Stable limits of martingale transforms with application to the estimation of garch parameters

By Thomas Mikosch, D. Straumann. 2006

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In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index � 2 (2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and p n-rates breaks down. This was recently observed by Hall and Yao [Econometrica 71 (2003) 285–317]. It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy-tailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes. It is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than GARCH.

Subfields: Artificial Intelligence, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/cba079ba3b344222c4c4afdcbf4651240b6a9236

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Citations in the corpus

Stationarity, Mixing, Distributional Properties and Moments of GARCH(p, q)-Processes (2009, background): “See also Basrak et al. (2002), Corollary 3.5, and Mikosch and Straumann (2006), Theorem 4.5 and Proposition 4.10.”
Stable limits for sums of dependent infinite variance random variables (2009, unknown): “Various other financial time series models fit into the framework of stochastic recurrence equations, such as the AGARCH and EGARCH models; see e.g. the treatment in Straumann and Mikosch (53) and the lecture notes by Straumann (52).”
Risk-parameter estimation in volatility models (2012, unknown): “…of the QML estimation for volatility parameters has been extensively studied, in particular for the GARCH(p, q) by Berkes, Horváth and Kokoszka (2003) and Francq and Zakoïan (2004), for general models by Mikosch and Straumann (2006), Straumann and Mikosch (2006), Bardet and Wintenberger (2009).”
Break detection in the covariance structure of multivariate time series models (2009, background): “Theoretical results such as limit theorems for the sample autocorrelation function and the quasi-likelihood estimation of the unknown model parameters has for heavy-tailed univariate ARCH and GARCH processes been discussed in [21, 27, 41], among others.”