ESTIMATION RISK IN GARCH VaR AND ES ESTIMATES

Gao, Feng; Fu, Song

doi:10.1017/s0266466608080559

Cited by 34 publications

(33 citation statements)

References 30 publications

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“…whereβ 0 = .0, : : : , 0, β 01 , : : : , β 0p / ∈ R p+q+1 and Σ 0 = θ 0β 0 +β 0 θ 0 . If the conditions in theorem 1 hold, we can show that √ n.θ τ n − θ τ 0 / → d N.0, Σ 3 /; see also Gao and Song (2008) and Francq and Zakoïan (2015). In particular, for ARCH models, Σ 1 and Σ 3 reduce to…”

Section: Relationship With Existing Methodsmentioning

confidence: 93%

“…Let

{normalΣ}_{3} = \frac{τ - τ^{2}}{f^{2} false(b_{τ} false)} θ_{0} θ_{0}^{'} + \frac{κ_{1} b_{τ}}{f (b_{τ})} {normalΣ}_{0} + κ_{2} b_{τ}^{2} false(Σ_{0} + J^{- 1} - θ_{0} θ_{0}^{'} false),

where

{\overset{β}{true}}_{0} = {(0, \dots, 0, β_{01}, \dots, β_{0 p})}^{'} \in R^{p + q + 1}

and

{normalΣ}_{0} = θ_{0} {\overset{β}{true}}_{0}^{'} + {\overset{β}{true}}_{0} θ_{0}^{'}

. If the conditions in theorem 1 hold, we can show that

\sqrt n false({\overset{false~}{θ}}_{τ n} - θ_{τ 0} false) {false\to}_{normald} N false(0, Σ_{3} false)

; see also Gao and Song () and Francq and Zakoïan (). In particular, for ARCH models, Σ 1 and Σ 3 reduce to ( τ − τ 2 ) J −1 / f 2 ( b τ ) and

false(τ - τ^{2} false) θ_{0} θ_{0}^{'} / f^{2} false(b_{τ} false) + κ_{2} b_{τ}^{2} false(J^{- 1} - θ_{0} θ_{0}^{'} false)

respectively.…”

Section: Hybrid Conditional Quantile Estimationmentioning

confidence: 99%

See 1 more Smart Citation

Hybrid Quantile Regression Estimation for Time Series Models with Conditional Heteroscedasticity

Zheng

Zhu

et al. 2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Estimating conditional quantiles of financial time series is essential for risk management and many other financial applications. For time series models with conditional heteroscedasticity, although it is the generalized auto‐regressive conditional heteroscedastic (GARCH) model that has the greatest popularity, quantile regression for this model usually gives rise to non‐smooth non‐convex optimization which may hinder its practical feasibility. The paper proposes an easy‐to‐implement hybrid quantile regression estimation procedure for the GARCH model, where we overcome the intractability due to the square‐root form of the conditional quantile function by a simple transformation. The method takes advantage of the efficiency of the GARCH model in modelling the volatility globally as well as the flexibility of quantile regression in fitting quantiles at a specific level. The asymptotic distribution of the estimator is derived and is approximated by a novel mixed bootstrapping procedure. A portmanteau test is further constructed to check the adequacy of fitted conditional quantiles. The finite sample performance of the method is examined by simulation studies, and its advantages over existing methods are illustrated by an empirical application to value‐at‐risk forecasting.

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Section: Relationship With Existing Methodsmentioning

confidence: 93%

“…Let

{normalΣ}_{3} = \frac{τ - τ^{2}}{f^{2} false(b_{τ} false)} θ_{0} θ_{0}^{'} + \frac{κ_{1} b_{τ}}{f (b_{τ})} {normalΣ}_{0} + κ_{2} b_{τ}^{2} false(Σ_{0} + J^{- 1} - θ_{0} θ_{0}^{'} false),

where

{\overset{β}{true}}_{0} = {(0, \dots, 0, β_{01}, \dots, β_{0 p})}^{'} \in R^{p + q + 1}

and

{normalΣ}_{0} = θ_{0} {\overset{β}{true}}_{0}^{'} + {\overset{β}{true}}_{0} θ_{0}^{'}

. If the conditions in theorem 1 hold, we can show that

\sqrt n false({\overset{false~}{θ}}_{τ n} - θ_{τ 0} false) {false\to}_{normald} N false(0, Σ_{3} false)

; see also Gao and Song () and Francq and Zakoïan (). In particular, for ARCH models, Σ 1 and Σ 3 reduce to ( τ − τ 2 ) J −1 / f 2 ( b τ ) and

false(τ - τ^{2} false) θ_{0} θ_{0}^{'} / f^{2} false(b_{τ} false) + κ_{2} b_{τ}^{2} false(J^{- 1} - θ_{0} θ_{0}^{'} false)

respectively.…”

Section: Hybrid Conditional Quantile Estimationmentioning

confidence: 99%

Hybrid Quantile Regression Estimation for Time Series Models with Conditional Heteroscedasticity

