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Journal of Probability and Statistics
Volume 2011, Article ID 906212, 17 pages
http://dx.doi.org/10.1155/2011/906212
Review Article

Semi- and Nonparametric ARCH Processes

1Department of Economics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK
2Department of Statistics, The London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK

Received 27 February 2010; Accepted 28 June 2010

Academic Editor: Tak Kuen Siu

Copyright © 2011 Oliver B. Linton and Yang Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Mandelbrot, “The variation of certain speculative prices,” Journal of Business, vol. 36, pp. 394–419, 1963. View at Google Scholar
  2. E. F. Fama, “The behavior of stock market prices,” Journal of Business, vol. 38–, pp. 34–105, 1965. View at Google Scholar
  3. R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation,” Econometrica, vol. 50, no. 4, pp. 987–1007, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” Journal of Econometrics, vol. 31, no. 3, pp. 307–327, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  5. T. Bollerslev, R. F. Engle, and D. B. Nelson, “Arch models,” in Handbook of Econometrics, Vol. IV, R. F. Engle and D. L. McFadden, Eds., vol. 2 of Handbooks in Econom., pp. 2959–3038, North-Holland, Amsterdam, The Netherlands, 1994. View at Google Scholar · View at MathSciNet
  6. F. C. Drost and T. E. Nijman, “Temporal aggregation of GARCH processes,” Econometrica, vol. 61, no. 4, pp. 909–927, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. Tsay, “A critique of GARCH models,” 2007. View at Google Scholar
  8. A. A. Weiss, “Asymptotic theory for ARCH models: estimation and testing,” Econometric Theory, vol. 2, no. 1, pp. 107–131, 1986. View at Google Scholar
  9. T. Bollerslev and J. M. Wooldridge, “Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances,” Econometric Reviews, vol. 11, no. 2, pp. 143–172, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Engle, “Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models,” Journal of Business and Economic Statistics, vol. 20, no. 3, pp. 339–350, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. O. Linton, “Adaptive estimation in ARCH models,” Econometric Theory, vol. 9, no. 4, pp. 539–569, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  12. F. C. Drost and C. A. J. Klaassen, “Efficient estimation in semiparametric GARCH models,” Journal of Econometrics, vol. 81, no. 1, pp. 193–221, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. B. Nelson, “Conditional heteroskedasticity in asset returns: a new approach,” Econometrica, vol. 59, no. 2, pp. 347–370, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. L. R. Glosten, R. Jagannathan, and D. E. Runkle, “On the relation between the expected value and the volatility of the nominal excess returns on stocks,” Journal of Finance, vol. 48, pp. 1779–1801, 1993. View at Google Scholar
  15. A. R. Pagan and G. W. Schwert, “Alternative models for conditional stock volatility,” Journal of Econometrics, vol. 45, no. 1-2, pp. 267–290, 1990. View at Google Scholar · View at Scopus
  16. A. R. Pagan and Y. S. Hong, “Nonparametric estimation and the risk premium,” in Nonparametric and Semiparametric Methods in Econometrics and Statistics, W. Barnett, J. Powell, and G. E. Tauchen, Eds., Cambridge University Press, Cambridge, UK, 1991. View at Google Scholar
  17. W. Härdle and A. Tsybakov, “Local polynomial estimators of the volatility function in nonparametric autoregression,” Journal of Econometrics, vol. 81, no. 1, pp. 223–242, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. W. Härdle, A. Tsybakov, and L. Yang, “Nonparametric vector autoregression,” Journal of Statistical Planning and Inference, vol. 68, no. 2, pp. 221–245, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. Masry and D. Tjøstheim, “Nonparametric estimation and identification of nonlinear ARCH time series,” Econometric Theory, vol. 11, no. 2, pp. 258–289, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Z. Lu and O. Linton, “Local linear fitting under near epoch dependence,” Econometric Theory, vol. 23, no. 1, pp. 37–70, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Fan and Q. Yao, “Efficient estimation of conditional variance functions in stochastic regression,” Biometrika, vol. 85, no. 3, pp. 645–660, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. P. Avramidis, “Local maximum likelihood estimation of volatility function,” manuscript.
