Risks and Non-Linear Dynamics

  • Dominique Guégan
  • Bertrand K. Hassani


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.


  1. Andel, Jiri. 1993. “A time series model with suddenly changing parameters”. Journal of Time Series Analysis 14, no. 2: 111–123.CrossRefGoogle Scholar
  2. Andersson, Jonas. 2001. “On the normal inverse Gaussian stochastic volatility model”. Journal of Business & Economic Statistics 19, no. 1: 44–54.CrossRefGoogle Scholar
  3. 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.CrossRefGoogle Scholar
  4. 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.CrossRefGoogle Scholar
  5. 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.CrossRefGoogle Scholar
  6. Barndorff-Nielsen, Ole E. 1997. “Normal inverse Gaussian distributions and stochastic volatility modelling”. Scandinavian Journal of Statistics 24, no. 1: 1–13.CrossRefGoogle Scholar
  7. Berkes, I., L. Horváth, Piotr Kokoszka. 2003. “GARCH processes: Structure and estimation”. Bernoulli 9, no. 370: 201–227.CrossRefGoogle Scholar
  8. 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.Google Scholar
  9. Bollerslev, Tim. 1986. “Generalized autoregressive conditional heteroskedasticity”. Journal of Econometrics 31, no. 3: 307–327.CrossRefGoogle Scholar
  10. –. 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.Google Scholar
  11. 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.Google Scholar
  12. –. 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.Google Scholar
  13. 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.CrossRefGoogle Scholar
  14. 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.CrossRefGoogle Scholar
  15. 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.CrossRefGoogle Scholar
  16. Caporin, Massimiliano, and Michael McAleer. 2006. “Dynamic asymmetric GARCH”. Journal of Financial Econometrics 4, no. 3: 385–412.CrossRefGoogle Scholar
  17. Carrasco, Marine, and Xiaohong Chen. 2002. “Mixing and moment properties of various GARCH and stochastic volatility models”. Econometric Theory 18, no. 1: 17–39.CrossRefGoogle Scholar
  18. 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.CrossRefGoogle Scholar
  19. 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.Google Scholar
  20. 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.Google Scholar
  21. Diongue, Abdou Kâ, and Dominique Guégan. 2004. “Estimating parameters of a k-factor GIGARCH process”. Comptes Rendus Mathematique 339, no. 6: 435–440.CrossRefGoogle Scholar
  22. Diongue, Abdou Kâ, and Dominique Guégan. 2007. “The stationary seasonal hyperbolic asymmetric power ARCH model”. Statistics & Probability Letters 77, no. 11: 1158–1164.CrossRefGoogle Scholar
  23. –. 2008. “Estimation of k-factor GIGARCH process: A Monte Carlo study”. Communications in Statistics-Simulation and Computation 37, no. 10: 2037–2049.Google Scholar
  24. 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.Google Scholar
  25. 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.Google Scholar
  26. 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.CrossRefGoogle Scholar
  27. 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.CrossRefGoogle Scholar
  28. Engel, Robert F. 1990. “Discussion: stock market volatility and the crash”. Review of Financial Studies 3, no. 1: 103–106.CrossRefGoogle Scholar
  29. Engle, Robert F, and Gloria Gonzalez-Rivera. 1991. “Semiparametric ARCH models”. Journal of Business & Economic Statistics 9, no. 4: 345–359.Google Scholar
  30. Engle, Robert F, and Kenneth F Kroner. 1995. “Multivariate simultaneous generalized ARCH”. Econometric Theory 11, no. 1: 122–150.Google Scholar
  31. 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.Google Scholar
  32. 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.Google Scholar
  33. 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.Google Scholar
  34. Ferrara, Laurent and Dominique Guégan. 2001. “Forecasting with k-factor Gegenbauer processes: Theory and applications”. Journal of Forecasting 20, no. 8: 581–601.CrossRefGoogle Scholar
