Advertisement

A Perturbation Method to Optimize the Parameters of Autoregressive Conditional Heteroscedasticity Model

  • Xuejie FengEmail author
  • Chiping Zhang
Article
  • 8 Downloads

Abstract

As a linear regression parameter estimation method, the least square method plays an indispensable role in the parameter estimation of ARCH model. Although the least squares solution can minimize the sum of the squared errors, it will cause uneven distribution of errors, that is, some fitting errors are too large, and some fitting errors are too small, which will lead to overfitting. In response to this situation, we adopt a novel perturbation method to solve this problem. The specific theoretical derivation of the perturbation method is given in this paper. It takes parameters estimated by the least squares method as its initial iteration value. The maximum fitting error will decrease continuously from a series of iterations to final convergence. Furthermore, based on the perturbation method, this paper finds a condition that makes the ARCH model satisfy the non-negative limit of parameters. The experimental process uses real stock fund’s fluctuation data for fitting analysis and prediction. The experimental results show that the perturbation method can achieve the expected effect in the parameter estimation of the ARCH model, it can also effectively ensure that the fitting errors fluctuate within the controllable range when predicting the price fluctuation of stock funds in the future.

Keywords

Perturbation Maximum error minimization Prediction Heteroscedasticity 

Mathematics Subject Classification

40A05 47N10 97M40 90C30 

Notes

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and Animal Rights

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. This article does not contain any studies with animals performed by any of the authors.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

References

  1. Berndt, E.R., Hall, B.H., Hall, R.E., & Hausman, J.A. (1974). Estimation and inference in nonlinear structural models. In Annals of economic and social measurement, volume 3, number 4, NBER (pp. 653–665).Google Scholar
  2. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.Google Scholar
  3. Bose, A., & Mukherjee, K. (2003). Estimating the arch parameters by solving linear equations. Journal of Time Series Analysis, 24(2), 127–136.Google Scholar
  4. Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1), 83–106.Google Scholar
  5. Drost, F. C., & Klaassen, C. A. (1997). Efficient estimation in semiparametric garch models. Social Science Electronic Publishing, 81(81), 193–221.Google Scholar
  6. Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica: Journal of the Econometric Society, 50(4), 987–1007. Google Scholar
  7. Francq, C., Zakoian, J. M., et al. (2004). Maximum likelihood estimation of pure garch and arma-garch processes. Bernoulli, 10(4), 605–637.Google Scholar
  8. Fryzlewicz, P., Sapatinas, T., Rao, S. S., et al. (2008). Normalized least-squares estimation in time-varying arch models. The Annals of Statistics, 36(2), 742–786.Google Scholar
  9. Giraitis, L., & Robinson, P. M. (2001). Whittle estimation of arch models. Econometric Theory, 17(3), 608–631.Google Scholar
  10. Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 48(5), 1779–1801.Google Scholar
  11. Hentschel, L. (1995). All in the family nesting symmetric and asymmetric garch models. Journal of Financial Economics, 39(1), 71–104.Google Scholar
  12. Horv, L., Kokoszka, P., et al. (2003). Garch processes: Structure and estimation. Bernoulli, 9(2), 201–227.Google Scholar
  13. Kristensen, D., & Linton, O. (2006). A closed-form estimator for the garch (1, 1) model. Econometric Theory, 22(2), 323–337.Google Scholar
  14. Ling, S., & McAleer, M. (2002). Stationarity and the existence of moments of a family of garch processes. Journal of Econometrics, 106(1), 109–117.Google Scholar
  15. Linton, O. (1993). Adaptive estimation in arch models. Econometric Theory, 9(4), 539–569.Google Scholar
  16. Nelson, D.B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, 59(2), 347–370.Google Scholar
  17. NetEase. (2018a). Csi 300 equal weight index. http://quotes.money.163.com/1399300.html, Last Retrived 8 June 2018.
  18. NetEase. (2018b). Fund index. http://quotes.money.163.com/0000011.html, Last Retrived 8 June 2018.
  19. NetEase. (2018c). Shanghai stock exchange composite index. http://quotes.money.163.com/0000001.html, Last Retrived 8 June 2018.
  20. NetEase. (2018d). Shenzhen stock exchange component index. http://quotes.money.163.com/1399001.html, Last Retrived 8 June 2018.
  21. Pantula, S.G. (1988). Estimation of autoregressive models with arch errors. Sankhyā: The Indian Journal of Statistics, Series B, 50(1), 119–138.Google Scholar
  22. Straumann, D. (2006). Estimation in conditionally heteroscedastic time series models. Annals of Statistics, 34(5), 2449–2495.Google Scholar
  23. Weiss, A. A. (1986). Asymptotic theory for arch models: Estimation and testing. Econometric Theory, 2(1), 107–131.Google Scholar
  24. Wong, H., & Li, W. K. (1997). On a multivariate conditional heteroscedastic model. Biometrika, 84(1), 111–123.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.School of International BusinessQingdao Huanghai UniversityQingdaoChina

Personalised recommendations