Stationarity Domains of Power Threshold GARCH Time Series with Heavy Tailed Generator Processes
Esmeralda Gonçalves, Joana Leite, Nazare Mendes-Lopes
Dep. Mathematics - FCTUC, CMUC, University of Coimbra, Coimbra, Portugal; Institute of Accounting and Administration of Coimbra, Coimbra, Portugal; Dep. Mathematics - FCTUC, CMUC, University of Coimbra, Coimbra, Portugal

Financial time series present a lot of features (stylised facts) reproduced by the several conditionally heteroscedastic (CH) models that, since Engle (1982), appeared in the literature.

A property of “long memory” in the shocks of the conditional variance was detected in this kind of data and led to the introduction of power GARCH models (Ding and al (1993) and Liu and Brorsen (1995)). The introduction of an exponent delta in the GARCH equation propagates not just conditional moments of order two but, more generally, absolute moments of order delta and, according with Ding and al (1993), the corresponding models incorporate the referred “long memory” property.

We consider here a natural extension of Threshold GARCH processes that takes into account both long memory property and asymmetry in the stochastic volatility. This class of power-transformed and threshold GARCH (delta-TGARGH) models was introduced by Pan, Wang and Tong (2008) with a slightly different parameterization and they called them PTTGARCH models.

The probabilistic structure of these models is analysed by discussing the strict and weak of order r stationarities (r>0) as well as the ergodicity.

Considering particular delta-TGARCH models with Gaussian and non-Gaussian generator processes we discuss and compare the corresponding stationarity domains. As this kind of models is adequate to heavy tailed data we give particular relevance to stable distributions generator processes.

An application of this probabilistic study is developed, bounding the marginal distribution of delta-TGARCH models by functions particularly dependent on generator process distributions. This study is preliminary to the application of these processes on the general study of time series control charts (Goncalves, Leite and Mendes-Lopes, 2010).


Ding, Z., Granger, C.W., Engle, R.F. (1993) A long memory property of stock market returns and a new model, J. Empir. Finance, 1, 83-106.

Engle, R.F. (1982) Autoregressive conditional heterskedasticity with estimates of the variance of the United Kingdom inflation, Econometrica 50, 987-1008.

Goncalves, E., Leite, J. and N. Mendes-Lopes (2010) The ARL of modified Shewhart control charts for conditionally heteroskedastic models (submitted).

Liu, S. and B.W. Brorsen (1995) Maximum likelihood estimation of a GARCH-stable model, Journal of Applied Econometrics, 10, 273-285.

Pan, J., Wang, H. and H. Tong (2008) Estimation and tests for power-transformed and threshold GARCH models, J. Econom., 142, 352-378.

MSC: 62M10, 62G20

Keywords: Power TGARCH models; Strict stationarity; Moments stationarity; Heavy tailed distributions

Biography: Esmeralda Gonçalves is Professor at the Faculty of Sciences and Technology of the University of Coimbra (UC). She is researcher of the Mathematical Center of the UC (CMUC) and Time Series Analysis or, more generally, Stochastic Processes are her main scientific interests.

Her most recent research and publications are dedicated to conditionally heteroskedastic time series models. In this talk Esmeralda Gonçalves presents a probabilistic study on power Threshold GARCH stochastic processes.