  35. 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.CrossRefGoogle Scholar
  36. 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.CrossRefGoogle Scholar
  37. Francq, Christian and J-M Zakoian. 2001. “Stationarity of multivariate Markov-switching ARMA models”. Journal of Econometrics 102, no. 2: 339–364.CrossRefGoogle Scholar
  38. Ghysels, Eric, Andrew C Harvey, and Eric Renault. 1996. “5 Stochastic volatility”. Handbook of Statistics 14: 119–191.Google Scholar
  39. Giraitis, Liudas, and Peter M Robinson. 2001. “Whittle estimation of ARCH models”. Econometric Theory 17, no. 3: 608–631.Google Scholar
  40. 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.Google Scholar
  41. Gonzalez-Rivera, Gloria. 1998. “Smooth-transition GARCH models”. Studies in Nonlinear Dynamics & Econometrics 3, no. 2: 61–78.CrossRefGoogle Scholar
  42. 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.CrossRefGoogle Scholar
  43. 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.CrossRefGoogle Scholar
  44. 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.Google Scholar
  45. Guégan, Dominique. 2005. “How can we define the concept of long memory? An econometric survey”. Econometric Reviews 24, no. 2: 113–149.CrossRefGoogle Scholar
  46. Haas, Markus, Stefan Mittnik, and Marc S Paolella. 2004. “Mixed normal conditional heteroskedasticity”. Journal of Financial Econometrics 2, no. 2: 211–250.Google Scholar
  47. Hansen, Bruce E. 1994. “Autoregressive conditional density estimation”. International Economic Review 35, no. 3: 705–730.CrossRefGoogle Scholar
  48. He, Changli, and Timo Teräsvirta. 1999. “Properties of moments of a family of GARCH processes”. Journal of Econometrics 92, no. 1: 173–192.CrossRefGoogle Scholar
  49. 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.CrossRefGoogle Scholar
  50. Hentschel, Ludger. 1995. “All in the family nesting symmetric and asymmetric GARCH models”. Journal of Financial Economics 39, no. 1: 71–104.CrossRefGoogle Scholar
  51. Higgins, Matthew L, and Anil K Bera. 1992. “A class of nonlinear ARCH models”. International Economic Review 33, no. 1: 137–158.Google Scholar
  52. Hosking, Jonathan R.M. 1981. “Fractional differencing”. Biometrika 68, no. 1: 165–176.CrossRefGoogle Scholar
  53. James Chu, Chia-Shang. 1995. “Detecting parameter shift in GARCH models”. Econometric Reviews 14, no. 2: 241–266.CrossRefGoogle Scholar
  54. Jeganathan, Pradeep. 1995. “Some aspects of asymptotic theory with applications to time series models”. Econometric Theory 11, no. 5: 818–887.CrossRefGoogle Scholar
  55. 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.CrossRefGoogle Scholar
  56. J.P.Morgan. 1996. “Riskmetrics Technical Document”.Google Scholar
  57. Kesten, Harry. 1973. “Random difference equations and renewal theory for products of random matrices”. Acta Mathematica 131, no. 1: 207–248.CrossRefGoogle Scholar
  58. 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.CrossRefGoogle Scholar
  59. Krolzig, Hans M. 1997. “Markov-switching vector autoregression”. Lecture Notes in Economic and Mathematical Systems. no. 454. Springer-Verlag, New York.Google Scholar
  60. 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.Google Scholar
  61. 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.CrossRefGoogle Scholar
  62. 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.Google Scholar
  63. 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.Google Scholar
  64. 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.CrossRefGoogle Scholar
  65. –. 2003. “Asymptotic theory for a vector ARMA-GARCH model”. Econometric Theory 19, no. 2: 280–310.Google Scholar
  66. Linton, Oliver. 1993. “Adaptive estimation in ARCH models”. Econometric Theory 9, no. 4: 539–569.CrossRefGoogle Scholar
  67. Lubrano, Michel. 2001. “Smooth transition GARCH models: A Bayesian perspective”. Recherches Economiques de Louvain/Louvain Economic Review 67, no. 3: 257–287.Google Scholar
