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Expected Shortfall LASSO

Sander Barendse

TL;DR

A nonasymptotic bound is derived on the prediction and estimator errors of the ES estimator, accounting for the estimation error in the dependent variable, and it is found that the nonlinear model outperforms an unpenalized and untransformed benchmark considerably.

Abstract

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.

Expected Shortfall LASSO

TL;DR

A nonasymptotic bound is derived on the prediction and estimator errors of the ES estimator, accounting for the estimation error in the dependent variable, and it is found that the nonlinear model outperforms an unpenalized and untransformed benchmark considerably.

Abstract

We propose an -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.
Paper Structure (27 sections, 6 theorems, 72 equations, 8 figures, 3 tables)

This paper contains 27 sections, 6 theorems, 72 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. The model
  4. The estimator
  5. Nonasymptotic bound
  6. Asymptotics
  7. Quantile bound
  8. Empirical study: Systemic risk of the banking sector
  9. Systemic risk measurement by CoES
  10. Model and estimation
  11. Data, sample split, and estimation procedure
  12. High-dimensional features
  13. Estimation results
  14. Simulation study
  15. Simulation design
  16. ...and 12 more sections

Key Result

Lemma 2.1

Let Conditions cond:Variances and cond:REC be satisfied. On $\mathcal{T} \cap \mathcal{S} \cap \mathcal{U}$, with $\lambda_0 \leq \frac{6C_0 - 10}{15(1+C_0)} \lambda$ and $\lambda_1 \leq \frac{\phi_0^2}{2s_0 (1+C_0)^2}$, there exists some universal constant $C>0$ such that

Figures (8)

  • Figure 1: $\Delta$CoES predictions. This plot shows predictions for $K=1$ (small) and $K=3$ (large) models.
  • Figure 2: Weekly market and industry return
  • Figure 3: ES predictions for $R^M_t$ conditional on $(R^I_t,Z_{t-1}')'$ at quantile level $\tau=0.025$. This plot shows predictions for $K=1$ (small) and $K=3$ (large) models.
  • Figure 4: ES predictions for $R^M_t$ conditional on $(R^I_t,Z_{t-1}')'$ at quantile level $\tau=0.5$. This plot shows predictions for $K=1$ (small) and $K=3$ (large) models.
  • Figure 5: VaR predictions for $R^M_t$ conditional on $(R^I_t,Z_{t-1}')'$ at quantile level $\tau=0.025$. This plot shows predictions for $K=1$ (small) and $K=3$ (large) models.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 5.1: $\beta$-mixing
  • Definition 5.2: Blocking strategy
  • Lemma 5.1
  • ...and 3 more