  23. J. Franke, M. H. Neumann, and J.-P. Stockis, “Bootstrapping nonparametric estimators of the volatility function,” Journal of Econometrics, vol. 118, no. 1-2, pp. 189–218, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. C. J. Stone, “Optimal rates of convergence for nonparametric estimators,” The Annals of Statistics, vol. 8, no. 6, pp. 1348–1360, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. B. Perron, “A Monte Carlo Comparison of Non-parametric Estimator of the Conditional Variance,” Mimeo, 1998.
  26. T. J. Hastie and R. J. Tibshirani, Generalized Additive Models, vol. 43 of Monographs on Statistics and Applied Probability, Chapman and Hall, London, UK, 1990. View at MathSciNet
  27. O. Linton and J. P. Nielsen, “A kernel method of estimating structured nonparametric regression based on marginal integration,” Biometrika, vol. 82, no. 1, pp. 93–100, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. D. Tjøstheim and B. H. Auestad, “Nonparametric identification of nonlinear time series: projections,” Journal of the American Statistical Association, vol. 89, no. 428, pp. 1398–1409, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  29. C. J. Stone, “Additive regression and other nonparametric models,” The Annals of Statistics, vol. 13, no. 2, pp. 689–705, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. L. Yang, W. Härdle, and J. P. Nielsen, “Nonparametric autoregression with multiplicative volatility and additive mean,” Journal of Time Series Analysis, vol. 20, no. 5, pp. 579–604, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. Kim and O. Linton, “The live method for generalized additive volatility models,” Econometric Theory, vol. 20, no. 6, pp. 1094–1139, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. O. Linton and E. Mammen, “Estimating semiparametric ARCH() models by kernel smoothing methods,” Econometrica, vol. 73, no. 3, pp. 771–836, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. R. F. Engle and V. K. Ng, “Measuring and testing the impact of news on volatility,” The Journal of Finance, vol. 48, pp. 1749–1778, 1993. View at Google Scholar
  34. L. Yang, “A semiparametric GARCH model for foreign exchange volatility,” Journal of Econometrics, vol. 130, no. 2, pp. 365–384, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  35. P. M. Robinson, “Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression,” Journal of Econometrics, vol. 47, no. 1, pp. 67–84, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. F. Audrino and P. Bühlmann, “Tree-structured generalized autoregressive conditional heteroscedastic models,” Journal of the Royal Statistical Society. Series B, vol. 63, no. 4, pp. 727–744, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. R. C. Merton, “An intertemporal capital asset pricing model,” Econometrica, vol. 41, pp. 867–887, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. R. F. Engle, D. M. Lilien, and R. P. Robins, “Estimating time varying risk premia in the term structure: the ARCH-M model,” Econometrica, vol. 19, pp. 3–29, 1987. View at Google Scholar
  39. A. R. Pagan and A. Ullah, “The econometric analysis of models with risk terms,” Journal of Applied Econometrics, vol. 3, pp. 87–105, 1988. View at Google Scholar
  40. O. Linton and B. Perron, “The shape of the risk premium: evidence from a semiparametric generalized autoregressive conditional heteroscedasticity model,” Journal of Business & Economic Statistics, vol. 21, no. 3, pp. 354–367, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  41. T. Bollerslev and H. O. Mikkelsen, “Modeling and pricing long memory in stock market volatility,” Journal of Econometrics, vol. 73, no. 1, pp. 151–184, 1996. View at Publisher · View at Google Scholar · View at Scopus
  42. T. Mikosch and C. Stărică, “Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process,” The Annals of Statistics, vol. 28, no. 5, pp. 1427–1451, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. L. Giraitis, R. Leipus, and D. Surgailis, “ARCH() models and long memory properties,” in Handbook of Financial Time Series, pp. 71–84, Springer, Berlin, Germany, 2009. View at Google Scholar
  44. R. Dahlhaus, “Fitting time series models to nonstationary processes,” The Annals of Statistics, vol. 25, no. 1, pp. 1–37, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. R. Dahlhaus and S. Subba Rao, “Statistical inference for time-varying ARCH processes,” The Annals of Statistics, vol. 34, no. 3, pp. 1075–1114, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. V. Spokoiny and P. Cizek, “Varying coefficient GARCH Models,” in Handbook of Financial Time Series, Springer, Berlin, Germany, 2007. View at Google Scholar
