Abstract
In this chapter, we introduce related GARCH processes and two-state Markov switching processes whose parameters are time-varying and are governed by an unobservable random variable, which is modelled by an ergodic Markov chain. We provide the risk measure associated to these dynamics.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
We consider here a portfolio composed by three assets as an illustration. x′ stands for transpose of vector x.
References
Andel, Jiri. 1993. “A time series model with suddenly changing parameters”. Journal of Time Series Analysis 14, no. 2: 111–123.
Andersson, Jonas. 2001. “On the normal inverse Gaussian stochastic volatility model”. Journal of Business & Economic Statistics 19, no. 1: 44–54.
Arvanitis, Stelios, and Antonis Demos. 2004. “Time dependence and moments of a family of time-varying parameter GARCH in mean models”. Journal of Time Series Analysis 25, no. 1: 1–25.
Badescu, Alexandru et al. 2011. “A comparison of pricing kernels for GARCH option pricing with generalized hyperbolic distributions”. International Journal of Theoretical and Applied Finance 14, no. 05: 669–708.
Baillie, R.T., and T. Bollerslev 1992. “Prediction in dynamic models with time dependent conditional variances”. Journal of Econometrics 52, no.1–2: 91–113.
Barndorff-Nielsen, Ole E. 1997. “Normal inverse Gaussian distributions and stochastic volatility modelling”. Scandinavian Journal of Statistics 24, no. 1: 1–13.
Berkes, I., L. Horváth, Piotr Kokoszka. 2003. “GARCH processes: Structure and estimation”. Bernoulli 9, no. 370: 201–227.
Black, Fischer. 1976. “Studies of stock price volatility changes”. In Proceedings of the 1976 meeting of the business and economic statistics section, American Statistical Association, Washington DC, 177–181.
Bollerslev, Tim. 1986. “Generalized autoregressive conditional heteroskedasticity”. Journal of Econometrics 31, no. 3: 307–327.
–. 1987. “A conditionally heteroskedastic time series model for speculative prices and rates of return”. The Review of Economics and Statistics 69, no. 3: 542–547.
Bollerslev, Tim, Robert F Engle, and Jeffrey M Wooldridge. 1988. A capital asset pricing model with time-varying covariances”. Journal of Political Economy 96, no. 1: 116–131.
–. 1990. “Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model”. The Review of Economics and Statistics 72, no. 3: 498–505.
Bougerol, Philippe, and Nico Picard. 1992. “Stationarity of GARCH processes and of some nonnegative time series”. Journal of Econometrics 52, no. 1–2: 115–127.
Breidt, F. Jay and Nan-Jung Hsu. 2002. “A class of nearly long-memory time series models”. International Journal of Forecasting 18, no. 2: 265–281.
Brockwell, Peter J. and Richard A. Davis. 1988. “Simple consistent estimation of the coefficients of a linear filter”. Stochastic Processes and their Applications 28, no. 1: 47–59.
Caporin, Massimiliano, and Michael McAleer. 2006. “Dynamic asymmetric GARCH”. Journal of Financial Econometrics 4, no. 3: 385–412.
Carrasco, Marine, and Xiaohong Chen. 2002. “Mixing and moment properties of various GARCH and stochastic volatility models”. Econometric Theory 18, no. 1: 17–39.
Chan, Kung-Sik. 1993. “Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model”. The Annals of Statistics 21, no. 1: 520–533.
Collet, J., D. Guégan, and P. Valdes. 2003. “How shall we determine the number and the location of the Gegenbauer frequencies”. An Empirical Approach, Note de Recherche IDHE-MORA 2003–09.
Ding, Zhuanxin, Clive WJ Granger, and Robert F Engle. 1993. “A long memory property of stock market returns and a new model”. Journal of Empirical Finance 1, no. 1: 83–106.
Diongue, Abdou Kâ, and Dominique Guégan. 2004. “Estimating parameters of a k-factor GIGARCH process”. Comptes Rendus Mathematique 339, no. 6: 435–440.
Diongue, Abdou Kâ, and Dominique Guégan. 2007. “The stationary seasonal hyperbolic asymmetric power ARCH model”. Statistics & Probability Letters 77, no. 11: 1158–1164.
