Some Applications of Univariate ARCH Models

  • Christian Gouriéroux
Part of the Springer Series in Statistics book series (SSS)


In this chapter, we present several applications of ARCH models proposed in the literature. The discussion covers the modelling aspects (sections 1, 4, 5), the random walk hypothesis tests (section 3) and the interpretation of ARCH models as discrete approximations of continuous time models (section 2). We emphasize the particular importance of these different questions in financial econometrics.


Stochastic Differential Equation Conditional Variance Threshold Model Arch Model Continuous Time Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Drost, E and T. Nijman (1993) “Temporal Aggregation of GARCH Processes”, Econometrica, 61, 909–927.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Taylor, S. (1985) Modelling Financial Time Series, Amsterdam: North-Holland.Google Scholar
  3. Thomas, A. (1989) Modèles à variance conditionnelle et applications aux modèles d’évaluation financière, Ph.D. thesis. Toulouse: Toulouse University.Google Scholar
  4. Duffle, D. and K. Singleton (1993) “Simulated Moments Estimation of Markov Models of Asset Prices”, Econometrica, 61, 929–952.MathSciNetCrossRefGoogle Scholar
  5. Gouriéroux, C. and A. Monfort (1996) Simulated Estimation Methods, Oxford: Oxford University Press.Google Scholar
  6. Nelson, D. (1990) “ARCH Models as Diffusion Approximations”, Journal of Econometrics, 45, 7–38.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Nelson, D. (1992) “Filtering and Forecasting with Misspecified ARCH Models: Getting the Right Variance with the Wrong Model”, Journal of Econometrics, 25, 61–90.CrossRefGoogle Scholar
  8. Nelson, D. (1996) “Asymptotically Optimal Smoothing with ARCH Models”, Econometrica, 64, 531–573.CrossRefGoogle Scholar
  9. Nelson, D. and D. Foster (1994) `Asymptotic Filtering Theory for Univariate ARCH Models“, Econometrica, 62, 1–41.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Stroock, D. and S. Varadhan (1979) “Multidimensional Diffusion Processes”, Berlin: Springer-Verlag.Google Scholar
  11. Bachelier, L. (1900) “Theory of Speculation”, The Random Character of Stock Market Prices (P. Cootner, Ed.), Cambridge: MIT Press. pp. 17–78.Google Scholar
  12. Diebold, F. (1986) “Testing for Serial Correlation in the Presence of ARCH”, Proceedings of the American Statistical Association, Business and Economic Statistic Section, 323–328.Google Scholar
  13. Fama, E.F. (1965) “The Behaviour of Stock Market Prices”, Journal of Business, 38, 34–105.CrossRefGoogle Scholar
  14. Fama, E.F. (1976) Foundations of Finance, Basil Blackwell: Oxford.Google Scholar
  15. Kim, K. and P. Schmidt (1989) “Unit Root Tests with Conditional Heteroskedasticity”, Discussion Paper. East Lansing: Michigan State University.Google Scholar
  16. Milhoj, A. (1985) “The Moment Structure of ARCH Processes”, Scandinavian Journal of Statistics, 12, 281–292.MathSciNetGoogle Scholar
  17. Samuelson, P. (1973) “Proof that Properly Discounted Present Values of Assets Vibrate Randomly”, Bell Journal, 4.Google Scholar
  18. Solnik, B. (1973) “Note on the Validity of the Random Walk for European Stock Prices”, Journal of Finance, 28, 1151–1160.CrossRefGoogle Scholar
  19. Wooldridge, J. (1991) “On the Application of Robust, Regression Based Diagnostics to Models of Conditional Means and Conditional Variances”, Journal of Econometrics, 47, 5–46.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Black, F. (1976) “Studies in Stock Price Volatility Changes”, Proceedings of the 1976 Business Meeting, American Statistical Association, pp. 177–181.Google Scholar
  21. Cai, J. (1994) “A Markov Model of Switching Regime ARCH”, Journal of Business and Econometric Statistics, 12, 309–316.Google Scholar
  22. Engle, R. and G. Gonzales Rivera (1991) “Semi-Parametric ARCH Models”, Journal of Business and Econometric Statistics, 9, 345–359.Google Scholar
  23. Gallant, R. (1981) “On the Bias in Flexible Functional Forms and an Essentially Unbiased Form: The Fourier Flexible Form”, Journal of Econometrics, 15, 211–244.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Gallant, R., Hansen, L.P. and G. Tauchen (1990) “Using Conditional Moments of Assets Payoffs to Infer the Volatility of Intertemporal Marginal Rates of Substitution”, Journal of Econometrics, 45, 141–180.zbMATHCrossRefGoogle Scholar
