Bootstrapping Nonparametric Prediction Intervals for Conditional Value-at-Risk with Heteroscedasticity

Emmanuel Torsen,Lema Logamou Seknewna

doi:10.1155/2019/7691841

Abstract

Using bootstrap method, we have constructed nonparametric prediction intervals for Conditional Value-at-Risk for returns that admit a heteroscedastic location-scale model where the location and scale functions are smooth, and the function of the error term is unknown and is assumed to be uncorrelated to the independent variable. The prediction interval performs well for large sample sizes and is relatively small, which is consistent with what is obtainable in the literature.

Highlights

The field of prediction intervals (PIs) has spanned for so many decades and could be traced back to the work of Baker in 1935
A single future observation from a population that is contained in a predictive interval is usually specified with a coverage probability; such a future observation is assumed to have a particular distribution in statistical modeling
The authors in [5] examined the problem of constructing Nonparametric Predictive Intervals (NPIs) for a future sample median, where asymptotic relative efficiency was performed to compare the NPIs and their parametric prediction intervals (PPIs) counterpart; the results showed NPIs to have better properties

Summary

Introduction

The field of prediction intervals (PIs) has spanned for so many decades and could be traced back to the work of Baker in 1935 (see [1] for details). Reference [3] studied the problem of constructing NPIs for future observations for mixed linear models, what the authors therein called distribution-free PIs in mixed linear models. In this paper, we have considered the problem of constructing NPIs for Conditional Value-at-Risk (CVaR) which admit a location-scale model with heteroscedasticity, when the distribution of the innovation is assumed to be unknown using bootstrap method. Consider the function estimator Ĉ VaR(x)τ in (5); the asymptotic distribution of a pivotal quantity is used to construct confidence intervals (CIs). With the proposed estimator of the residual distribution (23) by [12] and the estimators ((12) and (20)) of the mean and variance functions with their asymptotic properties, respectively, and the asymptotic normal distribution of (6), one can obtain NPIs for CVaR.

Simulation Study

Conclusion