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.
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References
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.
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.
Asai, M. (2006). Comparison of MCMC methods for estimating GARCH models. Journal of the Japan Statistical Society, 36, 199–212.
Bauwens, L., & Lubrano, M. (1998). Bayesian inference on GARCH models using the Gibbs sampler. Econometrics Journal, 1, 23–46.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.
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.
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.
Kim, S., Shephard, N., & Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies, 65, 361–393.
Lee, S. W., & Hansen, B. E. (1994). Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econometric Theory, 10, 29–52.
Lindley, D. V. (1980). Approximate bayes methods. Trabajos de Estadistica, 3, 281–288.
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.
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.
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).
Moore, D., & Papadopoulos, A. S. (2000). The Burr Type XII distribution as a failure model under various loss functions. Microelectronics Reliability, 40, 2117–2122.
Müller, P., & Pole, A. (1998). Monte Carlo posterior integration in GARCH models. Sankhya, Series, B, 60, 127–144.
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.
Nakatsuma, T. (2000). Bayesian analysis of ARMA-GARCH models: A Markov Chain sampling approach. Journal of Econometrics, 95, 57–69.
Tierney, L., & Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of American Statistical Association, 81, 82–86.
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.
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Arı, Y. (2018). Bayesian Estimation of GARCH(1,1) Model Using Tierney-Kadane’s Approximation. In: Tsounis, N., Vlachvei, A. (eds) Advances in Time Series Data Methods in Applied Economic Research. ICOAE 2018. Springer Proceedings in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-02194-8_24
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DOI: https://doi.org/10.1007/978-3-030-02194-8_24
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