Inference for conditional value-at-risk of a predictive regression
type: Item
instance of: scholarly article
title: Inference for conditional value-at-risk of a predictive regression
zbMATH Open document ID: 1469.62250
DOI: 10.1214/19-AOS1937
author: Yi He
author: Yanxi Hou
author: Liang Peng
author: Haipeng Shen
published in: The Annals of Statistics
publication date: 2021-02-26T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aos/1607677242
review text: The authors deal with the inference problem of conditional value-at-risk under a linear predictive regression model. Denote by \(Y\) the return of an asset and let \(X = (X_1, \dots, X_k)^{\top}\) be a collection of predictors (market variables or risk factors). Suppose that we have \(n\) observations (data points) \(\{Y_t,X_t\}, t=1,\dots,n\), with \(X_t = (X_{t,1},\dots, X_{t,k})^{\top}\) related to the linear predictive regression \[ Y_t = \beta_0 + \sum_{i=1}^{k}\beta_ iX_{t,i} + \varepsilon_t, \quad t = 1,\dots, n, \] where \(\{\varepsilon_t \}\) is a sequence of i.i.d. random variables with zero mean and finite variance. The conditional VaR of \(Y_t\) at level \(\alpha\in(0,1)\), given \(X_t = x :=(x_1,\dots, x_k)^{\top}\) is determined as the conditional quantile \[ \mathrm{VaR}_x(\alpha) := \inf\{q : P (Y_t \leq q|X_t = x) \geq\alpha\}= F^{-1}_{ \varepsilon } + Z^{\top}\beta, \] where \(\beta = (\beta_0, \beta_1, \dots , \beta_k) ^{\top}\), \( z = (1,x^{\top})^{\top}\), \( F_{ \varepsilon }\) denotes the distribution function of \( \varepsilon_t\) and \(F^{-1}_{ \varepsilon }\) denotes the generalized inverse of \(F_{ \varepsilon }\). A simple estimator for the above conditional VaR is \[ \widehat{\mathrm{VaR}}_ x(\alpha) =\widehat{\varepsilon}_{n,[n\alpha] }+ Z^{\top}\hat{\beta}, \] where \(\hat{\beta}\) is a consistent estimator of \({\beta}\), \(\widehat{\varepsilon}_t=Y_t- \hat{\beta}^{\top}Z_t,\) \( Z_t= (1,X_t^{\top})^{\top}\), and where \(\widehat{\varepsilon}_{n,1}\leq\cdots\leq \widehat{\varepsilon}_{n,n }\) denote the order statistics of \(\widehat{\varepsilon}_{1},\dots, \widehat{\varepsilon}_{n }\). \textit{S. Y. Chun} et al. [Oper. Res. 60, No. 4, 739--756 (2012; Zbl 1260.91121)] considered the least squares estimator \[ \hat{\beta} = \left\{ \frac{1}{n}\sum_{t=1}^{n}Z_tZ_t^{\top} \right\}^{-1} \left\{\frac{1}{n}\sum_{t=1}^{n}Y_tZ_t\right\}, \] and derived the asymptotic distribution of the corresponding \(\widehat{\mathrm{VaR}}_ x(\alpha)\) und…
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G08
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62P20
Mathematics Subject Classification ID: 91G70
zbMATH DE Number: 7315917
zbMATH Keywords: conditional risk measures
zbMATH Keywords: empirical likelihood
zbMATH Keywords: least squares estimation
zbMATH Keywords: interval estimation
zbMATH Keywords: quantile regression
zbMATH Keywords: risk management
zbMATH Keywords: value-at-risk
reviewed by: M. P. Moklyachuk
describes a project that uses: CAViaR
describes a project that uses: QRM
MaRDI profile type: Publication
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OpenAlex ID: W3112224177
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