Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 270–296 | Cite as

Empirical Likelihood Inference for Functional Coefficient ARCH-M Model

  • Hai Qing Zhao
  • Yuan LiEmail author
  • Yan Meng Zhao


Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β. Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation (MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.


ARCH-M model empirical likelihood relative risk aversion 

MR(2010) Subject Classification

62G05 62G20 


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  1. [1]
    Arvanitis, S., Demos, A.: Time dependence and moments of a family of time-varying parameter garch in mean models. Journal of Time Series analysis, 25, 1–25 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Backus, D., Gregory, A.: Theoretical relations between risk premiums and conditional variances. Journal of Business and Economic Statistics, 11, 177–185 (1988)Google Scholar
  3. [3]
    Buckholder, D. L.: Distribution function inequalities for martingales. Annals of Probability, 1(1), 19–42 (1973)MathSciNetCrossRefGoogle Scholar
  4. [4]
    Chan, N. H., Ling, S. Q.: Empirical likelihood for garch models. Econometric Theory, 22, 403–428 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Chen, J. h., Variyath, A. M., Abraham, B.: Adjusted Empirical Likelihood and its Properties. Journal of Computational and Graphical Statistics, 17(2), 426–443 (2008)MathSciNetCrossRefGoogle Scholar
  6. [6]
    Chen, S. X., Cui, H. J.: An extended empirical likelihood for generalized linear models. Statistica Sinica, 13, 69–81 (2003)MathSciNetzbMATHGoogle Scholar
  7. [7]
    Chou, R., Kane, A., Engle, R. F.: Measuring risk aversion from excess returns on a stock index. Journal of Econometrics, 52, 201–224 (1992)CrossRefGoogle Scholar
  8. [8]
    Christensen, B. J., Christian, M. D., Iglesias, E. M.: Semiparametric inference in a garch-in-mean model. Journal of Econometrics, 167(2), 458–472 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Conrad, C., Karanasos, M.: The impulse reponse function of the long memory garch process. Economic Letters, 90, 34–41 (2006)CrossRefzbMATHGoogle Scholar
  10. [10]
    Ding, J., Liu, Y.: Semiparametric Empirical Likelihood Estimation for Two-stage Outcome-dependent Sampling under the Frame of Generalized Linear Models. Acta Mathematica Sinica, English Series, 30(3), 663–676 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Engle, R. F., Lilien, D. M., Robins, R. P.: Estimating time varying risk premia in the term structure: The arch-m model. Econometrica, 55(2), 391–407 (1987)CrossRefGoogle Scholar
  12. [12]
    Fan, J. Q., Yao, Q. W.: Nonlinear Time Series: Nonparametric and Parametric Methods, Springer-Verlag, New York, 2003CrossRefzbMATHGoogle Scholar
  13. [13]
    Hall, P., Heyde, C. C.: Martingale Limit Theory and Its Application, Academic Press, London, 1980zbMATHGoogle Scholar
  14. [14]
    Harvey, C. R.: Time-varying conditional covariances in tests of asset pricing models. Journal of Financial Economics, 24(2), 289–317 (1989)MathSciNetCrossRefGoogle Scholar
  15. [15]
    Hodgson, D. J., Vorkink, K. P.: Efficient estimation of conditional asset-pricing models. Journal of Business and Economic Statistics, 21(2), 269–283 (2012)MathSciNetCrossRefGoogle Scholar
  16. [16]
    Hong, E. P.: The autocorrelation structure for the garch-m process. Economics Letters, 37, 129–132 (1991)CrossRefzbMATHGoogle Scholar
  17. [17]
    Hughes, A. W., Maxwell, L. K., Kain, T. K.: Selecting the order of an arch models. Economics Letters, 83, 269–275 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Kitamura, Y.: Empirical likelihood methods with weakly dependent processes. Ann. Statist., 25, 2084–2102 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Li, R. G., Tian, P., Xue, L. G.: Generalized Empirical Likelihood Inference in Semiparametric Regression Model for Longitudinal Data. Acta Mathematica Sinica, English Series, 24(12), 2029–2040 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Ling, S. Q.: Estimation and testing stationarity for double-autoregressive models. J. R. Statist, 66, 63–78 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Owen, A. B.: Empirical likelihood ratio confidence intervals for a single function. Biometrika, 75, 237–249 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Qin, G. S., Jing, B. Y.: Empirical likelihood for censored linear regression. Scandinavian Journal of statistics, 78, 37–61 (2001)zbMATHGoogle Scholar
  23. [23]
    Qin, J., Lawless, J.: Empirical likelihood and general estimating equations. Ann. Statist., 22, 300–325 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Song, Z. F., Zhang, X. F., Li, Y., et al.: A linear varying coefficient ARCH-M model with a latent variable. Science China Mathematics, 59(9), 1795–1814 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Wang, Q. H., Jing, B. Y.: Empirical likelihood for partially linear models with fixed designs. Statist. Probab. Lett., 41, 425–433 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Xiong, Q., Li, Y., Zhang, X. F.: The profile likelihood estimation for single-index arch(p)-m model. Mathematical Problems in Engineering, 2014(2), 1–17 (2014)MathSciNetGoogle Scholar
  27. [27]
    Yang, Y. P., Xue, L. G., Cheng, W. H.: An Empirical Likelihood Method in a Partially Linear Single-index Model with Right Censored Data. Acta Mathematica Sinica English Series, 28(5), 1041–1060 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Zhang, X. F.: Some parametric and semiparameric models for financial time series anlysis, Phd. thesis, The Hong Kong Polytechnic University, Hong Kong, 2012Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouP. R. China
  2. 2.School of Mathematics and StatisticsLingnan Normal UniversityZhanjiangP. R. China
  3. 3.School of Economics and StatisticsGuangzhou UniversityGuangzhouP. R. China
  4. 4.School of Mathematics and StatisticsShenzhen UniversityShenzhenP. R. China

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