–. 2008. “Estimation of k-factor GIGARCH process: A Monte Carlo study”. Communications in Statistics-Simulation and Computation 37, no. 10: 2037–2049.
Diongue, Abdou Kâ, Dominique Guégan, and Rodney C Wolff. 2010. “BL-GARCH models with elliptical distributed innovations”. Journal of Statistical Computation and Simulation 80, no. 7: 775–791.
Drost, Feike C, Chris AJ Klaassen, Bas JM Werker, et al. 1997. “Adaptive estimation in time-series models”. The Annals of Statistics 25, no. 2: 786–817.
Dufrénot, Gilles, Dominique Guégan, and Anne Peguin-Feissolle. 2005. “Modelling squared returns using a SETAR model with long-memory dynamics”. Economics Letters 86, no. 2: 237–243.
Engle, Robert F. 1982. “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation”. Econometrica: Journal of the Econometric Society 50, no. 4: 987–1007.
Engel, Robert F. 1990. “Discussion: stock market volatility and the crash”. Review of Financial Studies 3, no. 1: 103–106.
Engle, Robert F, and Gloria Gonzalez-Rivera. 1991. “Semiparametric ARCH models”. Journal of Business & Economic Statistics 9, no. 4: 345–359.
Engle, Robert F, and Kenneth F Kroner. 1995. “Multivariate simultaneous generalized ARCH”. Econometric Theory 11, no. 1: 122–150.
Engle, Robert F, and Victor K Ng. 1993. “Measuring and testing the impact of news on volatility”. The Journal of Finance 48, no. 5: 1749–1778.
Engle, Robert F, David M Lilien, and Russell P Robins. 1987. “Estimating time varying risk premia in the term structure: The ARCH-M model”. Econometrica: Journal of the Econometric Society 55, no. 2: 391–407.
Fernandez Rodriguez, Fernando, Simon Sosvilla Rivero, and Julian Andrada Félix. 1999. “Technical analysis in the Madrid stock exchange”. In FEDEA working paper no. 99-05.
Ferrara, Laurent and Dominique Guégan. 2001. “Forecasting with k-factor Gegenbauer processes: Theory and applications”. Journal of Forecasting 20, no. 8: 581–601.
Forsberg, Lars, and Tim Bollerslev. 2002. “Bridging the gap between the distribution of realized (ECU) volatility and ARCH modelling (of the Euro): The GARCH-NIG model”. Journal of Applied Econometrics 17, no. 5: 535–548.
Francq, Christian, and Jean-Michel Zakïan. 2006. “Mixing properties of a general class of GARCH (1, 1) models without moment assumptions on the observed process”. Econometric Theory 22, no. 5: 815–834.
Francq, Christian and J-M Zakoian. 2001. “Stationarity of multivariate Markov-switching ARMA models”. Journal of Econometrics 102, no. 2: 339–364.
Ghysels, Eric, Andrew C Harvey, and Eric Renault. 1996. “5 Stochastic volatility”. Handbook of Statistics 14: 119–191.
Giraitis, Liudas, and Peter M Robinson. 2001. “Whittle estimation of ARCH models”. Econometric Theory 17, no. 3: 608–631.
Glosten, Lawrence R, Ravi Jagannathan, and David E Runkle. 1993. “On the relation between the expected value and the volatility of the nominal excess return on stocks”. The Journal of Finance 48, no. 5: 1779–1801.
Gonzalez-Rivera, Gloria. 1998. “Smooth-transition GARCH models”. Studies in Nonlinear Dynamics & Econometrics 3, no. 2: 61–78.
Granger, Clive W.J. and Roselyne Joyeux. 1980. “An introduction to long-memory time series models and fractional differencing”. Journal of Time Series Analysis 1, no. 1: 15–29.
Gray, Henry L., Nien-Fan Zhang, and Wayne A. Woodward. 1989. “On generalized fractional processes”. Journal of Time Series Analysis 10, no. 3: 233–257.
Guégan, Dominique. 2003. “A prospective study of the k-factor Gegenbauer processes with heteroscedastic errors and an application to inflation rates”. Finance India 17, no. 1: 165–197.
