Risk-parameter estimation in volatility models |
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Authors: | Christian Francq Jean-Michel Zakoïan |
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Institution: | 1. CREST and University Lille 3 (EQUIPPE), BP 60149, 59653 Villeneuve d’Ascq cedex, France;2. EQUIPPE (University Lille 3) and CREST, 15 boulevard Gabriel Péri, 92245 Malakoff Cedex, France |
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Abstract: | This paper introduces the concept of risk parameter in conditional volatility models of the form ?t=σt(θ0)ηt and develops statistical procedures to estimate this parameter. For a given risk measure r, the risk parameter is expressed as a function of the volatility coefficients θ0 and the risk, r(ηt), of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures, as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate the superiority of the one-step approach for financial series. |
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Keywords: | C13 C22 C58 |
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