Zheng

Zhu

et al. 2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…Also, in the computation of the CVaR, K' and µ replace K and r in Equation (5). The change of measure is equivalent to the introduction of a continuous dividend yield r -µ.…”

Section: Change Of Measurementioning

confidence: 99%

“…See [5,6], for a study of the properties of the filtered historical simulation [7] or alternative estimation methods. Kuester, Mittnik and Paolella [8] provide an extensive comparison of empirical performances of several methods.…”

Section: Introductionmentioning

confidence: 99%

VaR and CVaR Implied in Option Prices

Adesi

2016

JRFM

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VaR (Value at Risk) and CVaR (Conditional Value at Risk) are implied by option prices. Their relationships to option prices are derived initially under the pricing measure. It does not require assumptions about the distribution of portfolio returns. The effects of changes of measure are modest at the short horizons typically used in applications. The computation of CVaR from option price is very convenient, because this measure is not elicitable, making direct comparisons of statistical inferences from market data problematic.

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“…Spierdijk [2014] menciona que algunos de estos métodos fallan cuando el supuesto de normalidad asintótica no se tiene o cuando el tamaño de muestra no es lo suficientemente grande. Chan et al [2007], Gao y Song [2008] y Francq y Zakoïan [2015] emplean diferentes aproximaciones basadas en Quasi Máxima Verosimilitud (QML por sus siglas en inglés) con el objeto de estimar el cuantil de los residuales estandarizados sin hacer uso del supuesto de normalidad asintótica. Gao y Song [2008] usan simulación histórica filtrada, Francq y Zakoïan [2015] proponen una reparametrización de los errores estandarizados del GARCH, mientras Chan et al [2007] se basan en teoría del valor extremo.…”

Section: Introductionunclassified

Comparación de métodos para la estimación de la incertidumbre del valor en riesgo

Gamba-Santamaría¹,

Jaulín-Méndez²,

Melo‐Velandia³

et al. 2016

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Diciembre de 2015, no. 83 COMPARACIÓN DE MÉTODOS PARA LA ESTIMACIÓN DE LA INCERTIDUMBRE DEL VALOR EN RIESGO SANTIAGO GAMBA SANTAMARÍA OSCAR FERNANDO JAULÍN MÉNDEZ † LUIS FERNANDO MELO VELANDIA £ CARLOS ANDRÉS QUICAZÁN MORENO ‡RESUMEN. El Valor en Riesgo (VaR) es una medida de riesgo de mercado ampliamente usada por administradores de riesgo y autoridades regulatorias. Sin embargo, a pesar de que existe una gran variedad de metodologías propuestas en la literatura para la estimación del VaR, pocas de ellas dicen algo acerca de su distribución o sus intervalos de confianza. Este artículo compara distintas metodologías para calcular esos intervalos. Se utilizaron métodos basados en normalidad asintótica, teoría del valor extremo y bootstrap de submuestra. Usando simulaciones de Monte Carlo, se encontró que estas aproximaciones son válidas sólo para cuantiles altos. Particularmente, en términos de porcentaje de cobertura, estas metodologías presentan un buen desempeño para el VaR(99 %) y un bajo desempeño para el VaR(95 %) y el VaR(90 %). En general, estos resultados se confirman a través de un ejercicio empírico aplicado a los bonos de deuda publica colombiana.Palabras clave: Valor en Riesgo, intervalos de confianza, data tilting, bootstrap de submuestra.JEL Codes: C51, C52, C53, G32.ABSTRACT. Value at Risk (VaR) is a market risk measure widely used by risk managers and market regulatory authorities. There is a variety of methodologies proposed in the literature for the estimation of VaR. However, few of them get to say something about its distribution or its confidence intervals. This paper compares different methodologies for computing such intervals. Several methods, based on asymptotic normality, extreme value theory and subsample bootstrap, are implemented. Using Monte Carlo simulations, it is found that these approaches are only valid for high quantiles. Particularly, there is a good performance at VaR(99%), in terms of coverage rates, and bad performance for VaR(95%) and VaR(90%). In general, these results are confirmed by conducting an empirical exercise using Colombian public debt bonds.Las opiniones expresadas aquí pertenecen a los autores y no necesariamente representan las opiniones del Banco de la República o su junta directiva, por lo tanto se eximen de toda responsabilidad. Los autores agradecen a Daniel Mariño y Esteban Gómez por sus comentarios y sugerencias.

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ESTIMATION RISK IN GARCH VaR AND ES ESTIMATES

Cited by 34 publications

References 30 publications

Hybrid Quantile Regression Estimation for Time Series Models with Conditional Heteroscedasticity

Hybrid Quantile Regression Estimation for Time Series Models with Conditional Heteroscedasticity

VaR and CVaR Implied in Option Prices

Comparación de métodos para la estimación de la incertidumbre del valor en riesgo

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