的变换函数。利用稀疏性条件，我们推导了ES估计量的预测误差和估计误差非渐近界，其中考虑了因变量中的估计误差，并给出了估计量一致性的充分条件。该估计量适用于厚尾时间序列数据，我们发现模型中参数数量可随样本量以依赖于数据相关性与厚尾性特征的速度增长。在实证应用中，我们以系统性风险测度CoES为研究对象，使用包含一组状态变量非线性变换的回归因子集。结果表明，非线性模型显著优于未使用惩罚及未经变换的基准模型。