Inference for conditional value-at-risk of a predictive regression
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 \(\widehat{\mathrm{VaR}}_ x(\alpha)\) under the condition that \(X_t\) are independent and identically distributed random vectors. In this paper the authors derive the asymptotic distribution of the conditional \(\mathrm{VaR}_x(\alpha)\) based on the indicated least squares estimator. Their estimator of the conditional quantile is different from the one obtained via quantile regression proposed by \textit{R. Koenker} and \textit{G. Bassett jun.} [Econometrica 46, 33--50 (1978; Zbl 0373.62038)]. It is shown that the obtained estimator is asymptotically more efficient than the quantile regression estimator in regular cases. The authors generalize the described model assumptions by allowing the predictors in the model to be a stationary sequence, instead of being just an independent sequence and, furthermore, incorporating scenarios where some predictors have infinite variance. They propose an empirical likelihood method for deriving the corresponding confidence intervals via least squares estimation. The performance of the proposed method is demonstrated by a simulation study as well as a real data application.
scientific article; zbMATH DE number 7315917
scientific article