  47. R. F. Engle and J. G. Rangel, “The spline-GARCH model for low-frequency volatility and its global macroeconomic causes,” Review of Financial Studies, vol. 21, no. 3, pp. 1187–1222, 2008. View at Publisher · View at Google Scholar · View at Scopus
  48. D. Bosq, Nonparametric Statistics for Stochastic Processes. Estimation and Prediction, vol. 110 of Lecture Notes in Statistics, Springer, New York, NY, USA, 2nd edition, 1998. View at MathSciNet
  49. R. F. Engle, “The econometrics of ultra-high-frequency data,” Econometrica, vol. 68, no. 1, pp. 1–22, 2000. View at Google Scholar · View at Scopus
  50. K. Back, “Asset pricing for general processes,” Journal of Mathematical Economics, vol. 20, no. 4, pp. 371–395, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. T. Bollerslev, “Modeling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model,” Review of Economics and Statistics, vol. 72, pp. 498–505, 1990. View at Google Scholar
  52. R. Engle, “Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models,” Journal of Business & Economic Statistics, vol. 20, no. 3, pp. 339–350, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  53. C. M. Hafner and J. V. K. Rombouts, “Semiparametric multivariate volatility models,” Econometric Theory, vol. 23, no. 2, pp. 251–280, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  54. A. R. Gallant and G. Tauchen, “Seminonparametric estimation of conditionally constrained heterogeneous processes: asset pricing applications,” Econometrica, vol. 57, no. 5, pp. 1091–1120, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  55. J. E. Ingersoll, Theory of Financial Decision Making, Rowman & Littlefield, Totowa, NJ, USA, 1987.
  56. X. Chen and Y. Fan, “Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification,” Journal of Econometrics, vol. 135, no. 1-2, pp. 125–154, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  57. P. Embrechts, A. J. McNeil, and D. Straumann, “Correlation and dependence in risk management: properties and pitfalls,” in Risk Management: Value at Risk and Beyond (Cambridge, 1998), M. Dempster, M. Alan, and H. Dempster, Eds., pp. 176–223, Cambridge University Press, Cambridge, UK, 2002. View at Google Scholar · View at MathSciNet
  58. A. J. Patton, “Estimation of multivariate models for time series of possibly different lengths,” Journal of Applied Econometrics, vol. 21, no. 2, pp. 147–173, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  59. A. J. Patton, “Modelling asymmetric exchange rate dependence,” International Economic Review, vol. 47, no. 2, pp. 527–556, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  60. E. Jondeau and M. Rockinger, “The Copula-GARCH model of conditional dependencies: an international stock market application,” Journal of International Money and Finance, vol. 25, no. 5, pp. 827–853, 2006. View at Publisher · View at Google Scholar · View at Scopus
  61. V. Panchenko, “Estimating and evaluating the predictive abilities of semiparametric multivariate models with application to risk management,” Computing in Economics and Finance, No.382, 2006.
  62. J. C. Rodriguez, “Measuring financial contagion: a Copula approach,” Journal of Empirical Finance, vol. 14, no. 3, pp. 401–423, 2007. View at Publisher · View at Google Scholar · View at Scopus
  63. T. Okimoto, “New evidence of asymmetric dependence structures in international equity markets,” Journal of Financial and Quantitative Analysis, vol. 43, no. 3, pp. 787–815, 2008. View at Publisher · View at Google Scholar · View at Scopus
  64. L. Chollette, A. Heinen, and A. I. Valdesogo, “Modelling international financial returns with a multivariate regime switching copula,” CORE Discussion paper 2008/13, 2008. View at Google Scholar
  65. D. Pelletier, “Regime switching for dynamic correlations,” Journal of Econometrics, vol. 131, no. 1-2, pp. 445–473, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  66. C. M. Hafner, D. van Dijk, and P. H. Franses, “Semi-parametric modelling of correlation dynamics,” in Econometric Analysis of Financial and Economic Time Series. Part A, T. Fomby, C. Hill, and D. Terrell, Eds., vol. 20 of Advanced in Econometrics, pp. 59–103, Emerald/JAI, Bingley, UK, 2005. View at Google Scholar · View at MathSciNet
  67. C. M. Hafner and O. B. Linton, “Efficient estimation of a multivariate multiplicative volatility model,” working paper, 2009. View at Google Scholar
  68. Y. Feng, “A local dynamic conditional correlation model,” MPRA paper 1592, 2007. View at Google Scholar
