The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

ARCH Models

  • Oliver B. Linton
Reference work entry


The ARCH model and its many generalizations are very important in analysing discrete time financial data. We review the properties of the original model and discuss many of the subsequent developments.


ARCH models ARMA models Estimation Exponentially weighted moving average model Factor models GARCH models Generalized error distribution Heteroskedasticity IGARCH models Linear models Long memory models Multivariate models News impact curve Nonparametric models Semiparametric models Stationarity Time series analysis Unit roots 

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The author would like to thank the Economic and Social Science Research Council of the United Kingdom for financial support through a research fellowship.


  1. Bollerslev, T. 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31: 307–327.CrossRefGoogle Scholar
  2. Bollerslev, T. 1990. Modelling the coherence in short-run nominal exchange rates: A multivariate generalized autoregressive conditional heteroskedasticity. Review of Economics and Statistics 72: 498–505.CrossRefGoogle Scholar
  3. Bollerslev, T., R.F. Engle, and J.M. Wooldridge. 1988. A capital asset pricing model with time varying covariances. Journal of Political Economy 96: 116–131.CrossRefGoogle Scholar
  4. Bollerslev, T., and H.O. Mikkelson. 1996. Modelling and pricing long memory in stock market volatility. Journal of Econometrics 73: 151–84, 498–505.Google Scholar
  5. Bollerslev, T., and J.M. Wooldridge. 1992. Quasi maximum likelihood estimation and inference in dynamic models with time varying covariances. Econometric Reviews 11: 143–172.CrossRefGoogle Scholar
  6. Bollerslev, T., R.Y. Chou, and K. Kroner. 1992. ARCH modelling in finance. Journal of Econometrics 52: 5–59.CrossRefGoogle Scholar
  7. Bollerslev, T., R.F. Engle, and D. Nelson. 1994. ARCH models. In The handbook of econometrics, ed. D.F. McFadden and R.F. Engle III, Vol. 4. Amsterdam: North-Holland.Google Scholar
  8. Brooks, C., S.P. Burke, and G. Persand. 2001. Benchmarks and the accuracy of GARCH model estimation. International Journal of Forecasting 17: 45–56.CrossRefGoogle Scholar
  9. Carrasco, M., and X. Chen. 2002. Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18: 17–39.CrossRefGoogle Scholar
  10. Dahlhaus, R. 1997. Fitting time series models to nonstationary processes. Annals of Statistics 25: 1–37.CrossRefGoogle Scholar
  11. Diebold, F.S., and M. Nerlove. 1989. The dynamics of exchange-rate volatility: A multivairate latent-factor ARCH model. Journal of Applied Econometrics 4: 1–22.CrossRefGoogle Scholar
  12. Drost, F.C., and C.A.J. Klaassen. 1997. Efficient estimation in semiparametric GARCH models. Journal of Econometrics 81: 193–221.CrossRefGoogle Scholar
  13. Drost, F.C., and T.E. Nijman. 1993. Temporal aggregation of GARCH processes. Econometrica 61: 909–927.CrossRefGoogle Scholar
  14. Engle, R.F. 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50: 987–1008.CrossRefGoogle Scholar
  15. Engle, R.F., and T. Bollerslev. 1986. Modeling the persistence of conditional variances. Econometric Reviews 5: 1–50.CrossRefGoogle Scholar
  16. Engle, R.F., and G. González-Rivera. 1991. Semiparametric ARCH models. Journal of Business and Economic Statistics 9: 345–359.Google Scholar
  17. Engle, R.F., D.M. Lilien, and R.P. Robins. 1987. Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica 19: 3–29.Google Scholar
  18. Engle, R.F., and V.K. Ng. 1993. Measuring and testing the impact of news on volatility. Journal of Finance 48: 1749–1778.CrossRefGoogle Scholar
  19. Engle, R.F., V.K. Ng, and M. Rothschild. 1990. Asset pricing with a FACTOR-ARCH covariance structure: Empirical estimates for treasury bills. Journal of Econometrics 45: 213–237.CrossRefGoogle Scholar
  20. Engle, R.F., and K. Sheppard. 2001. Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH. Working Paper No. 8554. Cambridge, MA: NBER.Google Scholar
  21. Fama, E.F. 1965. The behavior of stock market prices. Journal of Business 38: 34–105.CrossRefGoogle Scholar
  22. Glosten, L.R., R. Jagannathan, and D.E. Runkle. 1993. On the relation between the expected value and the volatility of the nominal excess returns on stocks. Journal of Finance 48: 1779–1801.CrossRefGoogle Scholar
