Abstract
The aim of this chapter is to give a contribution for the estimation of the law of stationary generalized threshold ARCH (GTARCH) processes. Firstly we present bounds for the marginal distribution of a threshold ARCH process, \(\varepsilon\), with an independent generator process Z, as well as for the laws of finite dimension of the absolute value of this process. The results are illustrated by a simulation study considering several distributions for Z, in particular with different behavior in what concerns the tails’ height, and estimating the distribution function of the model by the empirical one. Secondly, with the same goal we establish a dependence property for strictly stationary GTARCH processes from which, as an application of Berkes and Horváth [Ann. Appl. Probab. 11(2), 789–809 (2001)] results, the convergence in law and the almost sure uniform convergence of the empirical process are obtained.
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Notes
- 1.
If q = 1 we take \(\prod _{t=2}^{q}g(t) = 1.\)
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We thank the referees for the constructive suggestions.
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Gonçalves, E., Mendes-Lopes, N. (2013). Distributional Properties of Generalized Threshold ARCH Models. In: Lita da Silva, J., Caeiro, F., Natário, I., Braumann, C. (eds) Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications. Studies in Theoretical and Applied Statistics(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34904-1_22
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DOI: https://doi.org/10.1007/978-3-642-34904-1_22
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