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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A FLEXIBLE COEFFICIENT GARCH(1,1) MODEL

Published online by Cambridge University Press:  01 February 2009

Marcelo C. Medeiros and
Alvaro Veiga
Marcelo C. Medeiros*
Affiliation:
Pontifical Catholic University of Rio de Janeiro
Alvaro Veiga
Affiliation:
Pontifical Catholic University of Rio de Janeiro
*
*Address correspondence to Marcelo C. Medeiros, Department of Economics, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil, e-mail: mcm@econ.puc-rio.br.
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Abstract

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In this paper a flexible multiple regime GARCH(1,1)-type model is developed to describe the sign and size asymmetries and intermittent dynamics in financial volatility. The results of the paper are important to other nonlinear GARCH models. The proposed model nests some of the previous specifications found in the literature and has the following advantages. First, contrary to most of the previous models, more than two limiting regimes are possible, and the number of regimes is determined by a simple sequence of tests that circumvents identification problems that are usually found in nonlinear time series models. The second advantage is that the novel stationarity restriction on the parameters is relatively weak, thereby allowing for rich dynamics. It is shown that the model may have explosive regimes but can still be strictly stationary and ergodic. A simulation experiment shows that the proposed model can generate series with high kurtosis and low first-order autocorrelation of the squared observations and exhibit the so-called Taylor effect, even with Gaussian errors. Estimation of the parameters is addressed, and the asymptotic properties of the quasi-maximum likelihood estimator are derived under weak conditions. A Monte-Carlo experiment is designed to evaluate the finite-sample properties of the sequence of tests. Empirical examples are also considered.

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Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 1 , February 2009, pp. 117 - 161
Copyright
Copyright © Cambridge University Press 2009

