Advertisement

Bayesian Estimation of GARCH(1,1) Model Using Tierney-Kadane’s Approximation

  • Yakup ArıEmail author
Conference paper
Part of the Springer Proceedings in Business and Economics book series (SPBE)

Abstract

The Generalized Autoregressive Conditionally Heteroscedastic (GARCH) process models the dependency of conditional second moments of financial time series. The maximum likelihood estimation (MLE) procedure is most commonly used for estimating the unknown parameters of a GARCH model. In this study, the parameters of the GARCH models with normal innovations are discussed for estimations using the Bayesian approach in which the parameters of the GARCH model are assumed as random variables having known prior probability density functions. The prior probability density functions of the parameters satisfy the conditions on GARCH parameters such as positivity and stationarity. The Bayesian estimators are not in a closed form. Thus Tierney-Kadane’s approximation that is a numerical integration method to calculate the ratio of two integrals is used to estimate. The Bayesian estimators are derived under squared error loss function. Finally, simulations are performed in order to compare the ML estimates to the Bayesian ones and furthermore, an example is given in order to illustrate the findings.

Keywords

Bayes Tierney-Kadane GARCH MLE 

References

  1. Ardia, D., & Hoogerheide, L. F. (2010). Bayesian estimation of the GARCH (1,1) model with student-t innovations in R. The R Journal, 2(2), 41–47.Google Scholar
  2. Ari, Y. & Papadopoulos, S. A. (2016). Bayesian estimation of the parameters of the ARCH model with Normal Innovations using Lindley’s approximation. Journal of Economic Computation and Economic Cybernetics Studies and Research, 50(4), 217–234.Google Scholar
  3. Asai, M. (2006). Comparison of MCMC methods for estimating GARCH models. Journal of the Japan Statistical Society, 36, 199–212.CrossRefGoogle Scholar
  4. Bauwens, L., & Lubrano, M. (1998). Bayesian inference on GARCH models using the Gibbs sampler. Econometrics Journal, 1, 23–46.CrossRefGoogle Scholar
  5. Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.CrossRefGoogle Scholar
  6. Bollerslev, T., Engle, R. F., & Nelson, D. B. (1994). ARCH models. In R. F. Engle & D. McFadden (Eds.), Handbook of econometrics (Vol. IV, pp. 2959–3038). North-Holland, Amsterdam.Google Scholar
  7. Gilks, W. R., Best, N. G., & Tan, K. K. C. (1995). Adaptive rejection metropolis sampling within Gibbs sampling. Journal of the Royal Statistical Society Series, C, 44, 455–472.Google Scholar
  8. Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies, 65, 361–393.CrossRefGoogle Scholar
  9. Lee, S. W., & Hansen, B. E. (1994). Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory, 10, 29–52.CrossRefGoogle Scholar
  10. Lindley, D. V. (1980). Approximate bayes methods. Trabajos de Estadistica, 3, 281–288.Google Scholar
  11. Marín, J. M., Rodríguez-Bernal, M. T., & Romero, E. (2015). Data cloning estimation of GARCH and COGARCH models. Journal of Statistical Computation and Simulation, 85(9), 1818–1831.CrossRefGoogle Scholar
  12. Metropolis, N., Rosenbluthv, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of State Calculations by Fast Computing Machines. The Journal of Chemical Physics, 21, 1087–1092.CrossRefGoogle Scholar
  13. Mitsui, H., & Watanabe, T. (2003). Bayesian analysis of GARCH option pricing models. The Journal of the Japan Statistical Society (Japanese Issue), 33, 307–324 (in Japanese).Google Scholar
  14. Moore, D., & Papadopoulos, A. S. (2000). The Burr Type XII distribution as a failure model under various loss functions. Microelectronics Reliability, 40, 2117–2122.CrossRefGoogle Scholar
  15. Müller, P., & Pole, A. (1998). Monte Carlo posterior integration in GARCH models. Sankhya, Series, B, 60, 127–144.Google Scholar
  16. Nadar, M., & Kızılaslan, F. (2015). Estimation and prediction of the Burr type XII distribution based on record values and inter-record times. Journal of Statistical Computation and Simulation, 85(16), 3297–3321.CrossRefGoogle Scholar
  17. Nakatsuma, T. (2000). Bayesian analysis of ARMA-GARCH models: A Markov Chain sampling approach. Journal of Econometrics, 95, 57–69.CrossRefGoogle Scholar
  18. Tierney, L., & Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of American Statistical Association, 81, 82–86.CrossRefGoogle Scholar
  19. Virbickaite, A., Ausin, M. C., & Galeano, P. (2015). Bayesian inference methods for univariate and multivariate GARCH models: A survey. Journal of Economic Surveys, 29(1), 76–79.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Alanya Alaaddin Keykubat UniversityAlanyaTurkey

Personalised recommendations