  23. Hafner, C. 1998. Nonlinear time series analysis with applications to foreign exchange rate volatility. Heidelberg: Physica.CrossRefGoogle Scholar
  24. Hall, P., and Q. Yao. 2003. Inference in ARCH and GARCH models with heavy tailed errors. Econometrica 71: 285–317.CrossRefGoogle Scholar
  25. Hansen, B.A. 1991. GARCH(1,1) processes are near epoch dependent. Economics Letters 36: 181–186.CrossRefGoogle Scholar
  26. Hansen, P.R., and A. Lunde. 2005. A forecast comparison of volatility models: Does anything beat a GARCH(1,1). Journal of Applied Econometrics 20: 873–889.CrossRefGoogle Scholar
  27. Härdle, W., and A.B.. Tsybakov. 1997. Locally polynomial estimators of the volatility function. Journal of Econometrics 81: 223–242.Google Scholar
  28. Härdle, W., A.B.. Tsybakov, and L. Yang. 1996. Nonparametric vector autoregression. Discussion Paper, SFB 373. Berlin: Humbodt-Universität.Google Scholar
  29. Jensen, S.T., and A. Rahbek. 2004. Asymptotic normality of the QMLE of ARCH in the nonstationary case. Econometrica 72: 641–646.CrossRefGoogle Scholar
  30. Kim, W., and O. Linton. 2004. A local instrumental variable estimation method for generalized additive volatility models. Econometric Theory 20: 1094–1139.CrossRefGoogle Scholar
  31. Lee, S.-W., and B.E. Hansen. 1994. Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10: 29–52.CrossRefGoogle Scholar
  32. Linton, O.B. 1993. Adaptive estimation in ARCH models. Econometric Theory 9: 539–569.CrossRefGoogle Scholar
  33. Linton, O.B., and E. Mammen. 2005. Estimating semiparametric ARCH(∞) models by kernel smoothing methods. Econometrica 73: 771–836.CrossRefGoogle Scholar
  34. 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.Google Scholar
  35. Lumsdaine, R.L. 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
  36. Mandelbrot, B. 1963. The variation of certain speculative prices. Journal of Business 36: 394–419.CrossRefGoogle Scholar
  37. Masry, E., and D. Tjøstheim. 1995. Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11: 258–289.CrossRefGoogle Scholar
  38. Meitz, M., and P. Saikkonen. 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 No. 573. Stockholm School of Economics.Google Scholar
  39. Merton, R.C. 1973. An intertemporal capital asset pricing model. Econometrica 41: 867–887.CrossRefGoogle Scholar
  40. Mikosch, T., and C. Starica. 2000. Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Annals of Statistics 28: 1427–1451.CrossRefGoogle Scholar
  41. Nelson, D.B. 1990. Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6: 318–334.CrossRefGoogle Scholar
  42. Nelson, D.B. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59: 347–370.CrossRefGoogle Scholar
  43. Nelson, D.B., and C.Q. Cao. 1992. Inequality constraints in the univariate GARCH model. Journal of Business and Economic Statistics 10: 229–235.Google Scholar
  44. Newey, W.K., and D.G. Steigerwald. 1997. Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models. Econometrica 65: 587–599.CrossRefGoogle Scholar
  45. Pagan, A.R., and G.W. Schwert. 1990. Alternative models for conditional stock volatility. Journal of Econometrics 45: 267–290.CrossRefGoogle Scholar
  46. Pagan, A.R., and Y.S. Hong. 1991. Nonparametric estimation and the risk premium. In Nonparametric and semiparametric methods in econometrics and statistics, ed. W. Barnett, J. Powell, and G.E. Tauchen. Cambridge: Cambridge University Press.Google Scholar
  47. Robinson, P.M. 1991. Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47: 67–84.CrossRefGoogle Scholar
  48. Robinson, P.M., and P. Zaffaroni. 2006. Pseudo-maximum likelihood estimation of ARCH(∞) models. Annals of Statistics 34: 1049–1074.CrossRefGoogle Scholar
  49. Sentana, E. 1998. The relation between conditionally heteroskedastic factor models and factor GARCH models. Econometrics Journal 1: 1–9.CrossRefGoogle Scholar
  50. Yang, L., W. Härdle, and J.P. Nielsen. 1999. Nonparametric autoregression with multiplicative volatility and additive mean. Journal of Time Series Analysis 20: 579–604.CrossRefGoogle Scholar

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© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Oliver B. Linton
    • 1
  1. 1.