References

REFERENCES

Anderson, H.M., Nam, K., & Vahid, F. (1999) Asymmetric nonlinear smooth transition GARCH models. In Rothman, P. (ed.) Nonlinear Time Series Analysis of Economic and Financial Data, pp. 191–207. Kluwer.CrossRefGoogle Scholar
Audrino, F. & Bühlmann, P. (2001) Tree-structured GARCH models. Journal of the Royal Statistical Society, Series B 63, 727–744.CrossRefGoogle Scholar
Audrino, F. & Trojani, F. (2006) Estimating and predicting multivariate volatility regimes in global stock markets. Journal of Applied Econometrics 21, 345–369.CrossRefGoogle Scholar
Bacon, D.W. & Watts, D.G. (1971) Estimating the transition between two intersecting lines. Biometrika 58, 525–534.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Black, F. (1976) Studies in stock price volatility changes. ASA Proceedings, Business and Economic Statistics, pp. 177–181.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 21, 307–328.CrossRefGoogle Scholar
Boussama, F. (2000) Normalité asymptotique de l'estimateur du pseudo-maximum de vraisemblance d’un modéle GARCH. Comptes Rendus de l’Academie des Sciences, Série I 331, 81–84.Google Scholar
Cai, J. (1994) A Markov model of switching-regime ARCH. Journal of Business & Economic Statistics 12, 309–316.Google Scholar
Carnero, M.A., Peña, D., & Ruiz, E. (2004) Persistence and kurtosis in GARCH and stochastic volatility models. Journal of Financial Econometrics 2, 319–342.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Chan, K.S. & Tong, H. (1986) On estimating thresholds in autoregressive models. Journal of Time Series Analysis 7, 179–190.CrossRefGoogle Scholar
Chen, C.W.S., Chiang, T.C., & So, M.K.P. (2003) Asymmetrical reaction to US stock-return news: Evidence from major stock markets based on a double-threshold model. Journal of Economics and Business 55, 487–502.CrossRefGoogle Scholar
Christoffersen, P.F. (1998) Evaluating interval forecasts. International Economic Review 39, 841–862.CrossRefGoogle Scholar
Comte, F. & Lieberman, O. (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.CrossRefGoogle Scholar
Ding, Z., Granger, C.W.J., & Engle, R.F. (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
Engle, R.F. & Ng, V.K. (1993) Measuring and testing the impact of news on volatility. Journal of Finance 48, 1749–1778.CrossRefGoogle Scholar
Fornari, F. & Mele, A. (1997) Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 49–65.3.0.CO;2-6>CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed processes. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Glosten, L., Jagannanthan, R., & Runkle, R. (1993) On the relationship between the expected value and the volatility of the nominal excess returns on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
Godfrey, L.G. (1988) Misspecification Tests in Econometrics, 2nd ed.Econometric Society Monographs 16. Cambridge University Press.Google Scholar
Goetzmann, W.N., Ibbotson, R.G., & Peng, L. (2001) A new historical database for the NYSE 1815 to 1925: Performance and predictability. Journal of Financial Markets 4, 1–32.CrossRefGoogle Scholar
Gonzalez-Rivera, G. (1998) Smooth transition GARCH models. Studies in Nonlinear Dynamics and Econometrics 3, 61–78.Google Scholar
Gourieroux, C. & Monfort, A. (1995) Statistics and Econometric Models, vol. 2. Cambridge University Press.Google Scholar
Granger, C.W.J. & Ding, Z. (1995) Some properties of absolute returns: An alternative measure of risk. Annales d’Economie et de Statistique 40, 67–95.CrossRefGoogle Scholar
Hagerud, G.E. (1997) A new non-Linear GARCH model. Ph.D. Dissertation, Stockholm School of Economics.Google Scholar
Hamilton, J.D. & Susmel, R. (1994) Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307–333.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, P.R. (2005) A test for superior predictive ability. Journal of Business & Economic Statistics 23, 365–380.CrossRefGoogle Scholar
He, C. & Teräsvirta, T. (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.CrossRefGoogle Scholar
Hwang, J.T.G. & Ding, A.A. (1997) Prediction intervals for artificial neural networks. Journal of the American Statistical Association 92, 109–125.CrossRefGoogle Scholar
Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.CrossRefGoogle Scholar
Jensen, S.T. & Rahbek, A. (2004) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.CrossRefGoogle Scholar
Lanne, M. & Saikkonen, P. (2005) Nonlinear GARCH models for highly persistent volatility. Econometrics Journal 8, 251–276.CrossRefGoogle Scholar
Lee, S.-W. & Hansen, B.E. (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Li, C.W. & Li, W.K. (1996) On a double threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics 11, 253–274.3.0.CO;2-8>CrossRefGoogle Scholar
Li, W.K., Ling, S., & McAleer, M. (2002) Recent theoretical results for time series models with GARCH errors. Journal of Economic Surveys 16, 245–269.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2002) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109–117.CrossRefGoogle Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310.CrossRefGoogle Scholar
Liu, J., Li, W.K. & Li, C.W. (1997) On a threshold autoregression with conditional heteroscedastic variances. Journal of Statistical Planning and Inference 62, 279–300.CrossRefGoogle Scholar
Lumsdaine, R. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575–596.CrossRefGoogle Scholar
Lundbergh, S. & Teräsvirta, T. (2002) Evaluating GARCH models. Journal of Econometrics 110, 417–435.CrossRefGoogle Scholar
Luukkonen, R., Saikkonen, P., & Teräsvirta, T. (1988) Testing linearity against smooth transition autoregressive models. Biometrika 75, 491–499.CrossRefGoogle Scholar
Malmsten, H. & Teräsvirta, T. (2004) Stylized Facts of Financial Time Series and Three Popular Models of Volatility. Working Paper Series in Economics and Finance 563, Stockholm School of Economics.Google Scholar
McAleer, M. (2005) Automated inference and learning in modeling financial volatility. Econometric Theory 21, 232–261.CrossRefGoogle Scholar
McAleer, M., Chan, F., & Marinova, D. (2007) An econometric analysis of asymmetric volatility: Theory and application to patents. Journal of Econometrics 139, 259–284.CrossRefGoogle Scholar
Medeiros, M.C., Teräsvirta, T., & Rech, G. (2006) Building neural network models for time series: A statistical approach. Journal of Forecasting 25, 49–75.CrossRefGoogle Scholar
Medeiros, M.C. & Veiga, A. (2005) A flexible coefficient smooth transition time series model. IEEE Transactions on Neural Networks 16, 97–113.CrossRefGoogle ScholarPubMed
Meitz, M. (2005) A necessary and sufficient condition for the strict stationarity of a family of GARCH processes. Econometric Theory 22, 985–988.Google Scholar
Meitz, M. & Saikkonen, P. (2004) Ergodicity, Mixing, and Existence of Moments of a Class of Markov Models with Applications to GARCH and ACD Models. Working Paper Series in Economics and Finance 573, Stockholm School of Economics.Google Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Newey, W. & McFadden, D. (1994) Large sample estimation and hypothesis testing. In Engle, R.F. and McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2111–2245. Elsevier Science.Google Scholar
Rabemananjara, R. & Zakoian, J.M. (1993) Threshold ARCH models and asymmetries in volatility. Journal of Applied Econometrics 8, 31–49.CrossRefGoogle Scholar
Saikkonen, P. & Choi, I. (2004) Cointegrating smooth transition regressions. Econometric Theory 20, 301–340.CrossRefGoogle Scholar
Scharth, M. & Medeiros, M.C. (2006) Asymmetric Effects and Long Memory in the Volatility of Dow Jones Stocks. Discussion paper 532, Pontifical Catholic University of Rio de Janeiro.Google Scholar
Schwert, G.W. (1990) Stock volatility and the crash of ‘87. Review of Financial Studies 3, 77–102.CrossRefGoogle Scholar
Stout, W.F. (1974) Almost Sure Convergence Academic Press.Google Scholar
Straumann, D. & Mikosch, T. (2006) Quasi-MLE in heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics 34, 2449–2495.CrossRefGoogle Scholar
Sussman, H.J. (1992) Uniqueness of the weights for minimal feedforward nets with a given input-output map. Neural Networks 5, 589–593.CrossRefGoogle Scholar
Teräsvirta, T. (1994) Specification, estimation, and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association 89, 208–218.Google Scholar
Teräsvirta, T. & Mellin, I. (1986) Model selection criteria and model selection tests in regression models. Scandinavian Journal of Statistics 13, 159–171.Google Scholar
Trapletti, A., Leisch, F., & Hornik, K. (2000) Stationary and integrated autoregressive neural network processes. Neural Computation 12, 2427–2450.CrossRefGoogle ScholarPubMed
van Dijk, D. & Franses, P.H. (1999) Modelling multiple regimes in the business cycle. Macroeconomic Dynamics 3, 311–340.CrossRefGoogle Scholar
van Dijk, D., Franses, P.H., & Lucas, A. (1999a) Testing for ARCH in the presence of additive outliers. Journal of Business & Economic Statistics 17, 217–235.Google Scholar
van Dijk, D., Franses, P.H., & Lucas, A. (1999b) Testing for smooth transition nonlinearity in the presence of outliers. Journal of Applied Econometrics 14, 539–562.3.0.CO;2-W>CrossRefGoogle Scholar
White, H. (1994) Estimation, Inference and Specification Analysis. Cambridge University Press.CrossRefGoogle Scholar
White, H. (2001) Asymptotic Theory for Econometricians. Academic Press.Google Scholar
Wooldridge, J.M. (1990) A unified approach to robust, regression-based specification tests. Econometric Theory 6, 17–43.CrossRefGoogle Scholar
Wooldridge, J.M. (1994) Estimation and inference for dependent process. In Engle, R.F. & McFadden, D.L. (eds.) Handbook of Econometrics, vol. 4, pp. 2639–2738. Elsevier Science.Google Scholar
Zakoïan, J.M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar