Risk-parameter estimation in volatility models

This paper introduces the concept of risk parameter in conditional volatility models of the form epsilon(t) = sigma(t)(theta(0))eta(t) and develops statistical procedures to estimate this parameter. For a given risk measurer, the risk parameter is expressed as a function of the volatility coefficients theta(0) and the risk, r (eta(t)), 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. (C) 2014 Elsevier B.V. All rights reserved.