  25. Gallant, R. and G. Tauchen (1989) “Semi-nonparametric Estimation of Conditionally Constrained Heterogeneous Processes: Asset Pricing Applications”, Econometrica, 57, 1091–1120.MathSciNetzbMATHCrossRefGoogle Scholar
  26. Gouriéroux, C. and A. Monfort (1992) “Qualitative Threshold ARCH Models”, Journal of Econometrics, 52, 159–200.MathSciNetzbMATHCrossRefGoogle Scholar
  27. Hafner, C. (1996) “Estimating High Frequency Foreign Exchange Rate Volatility with Nonparametric ARCH Models”, Discussion Paper. Berlin: Humboldt University.Google Scholar
  28. Hentschel, L. (1994) “Alternative Models of Asymmetric Volatility in Stock Returns”, Ph.D. dissertation, Princeton, NJ: Princeton University.Google Scholar
  29. Higgins, H.L. and A.K. Bera (1988) “Non Linear ARCH Models Properties, Testing and Applications”, Australian Meeting of the Econometric Society.Google Scholar
  30. Higgins, M. and A. Bera (1992) “A Class of Nonlinear ARCH Models”, International Economic Review, 33, 137–158.zbMATHCrossRefGoogle Scholar
  31. Pagan, A. and W. Schwert (1990) “Alternative Models for Conditional Stock Volatility”, Journal of Econometrics, 45, 267–290.CrossRefGoogle Scholar
  32. Rabemananjara, R. and J.M. Zakoïan (1993) “Threshold ARCH Models and Asymmetries in Volatility”, Journal of Applied Econometrics, 47, 67.Google Scholar
  33. Watt, W. and P. Yadav (1993) “An Empirical Analysis of Alternative Parametric ARCH Models”, Discussion Paper. Glasgow: University of Glasgow.Google Scholar
  34. Zakoïan, J.M. (1994) “Threshold Heteroskedastic Models”, Journal of Economics Dynamics and Control, 18, 931–956.CrossRefGoogle Scholar
  35. Bollerslev, T. (1988) “Integrated ARCH and Cointegration in Variance”, Evanston, IL: Northwestern University.Google Scholar
  36. Bollerslev, T. and Engle, R. (1989) “Common Persistence in Conditional Vari-ances”, Discussion Paper. La Jolla: University of California—San Diego.Google Scholar
  37. Chou, R.Y. (1987) “Persistent Volatility and Stock Returns”, Discussion Paper. La Jolla: University of California—San Diego.Google Scholar
  38. Engle, R.F. and R. Bollerslev (1986) “Modelling the Persistence of Conditional Variances”, Econometric Reviews, 5, 1–50.MathSciNetzbMATHCrossRefGoogle Scholar
  39. French, K., Schwert, G.W. and R. Stambaugh (1987) “Expected Stock Returns and Volatility”, Journal of Financial Economics, 19, 3–29.CrossRefGoogle Scholar
  40. Geweke, J. (1986) “Comment on Modeling the Persistence of Conditional Variances”, Econometric Reviews, 5, 57–62.CrossRefGoogle Scholar
  41. Hansen, B. (1990) “Regression Theory when Variances are Non Stationary”, Discussion Paper. Rochester, NY: University of Rochester.Google Scholar
  42. Hansen, B. (1990) Regression Theory When Variances are Non Stationary, Discussion Paper. Rochester: University of Rochester.Google Scholar
  43. Hendry, D.F. (1986) “Comment on Modeling the Persistence of Conditional Variances”, Econometric Reviews, 5, 63–70.CrossRefGoogle Scholar
  44. Hong, C.H. (1987) “The IGARCH-Model: the Process Estimation and Some Monte Carlo Experiments”, Report 87–32. La Jolla: University of California—San Diego.Google Scholar
  45. Lumsdaine, R. (1995) “Finite Sample Properties of the Maximum Likelihood Estimator in GARCH(1,1) and IGARCH(1,1) Models: A Monte Carlo Investigation”, Journal of Business and Economic Statistics, 13, 1–10.MathSciNetGoogle Scholar
  46. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  47. Lamoureux, C. and W. Lastrapes (1990) “Persistence in Variance Structure Change and the GARCH Model”, Journal of Business and Economic Statistics, 8, 225.Google Scholar
  48. Melvin, M. and B. Peiers (1996) “Volatility Persistence in High Frequency Data: Evidence from the Mark and the Yen”, Discussion Paper. Tempe: Arizona State University.Google Scholar
  49. Nelson, D. (1990) “Stationarity and Persistence in the GARCH (1,1) Models”, Econometric Theory, 6, 318–334.MathSciNetCrossRefGoogle Scholar
  50. Pantula, S. (1986) “Modelling the Persistence of Conditional Variances: a Comment”, Econometric Reviews, 5, 71–73.CrossRefGoogle Scholar
  51. Poterba, J. and L. Summers (1986) “The Persistence of Volatility and Stock Market Fluctuation”, American Economic Review, 76, 1142–1151.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Christian Gouriéroux
    • 1
  1. 1.Centre de Recherche en Economie et StatistiqueLaboratoire de Finance-AssuranceMalakoff CedexFrance

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