Guégan, Dominique. 2005. “How can we define the concept of long memory? An econometric survey”. Econometric Reviews 24, no. 2: 113–149.
Haas, Markus, Stefan Mittnik, and Marc S Paolella. 2004. “Mixed normal conditional heteroskedasticity”. Journal of Financial Econometrics 2, no. 2: 211–250.
Hansen, Bruce E. 1994. “Autoregressive conditional density estimation”. International Economic Review 35, no. 3: 705–730.
He, Changli, and Timo Teräsvirta. 1999. “Properties of moments of a family of GARCH processes”. Journal of Econometrics 92, no. 1: 173–192.
He, Changli, Timo Teräsvirta, and Hans Malmsten. 2002. “Moment structure of a family of first-order exponential GARCH models”. Econometric Theory 18, no. 4: 868–885.
Hentschel, Ludger. 1995. “All in the family nesting symmetric and asymmetric GARCH models”. Journal of Financial Economics 39, no. 1: 71–104.
Higgins, Matthew L, and Anil K Bera. 1992. “A class of nonlinear ARCH models”. International Economic Review 33, no. 1: 137–158.
Hosking, Jonathan R.M. 1981. “Fractional differencing”. Biometrika 68, no. 1: 165–176.
James Chu, Chia-Shang. 1995. “Detecting parameter shift in GARCH models”. Econometric Reviews 14, no. 2: 241–266.
Jeganathan, Pradeep. 1995. “Some aspects of asymptotic theory with applications to time series models”. Econometric Theory 11, no. 5: 818–887.
Jensen, Morten B, and Asger Lunde. 2001. The NIG-S&ARCH model: A fat-tailed, stochastic, and autoregressive conditional heteroskedastic volatility model”. The Econometrics Journal 4, no. 2: 319–342.
J.P.Morgan. 1996. “Riskmetrics Technical Document”.
Kesten, Harry. 1973. “Random difference equations and renewal theory for products of random matrices”. Acta Mathematica 131, no. 1: 207–248.
Koul, Hira L, Anton Schick, et al. 1996. “Adaptive estimation in a random coefficient autoregressive model”. The Annals of Statistics 24, no. 3: 1025–1052.
Krolzig, Hans M. 1997. “Markov-switching vector autoregression”. Lecture Notes in Economic and Mathematical Systems. no. 454. Springer-Verlag, New York.
Lee, Sang-Won, and Bruce E Hansen. 1994. “Asymptotic theory for the GARCH (1, 1) quasi-maximum likelihood estimator”. Econometric Theory 10, no. 1: 29–52.
Li, WK, and TK Mak. 1994. “On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity”. Journal of Time Series Analysis 15, no. 6: 627–636.
Lin, SJ, and J Yang. 1999. “Testing shift in financial models with conditional heteroskedasticity: An empirical distribution function approach”. Research Paper 30, University of Technology Sydney. Quantitative Finance Research Group.
Ling, Shiqing, and WK Li. 1997. “On fractionally integrated autoregressive moving-average time series models with conditional heteroscedasticity”. In: Journal of the American Statistical Association 92, no. 439: 1184–1194.
Ling, Shiqing, and Michael McAleer. 2002. “Stationarity and the existence of moments of a family of GARCH processes”. Journal of Econometrics 106, no. 1: 109–117.
–. 2003. “Asymptotic theory for a vector ARMA-GARCH model”. Econometric Theory 19, no. 2: 280–310.
Linton, Oliver. 1993. “Adaptive estimation in ARCH models”. Econometric Theory 9, no. 4: 539–569.
Lubrano, Michel. 2001. “Smooth transition GARCH models: A Bayesian perspective”. Recherches Economiques de Louvain/Louvain Economic Review 67, no. 3: 257–287.
Lumsdaine, Robin L. 1996. “Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH (1, 1) and covariance stationary GARCH (1, 1) models”. Econometrica: Journal of the Econometric Society 64, no. 3: 575–596.
Lundbergh, Stefan, and Timo Teräsvirta. 1999. Modelling economic high-frequency time series. Tech. rep. Tinbergen Institute Discussion Paper.
