Garch models without positivity constraints: exponential or log garch?

Francq, Christian and Wintenberger, Olivier and Zakoian, Jean-Michel (2012): Garch models without positivity constraints: exponential or log garch?

Abstract

This paper studies the probabilistic properties and the estimation of the asymmetric log-GARCH($p,q$) model. In this model, the log-volatility is written as a linear function of past values of the log-squared observations, with coefficients depending on the sign of the observations, and past log-volatility values. Conditions are obtained for the existence of solutions and finiteness of their log-moments. We also study the tail properties of the solution. Under mild assumptions, we show that the quasi-maximum likelihood estimation of the parameters is strongly consistent and asymptotically normal. Simulations illustrating the theoretical results and an application to real financial data are proposed.

• Item Type: MPRA Paper
• Language: English
• Keywords: log-GARCH: Quasi-Maximum Likelihood: Strict stationarity: Tail index
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Allen, D. , Chan, F., McAleer, M., and Peiris, S. (2008) Finite sample properties of the QMLE for the log-ACD model: Application to Australian stocks. Journal of Econometrics 147, 163--183.

Bardet, J.-M. and Wintenberger, O. (2009) Asymptotic normality of the Quasi Maximum Likelihood Estimator for multidimensional causal processes. Ann. Statist. 37, 2730--2759.

Bauwens, L. and Giot, P. (2000) The logarithmic ACD model: An application to the bidask quote process of three NYSE stocks. Annales D'Economie et de Statistique 60, 117--145.

Bauwens, L. and Giot, P. (2003) Asymmetric ACD models: introducing price information in ACD models. Empirical Economics 28, 709--731.

Bauwens, L., Galli, F. and Giot, P. (2008) The moments of log-ACD models. QASS 2, 1--28.

Bauwens, L., Giot, P., Grammig, J. and Veredas, D. (2004) A comparison of financial duration models via density forecast. International Journal of Forecasting 20, 589--604.

Berkes, I., Horv\'ath, L. and Kokoszka, P. (2003) GARCH processes: structure and estimation. Bernoulli 9, 201--227.

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307--327.

Bougerol, P. and Picard, N. (1992a) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714--1729.

Bougerol, P. and Picard, N. (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115--127.

Drost, F.C. and Nijman, T.E. (1993) Temporal Aggregation of GARCH Processes. Econometrica 61, 909--927.

Engle, R.F. (1982) Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U.K. Inflation. Econometrica 50, 987--1008.

Engle, R.F. and Russell, J.R. (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica 66, 1127--1162.

Francq, C. and Zakoïan, J-M. (2004) Maximum Likelihood Estimation of Pure GARCH and ARMA-GARCH. Bernoulli 10, 605--637.

Francq, C. and Zakoïan, J-M. (2010a) GARCH Models : structure, statistical inference and financial applications . John Wiley.

Francq, C. and Zakoïan, J-M. (2010b) Inconsistency of the MLE and inference based on weighted LS for LARCH models. Journal of Econometrics 159, 151--165.

Geweke, J. (1986) Modeling the Persistence of Conditional Variances: A Comment. Econometric Review 5, 57--61.

He, C., Terasvirta, T. and Malmsten, H. (2002) Moment structure of a family of first-order exponential GARCH models. Econometric Theory 18, 868--885.

Mikosch, T. and Rezapour, M. (2012) Stochastic volatility models with possible extremal clustering. Bernoulli To appear.

Mikosch, T. and Starica, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH (1,1) process. Ann. Statist. 28, 1427?1451.

Milhøj, A. (1987) A Multiplicative Parameterization of ARCH Models. Working paper, Department of Statistics, University of Copenhagen.

Nelson D.B. (1991) Conditional Heteroskedasticity in Asset Returns : a New Approach. Econometrica 59, 347--370.

Pantula, S.G. (1986) Modeling the Persistence of Conditional Variances: A Comment. Econometric Review 5, 71--74.

Sucarrat, G. and Escribano, A. (2010) The Power Log-GARCH Model. Working document, Economic Series 10-13, University Carlos III, Madrid.

Sucarrat, G. and Escribano, A. (2012) Automated Model Selection in Finance: General-to-Specific Modelling of the Mean and Volatility Specifications. Oxford Bulletin Of Economics And Statistics 74, 716--735.

Wintenberger, O. and Cai, S. (2011) Parametric inference and forecasting in continuously invertible volatility models. Preprint arXiv:1106.4983v3.