Volatility Fitting Performance of QGARCH(1,1) Model with Student-t, GED, and SGED Distributions

Authors

  • Didit Budi Nugroho Department of Mathematics & Data Science, Universitas Kristen Satya Wacana
  • Bintoro Ady Pamungkas Department of Mathematics & Data Science, Universitas Kristen Satya Wacana
  • Hanna Arini Parhusip Department of Mathematics & Data Science, Universitas Kristen Satya Wacana

DOI:

https://doi.org/10.21512/comtech.v11i2.6391

Keywords:

volatility fitting performance, Quadratic Generalized Autoregressive Conditional Heteroscedasticity (QGARCH), Student-t, General Error Distribution (GED), Skew GED (SGED)

Abstract

The research had two objectives. First, it compared the performance of the Generalized Autoregressive Conditional Heteroscedasticity (1,1) (GARCH) and Quadratic GARCH (1,1) (QGARCH)) models based on the fitting to real data sets. The model assumed that return error follows four different distributions: Normal (Gaussian), Student-t, General Error Distribution (GED), and Skew GED (SGED). Maximum likelihood estimation was usually employed in estimating the GARCH model, but it might not be easily applied to more complicated ones. Second, it provided two ways to evaluate the considered models. The models were estimated using the Generalized Reduced Gradient (GRG) Non-Linear method in Excel’s Solver and the Adaptive Random Walk Metropolis (ARWM) in the Scilab program. The real data in the empirical study were Financial Times Stock Exchange Milano Italia Borsa (FTSEMIB) and Stoxx Europe 600 indices over the daily period from January 2000 to December 2017 to test the conditional variance process and see whether the estimation methods could adapt to the complicated models. The analysis shows that GRG Non-Linear in Excel’s Solver and ARWM methods have close results. It indicates a good estimation ability. Based on the Akaike Information Criterion (AIC), the QGARCH(1,1) model provides a better fitting than the GARCH(1,1) model on each distribution specification. Overall, the QGARCH(1,1) with SGED distribution best fits both data.

Dimensions

Plum Analytics

Author Biographies

Didit Budi Nugroho, Department of Mathematics & Data Science, Universitas Kristen Satya Wacana

Department of Mathematics & Data Science

Bintoro Ady Pamungkas, Department of Mathematics & Data Science, Universitas Kristen Satya Wacana

Department of Mathematics & Data Science

Hanna Arini Parhusip, Department of Mathematics & Data Science, Universitas Kristen Satya Wacana

Department of Mathematics & Data Science

References

Abdalla, S. Z. S., & Winker, P. (2012). Modelling stock market volatility using univariate GARCH models: Evidence from Sudan and Egypt. International Journal of Economics and Finance, 4(8), 161-176.

Akaike, H. (1998). Information theory and an extension of the maximum likelihood principle. In Selected papers of Hirotugu Akaike (pp. 199-213). New York: Springer.

Ardia, D., & Hoogerheide, L. F. (2010). Bayesian estimation of the GARCH (1, 1) model with student-t innovations. The R Journal, 2(2), 41-47.

Bi, Z., Yousuf, A., & Dash, M. (2014). A study on options pricing using GARCH and Black-Scholes-Merton model. Asian Journal of Finance & Accounting, 6(1), 423-439.

Blangiardo, M., & Cameletti, M. (2015). Spatial and spatio-temporal Bayesian Models with R-INLA. Chichester: John Wiley & Sons.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.

Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics, 69(3), 542-547.

Chen, M. H., & Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69-92.

Chen, M. H., Shao, M. Q., & Ibrahim, J. G. (2012). Monte Carlo methods in Bayesian computation. Springer Science & Business Media.

Deschamps, P. J. (2006). A flexible prior distribution for Markov switching autoregressions with student-t errors. Journal of Econometrics, 133(1), 153-190.

Engle, R. F., & Ng, V. K. (1993). Measuring and testing the impact of news on volatility. The Journal of Finance, 48(5), 1749 -1778.

Francq, C., & Zakoian, J. M. (2019). GARCH models: Structure, statistical inference and financial applications. John Wiley & Sons.

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. The Journal of Finance, 48(5), 1779-1801.

Huang, Z., Wang, T., & Hansen, P. R. (2017). Option pricing with the realized GARCH model: An analytical approximation approach. Journal of Futures Markets, 37(4), 328-358.

Jarque, C. M. (2011). Jarque-Bera test. In M. Lovric (Ed.), International encyclopedia of statistical science (pp. 701-702). Berlin, Heidelberg: Springer.

Le, H., Pham, U., Nguyen, P., & Pham, T. B. (2020). Improvement on Monte Carlo estimation of HPD intervals. Communications in Statistics-Simulation and Computation, 49(8), 2164-2180.

Marin, J. M., & Robert, C. P. (2014). Bayesian essentials with R. New York: Springer.

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica: Journal of the Econometric Society, 59(2), 347-370.

Nugroho, D. B. (2018). Comparative analysis of three MCMC methods for estimating GARCH models. IOP Conference Series: Materials Science and Engineering, 403, 1-7.

Nugroho, D. B., Kurniawati, D., Panjaitan, L. P., Kholil, Z., Susanto, B., & Sasongko, L. R. (2019a). Empirical performance of GARCH, GARCH-M, GJR-GARCH and log-GARCH models for returns volatility. Journal of Physics: Conference Series, 1307, 1-7.

Nugroho, D. B., Susanto, B., Prasetia, K. N. P., & Rorimpandey, R. (2019b). Modeling of returns volatility using GARCH (1, 1) model under Tukey transformations. Jurnal Akuntansi dan Keuangan, 21(1), 12-20.

Nugroho, D. B., Susanto, B., & Rosely, M. M. M. (2018). Penggunaan MS Excel untuk estimasi model GARCH (1, 1). Jurnal Matematika Integratif, 14(2), 71-82.

Powell, S. G., & Batt, R. J. (2014). Modeling for insight: A master class for business analysts. Wiley.

Rothwell, A. (2017). Optimization methods in structural design. Springer International Publishing.

Roy, V. (2020). Convergence diagnostics for Markov chain Monte Carlo. Annual Review of Statistics and Its Application, 7, 387-412.

Sentana, E. (1995). Quadratic ARCH models. The Review of Economic Studies, 62(4), 639-661.

Snipes, M., & Taylor, D. C. (2014). Model selection and Akaike Information Criteria: An example from wine ratings and prices. Wine Economics and Policy, 3(1), 3-9.

Takaishi, T. (2009). Bayesian inference on QGARCH model using the adaptive construction scheme. In 2009 Eighth IEEE/ACIS International Conference on Computer and Information Science (pp. 525-529). IEEE.

Theodossiou, P. (2015). Skewed generalized error distribution of financial assets and option pricing. Multinational Finance Journal, 19(4), 223-266.

Tsikerdekis, M. (2016). Bayesian inference. In J. Robertson & M. Kaptein (Eds.), Modern statistical methods for HCI (pp. 173-197). Springer.

Turner, B. M., Sederberg, P. B., Brown, S. D., & Steyvers, M. (2013). A method for efficiently sampling from distributions with correlated dimensions. Psychological Methods, 18(3), 368-384.

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2020-12-16

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