  69. D. B. Nelson, “Stationarity and persistence in the GARCH(1,1) model,” Econometric Theory, vol. 6, no. 3, pp. 318–334, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  70. P. Bougerol and N. Picard, “Stationarity of GARCH processes and of some nonnegative time series,” Journal of Econometrics, vol. 52, no. 1-2, pp. 115–127, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  71. L. Giraitis, P. Kokoszka, and R. Leipus, “Stationary ARCH models: dependence structure and central limit theorem,” Econometric Theory, vol. 16, no. 1, pp. 3–22, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  72. S. Ling and M. McAleer, “Stationarity and the existence of moments of a family of GARCH processes,” Journal of Econometrics, vol. 106, no. 1, pp. 109–117, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  73. A. M. Lindner, “Stationarity, mixing, distributional properties and moments of GARCH(p,q) processes,” in Handbook of Financial Time Series, pp. 43–70, Springer, Berlin, Germany, 2009. View at Google Scholar
  74. V. Kazakevičius, R. Leipus, and M.-C. Viano, “Stability of random coefficient ARCH models and aggregation schemes,” Journal of Econometrics, vol. 120, no. 1, pp. 139–158, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  75. V. Kazakevičius and R. Leipus, “On stationarity in the ARCH() model,” Econometric Theory, vol. 18, no. 1, pp. 1–16, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  76. R. Douc, F. Roueff, and P. Soulier, “On the existence of some ARCH() processes,” Stochastic Processes and Their Applications, vol. 118, no. 5, pp. 755–761, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  77. X. Chen and Y. Fan, “Estimation of copula-based semiparametric time series models,” Journal of Econometrics, vol. 130, no. 2, pp. 307–335, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  78. B. Beare, “Copulas and temporal dependence,” UCSD Economics Working Paper 2008-10, 2008. View at Google Scholar
  79. X. Chen, W. B. Wu, and Y. Yi, “Efficient estimation of copula-based semiparametric Markov models,” The Annals of Statistics, vol. 37, no. 6B, pp. 4214–4253, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  80. R. L. Lumsdaine, “Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models,” Econometrica, vol. 64, no. 3, pp. 575–596, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  81. S.-W. Lee and B. E. Hansen, “Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator,” Econometric Theory, vol. 10, no. 1, pp. 29–52, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  82. P. Hall and Q. Yao, “Inference in ARCH and GARCH models with heavy-tailed errors,” Econometrica, vol. 71, no. 1, pp. 285–317, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  83. S. T. Jensen and A. Rahbek, “Asymptotic inference for nonstationary GARCH,” Econometric Theory, vol. 20, no. 6, pp. 1203–1226, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  84. L. Giraitis and P. M. Robinson, “Whittle estimation of arch models,” Econometric Theory, vol. 17, no. 3, pp. 608–631, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  85. L. Peng and Q. Yao, “Least absolute deviations estimation for ARCH and GARCH models,” Biometrika, vol. 90, no. 4, pp. 967–975, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  86. O. Linton, J. Pan, and H. Wang, “Estimation for a nonstationary semi-strong GARCH(1,1) with heavy-tailed errors,” Econometric Theory, vol. 26, no. 1, pp. 1–28, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  87. Y. Sun and T. Stengos, “Semiparametric efficient adaptive estimation of asymmetric GARCH models,” Journal of Econometrics, vol. 133, no. 1, pp. 373–386, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  88. W. Kim and O. Linton, “Estimation of a semiparametric IGARCH(1,1) model,” LSE STICERD Research Paper EM539, 2009. View at Google Scholar
  89. X. Chen, Y. Fan, and V. Tsyrennikov, “Efficient estimation of semiparametric multivariate copula models,” Journal of the American Statistical Association, vol. 101, no. 475, pp. 1228–1240, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  90. J.-D. Fermanian, D. Radulović, and M. Wegkamp, “Weak convergence of empirical copula processes,” Bernoulli, vol. 10, no. 5, pp. 847–860, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  91. J.-D. Fermanian and O. Scaillet, “Nonparametric estimation of copulas for time series,” Journal of Risk, vol. 5, pp. 25–54, 2003. View at Google Scholar
  92. S. X. Chen and T.-M. Huang, “Nonparametric estimation of copula functions for dependence modelling,” The Canadian Journal of Statistics, vol. 35, no. 2, pp. 265–282, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet