Julio A. Afonso Rodriguez, M.B.A., Institutional Economics, Economic Statistics, and Econometrics, UNIVERSITY OF LA LAGUNA & UNIVERSITARY INSTITUTE OF REGIONAL DEVELOPEMENT (IUDR), CAMINO LA HORNERA, S/N. SCHOOL OF ECONOMICS AND BUSINESS ADMINISTRATION. CAMPUS DE GUAJARA, LA Laguna, 38071, Spain
GARCH-type processes and models have become a very useful mechanism to represent and describe multiple characteristics of a great variety of high frequency financial time series. The kurtosis of the marginal distribution of the GARCH process is always greater than or equal to the kurtosis of the innovations or driving noise sequence. Recently, there has been presented formal proofs that the marginal and finite-dimensional strictly stationary distributions of several GARCH process are regularly varying with a negative index of regular variation, and thus belongs to the class of heavy-tailed distributions of the Pareto-type.
The main characteristic of such type of distributions is the so called tail index parameter, characterizing the tail behaviour of the distribution function and determining the maximum number of finite moments of the distribution. It is also crucial in determining the appropriate distributional and probability limits of the sample autocorrelations and quasi-maximum likelihood estimators of the GARCH parameters.
Because the Pareto-type representation of this stationary distribution is only a first-order approximate result, we consider a second-order regularly varying representation of a heavy-tailed distribution where the leading term is a second-order shape parameter (r<0) that allows to evaluate the rate of convergence to the first-order representation. This representation forms a generalization of the Pareto model know as the Hall model that imposes a second-order condition on the asymptotic behavior of the tail of the distribution function and at the same time robustifies the approach against deviations from the exact Pareto tail. The model holds for distributions such as the Fréchet, Burr and the Student-T. The smaller |r|, the slower is the slow varying part of the model. That is, as |r| increases, the rate of convergence in the first order approximation increases as well. This is also important to get more refined approximations to some characteristics of the process as extreme quantiles and Value-at-Risk measures.
For iid samples there are several different proposals to estimate this parameter, mainly related with the determination of the best threshold for Hill-type estimators of the tail index parameter. The existing estimators can be classified into three categories: (a) adaptive estimators, (b) semi-parametric estimators, and (c) maximum-likelihood (ML) based estimators.
In this paper we explore through simulation experiments the performance of some of such estimators when applied to dependent strictly stationary GARCH process under different distributions for the innovations, thin and heavy-tailed. For many of these distributions we can have the theoretical value of this parameter as a way to evaluate the performance of estimators when applied both to the innovations and GARCH processes. Preliminary numerical results indicate that the very flexible semi-parametric estimator of Fraga Alves et.al. (2003), when based on the GARCH(1,1) process performs as well as for the iid driving noise process and could serve as a way to get in practice some insight about possible departures from the pure Pareto tails of the marginal stationary distribution. Finally we consider the estimation of this two shape parameters for real exchange rate daily data.