–. 2002. “Evaluating GARCH models”. Journal of Econometrics 110, no. 2: 417–435.
Miguel, Jesus, and Pilar Olave. 2002. “Adjusting forecast intervals in arch-m models”. Journal of Time Series Analysis 23, no. 5: 587–598.
Milhøj, Anders. 1985. “The moment structure of ARCH processes”. Scandinavian Journal of Statistics 12, no. 4: 281–292.
Nelson, Daniel B. 1990. “Stationarity and persistence in the GARCH (1, 1) model”. Econometric Theory 6, no. 3: 318–334.
–. 1991. “Conditional heteroskedasticity in asset returns: A new approach”. Econometrica: Journal of the Econometric Society 59, no. 2: 347–370.
Nelson, Daniel B, and Charles Q Cao. 1992. “Inequality constraints in the univariate GARCH model”. Journal of Business & Economic Statistics 10, no. 2: 229–235.
Olave, Pilar, and Jesus Miguel. 2001. “The risk premium and volatility in the Spanish Stock Market. A forecasting approach”. Economie Appliquee 54, no. 4: 63–78.
Poskitt, D.S. and Shin-Ho Chung. 1996. “Markov chain models, time series analysis and extreme value theory”. Advances in Applied Probability 28, no. 2: 405–425.
Rabemananjara, Roger, and Jean-Michel Zakoian. 1993. “Threshold ARCH models and asymmetries in volatility”. Journal of Applied Econometrics 8, no. 1: 31–49.
Schreiber, Ulrich. 2000. “German tax reform-an international perspective”. FinanzArchiv/Public Finance Analysis 57, no. 4: 525–541.
Sentana, Enrique, and Gabriele Fiorentini. 2001. “Identification, estimation and testing of conditionally heteroskedastic factor models”. Journal of Econometrics 102, no. 2: 143–164.
Shephard, Neil. 1996. “Statistical aspects of ARCH and stochastic volatility”. Monographs on Statistics and Applied Probability 65: 1–68.
So, Mike KP, WK Li, and K Lam. 2002. “A threshold stochastic volatility model”. Journal of Forecasting 21, no. 7: 473–500.
Starica, Catalin. 2004. “Is GARCH(1,1) as good a model as the Nobel prize accolades would imply?” Econometrics 0411015, University Library of Munich, Germany.
Storti, Giuseppe, and Cosimo Vitale. 2003a. “BL-GARCH models and asymmetries in volatility”. Statistical Methods and Applications 12, no. 1: 19–39.
–. 2003b. “Likelihood inference in BL-GARCH models”. Computational Statistics 18, no. 3: 387–400.
Straumann, Daniel, Thomas Mikosch, et al. 2006. “Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach”. The Annals of Statistics 34, no. 5: 2449–2495.
Taylor, Stephen J. 1986. Modelling financial time series. New York: Wiley.
Timmermann, Allan. 2000. “Moments of Markov switching models”. Journal of Econometrics 96, no. 1: 75–111.
Tse, Yiu Kuen. 2002. “Residual-based diagnostics for conditional heteroscedasticity models”. The Econometrics Journal 5, no. 2: 358–374.
Weiss, Andrew A. 1986. “Asymptotic theory for ARCH models: Estimation and testing”. Econometric Theory 2, no. 1: 107–131.
Yang, Minxian. 2000. “Some properties of vector autoregressive processes with Markov-switching coefficients”. Econometric Theory 16, no. 1: 23–43.
Zakoian, Jean-Michel. 1994. “Threshold heteroskedastic models”. Journal of Economic Dynamics and Control 18, no. 5: 931–955.
Zhang, J. and R.A. Stine. 1999. “Autocovariance structure of Markov regime models and model selection”. Department of Statistics, The Wharton School of Business of the University of Pennsylvania.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Guégan, D., Hassani, B.K. (2019). Risks and Non-Linear Dynamics. In: Risk Measurement. Springer, Cham. https://doi.org/10.1007/978-3-030-02680-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-02680-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-02679-0
Online ISBN: 978-3-030-02680-6
eBook Packages: Economics and FinanceEconomics and Finance (R0)