  68. 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.CrossRefGoogle Scholar
  69. Lundbergh, Stefan, and Timo Teräsvirta. 1999. Modelling economic high-frequency time series. Tech. rep. Tinbergen Institute Discussion Paper.Google Scholar
  70. –. 2002. “Evaluating GARCH models”. Journal of Econometrics 110, no. 2: 417–435.Google Scholar
  71. Miguel, Jesus, and Pilar Olave. 2002. “Adjusting forecast intervals in arch-m models”. Journal of Time Series Analysis 23, no. 5: 587–598.CrossRefGoogle Scholar
  72. Milhøj, Anders. 1985. “The moment structure of ARCH processes”. Scandinavian Journal of Statistics 12, no. 4: 281–292.Google Scholar
  73. Nelson, Daniel B. 1990. “Stationarity and persistence in the GARCH (1, 1) model”. Econometric Theory 6, no. 3: 318–334.CrossRefGoogle Scholar
  74. –. 1991. “Conditional heteroskedasticity in asset returns: A new approach”. Econometrica: Journal of the Econometric Society 59, no. 2: 347–370.Google Scholar
  75. 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.Google Scholar
  76. 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.Google Scholar
  77. 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.CrossRefGoogle Scholar
  78. Rabemananjara, Roger, and Jean-Michel Zakoian. 1993. “Threshold ARCH models and asymmetries in volatility”. Journal of Applied Econometrics 8, no. 1: 31–49.CrossRefGoogle Scholar
  79. Schreiber, Ulrich. 2000. “German tax reform-an international perspective”. FinanzArchiv/Public Finance Analysis 57, no. 4: 525–541.CrossRefGoogle Scholar
  80. Sentana, Enrique, and Gabriele Fiorentini. 2001. “Identification, estimation and testing of conditionally heteroskedastic factor models”. Journal of Econometrics 102, no. 2: 143–164.CrossRefGoogle Scholar
  81. Shephard, Neil. 1996. “Statistical aspects of ARCH and stochastic volatility”. Monographs on Statistics and Applied Probability 65: 1–68.Google Scholar
  82. So, Mike KP, WK Li, and K Lam. 2002. “A threshold stochastic volatility model”. Journal of Forecasting 21, no. 7: 473–500.CrossRefGoogle Scholar
  83. 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.Google Scholar
  84. Storti, Giuseppe, and Cosimo Vitale. 2003a. “BL-GARCH models and asymmetries in volatility”. Statistical Methods and Applications 12, no. 1: 19–39.CrossRefGoogle Scholar
  85. –. 2003b. “Likelihood inference in BL-GARCH models”. Computational Statistics 18, no. 3: 387–400.Google Scholar
  86. 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.CrossRefGoogle Scholar
  87. Taylor, Stephen J. 1986. Modelling financial time series. New York: Wiley.Google Scholar
  88. Timmermann, Allan. 2000. “Moments of Markov switching models”. Journal of Econometrics 96, no. 1: 75–111.CrossRefGoogle Scholar
  89. Tse, Yiu Kuen. 2002. “Residual-based diagnostics for conditional heteroscedasticity models”. The Econometrics Journal 5, no. 2: 358–374.CrossRefGoogle Scholar
  90. Weiss, Andrew A. 1986. “Asymptotic theory for ARCH models: Estimation and testing”. Econometric Theory 2, no. 1: 107–131.CrossRefGoogle Scholar
  91. Yang, Minxian. 2000. “Some properties of vector autoregressive processes with Markov-switching coefficients”. Econometric Theory 16, no. 1: 23–43.CrossRefGoogle Scholar
  92. Zakoian, Jean-Michel. 1994. “Threshold heteroskedastic models”. Journal of Economic Dynamics and Control 18, no. 5: 931–955.CrossRefGoogle Scholar
  93. 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.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Dominique Guégan
    • 1
    • 2
  • Bertrand K. Hassani
    • 3
    • 4
    • 5
  1. 1.LabEx ReFi and IPAGUniversity Paris1 Panthéon-SorbonneParisFrance
  2. 2.University Ca’FoscariVeneziaItaly
  3. 3.Department of Computer ScienceUniversity College LondonLondonUK
  4. 4.Department of FinanceUniversité Paris 1 Panthéon-SorbonneParisFrance
  5. 5.Department of Financial RegulationLabeX Refi (ESCP - ENA - Paris 1)ParisFrance

Personalised recommendations