Modeling of Returns Volatility using GARCH(1,1) Model under Tukey Transformations
Keywords:Tukey transformation, Excel Solver, GARCH, Matlab, volatility
AbstractThis study proposed two new classes of GARCH(1,1) model by applying the Tukeytransformations to the returns and to the lagged variance. The behavior of return volatility was investigated on the basis of models with normal and Student-t distributions for return error. The competing models were estimated by using the Excel Solver and Matlab tools. The empirical analysis is based on simulated data, daily exchange rates of the IDR/USD, and daily stock indices of FTSE100 and TOPIX. This study recommends the use of Excel Solver for finance academics and practitioners working on volatility using GARCH(1,1) models. Our empirical findings conclude that GARCH(1,1) models under Tukey transformations should be considered in risk management decisions since the models are more appropriate than standard for describing returns and volatility of financial time series and its stylized facts including fat tails and mean reverting. The Tukey transformed returns imply a shorter volatility half-life, and thus this study suggests that investors should invest the observed assets in a shorter time period to obtain higher returns.
Ahmed, R. R., Vveinhardt, J., Streimikiene, D., and Channar, Z. A. (2018). Mean reversion in international markets: evidence from G.A.R.C.H. and half-life volatility models. Economic Research-Ekonomska Istraživanja, 31(1), 1198–1217.
Alexander, C., Market risk analysis II: Practical financial econometrics, Wiley. Chichester: John Wiley and Sons., 2008.
Anderson, D. R., Sweeney, D. J., and Williams, T. A. (2007). Essentials of Modern Business Statistics with Microsoft Excel (3rd ed.). Mason: Thomson South-Western.
Atchade, Y. F., and Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli, 11(5), 815–828.
Awalludin, S. A., Ulfah, S., and Soro, S. (2018). Modeling the stock price returns volatility using GARCH (1,1) in some Indonesia stock prices. Journal of Physics: Conference Series, 948(1), 1–7.
Bickel, P. J., and Doksum, K. A. (1981). An analysis of transformations revisited. Journal of the American Statistical Association, 76(374), 296–311.
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.
Bollerslev, T., Chou, R. Y., and Kroner, K. F. (1992). ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52(1–2), 5–59.
Box, G. E. P., and Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26, 211–252.
Braione, M., and Scholtes, N. (2016). Forecasting Value-at-Risk under different distributional assumptions. Econometrics, 4(1), 1–27.
Carnero, M. A., Pena, D., and Ruiz, E. (2004). Persistence and kurtosis in GARCH and stochastic volatility vodels. Journal of Financial Econometrics, 2(2), 319–342. https://acade-mic.oup.com/jfec/article-lookup/doi/10.1093/ jjfinec/nbh012
Charles, A., and Darné, O. (2014). Volatility persistence in crude oil markets. Energy Policy, 65, 729–742.
Chen, M.-H., and Shao, Q.-M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.
Christianti, A. (2018). Volatility shock persistence in investment decision making: A comparison between the consumer goods and property-real estate sectors of the Indonesian capital market. Journal of Indonesian Economy and Business, 33(2), 112–122.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987–1007.
Engle, R. F., and Bollerslev, T. 1986. Modelling the persistence of conditional variance. Econometric Reviews, 5, 1–50.
Engle, R. F., and Patton, A. J. (2001). What good is a volatility model? Quantitative Finance, 1(2), 237–245.
Hansen, P. R., and Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH (1,1)? Journal of Applied Econometrics, 20(7), 873–889.
Hatane, S. E. (2011). The predictability of GARCH-type models on the returns volatility of primary Indonesian exported agricultural commodities. Jurnal Akutansi dan Keuangan, 13(2), 87–97.
Hentschel, L. (1995). All in the family nesting symmetric and asymmetric GARCH models. Journal of Financial Economics, 39(1), 71–104.
Higgins, M. L., and Bera, A. K. (1992). A class of nonlinear ARCH models. International Economic Review, 33(1), 137–158.
Ho, K.-Y., and Tsui, K. C. 2004. Volatility dynamics of the Tokyo Stock Exchange: A sectoral analysis based on the multivariate GARCH approach (Money Macro and Finance (MMF) Research Group Conference 2004 No. 12).
Nugroho, D. B. 2018. Comparative analysis of three MCMC methods for estimating GARCH models. In IOP Conference Series: Materials Science and Engineering. IOP Publishing.
Nugroho, D. B., and Morimoto, T. (2014). Realized non-linear stochastic volatility models with asymmetric effects and generalized student’s t-distribution. Journal of The Japan Statistical Society, 44(1), 83–118.
Nugroho, D. B., and Morimoto, T. (2016). Box–Cox realized asymmetric stochastic volatility models with generalized Student’s t -error distributions. Journal of Applied Statistics, 43(10), 1906–1927.
Nugroho, D. B., and Susanto, B. 2017. Volatility modeling for IDR exchange rate through APARCH model with student-t distribution. In AIP Conference Proceedings (1868, 040005). AIP Publishing LLC.
Nugroho, D. B., Susanto, B., and Pratama, S. R. (2017). Estimation of exchange rate volatility using APARCH-type models: A case study of Indonesia (2010–2015). Jurnal Ekonomi Dan Ekonomi Studi Pembangunan, 9(1), 65–75.
Nugroho, D. B., Susanto, B., and Rosely, M. M. M. (2018). Penggunaan MS Excel untuk estimasi model GARCH (1,1). Jurnal Matematika Integratif, 14(2), 71–81.
Salvatore, D., International economics (11th ed.), John Wiley and Sons., New York, 2013.
Sarkar, N. 2000. ARCH model with Box–Cox transformed dependent variable. Statistics and Probability Letters, 50(4), 365–374.
Tsiotas, G. 2009. On the use of non-linear transformations in Stochastic Volatility models. Statistical Methods and Applications, 18(4), 555–583.
Tukey, J. W. 1957. On the comparative anatomy of transformations. The Annals of Mathematical Statistics, 28(3), 602–632.
Tung, H. K. K., Lai, D. C. F., and Wong, Mi. C. S., Professional financial computing using Excel and VBA, John Wiley and Sons., Singapore, 2010.
Utami, H., and Subanar. (2013). Second order least square estimation on ARCH(1) model with Box-Cox transformed dependent variable. Journal of the Indonesian Mathematical Society, 19(1), 99–110.
Vošvrda, M., and Žikeš, F. 2004. An application of the GARCH-t model on Central European Stock Returns. Prague Economic Papers, 13(1), 26–39.
Wei, C.-C. (2015). Does the SRI stock index return co‐movements : Evidence of the FTSE stock markets. Journal of Business, Economics and Finance, 4(4), 600–616.
Zivot, E. 2009. Practical issues in the analysis of univariate GARCH models. In T. G. Andersen, R. A. Davis, J.-P. Kreib, and T. Mikosch (Eds.), Handbook of Financial Time Series (p. 113). Berlin, Heidelberg: Springer-Verlag.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and publishing right, and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) followingthe publication of the article, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).<a href="http://creativecommons.org/lice