Improvement of Real-GJR Model using Jump Variables on High Frequency Data

Didit Budi Nugroho; Nadya Putri Wulandari; Yumita Cristin Alfagustina; Hanna Arini Parhusip; Faldy Tita; Bambang Susanto

doi:10.31764/jtam.v8i4.24294

Improvement of Real-GJR Model using Jump Variables on High Frequency Data

Didit Budi Nugroho, Nadya Putri Wulandari, Yumita Cristin Alfagustina, Hanna Arini Parhusip, Faldy Tita, Bambang Susanto

Abstract

Volatility is a key indicator in assessing risk when making investment decisions. In the world of financial markets, volatility reflects the degree to which the value of a financial asset fluctuates over a given period. The most common way to measure the future loss potential of an investment is through volatility. Focusing on the Realized GJR (RealGJR) volatility model, which consists of return, conditional volatility, and measurement equations, this study proposes the RealGJR-CJ model developed by decomposing the exogenous variable in the volatility equation of RealGJR into continuous C and discontinuous (jump) J variables. The decomposition of exogenous variables makes the RealGJR-CJ model follow realistic financial markets, where the asset volatility is a continuous process with some jump components. As an empirical illustration, the models are applied to an index in the Japanese stock market, namely Tokyo Stock Price Index, covering from January 2004 to December 2011. The observed exogenous variable in the volatility equation of RealGJR models is Realized Volatility (RV), which is calculated using intraday data with time intervals of 1 and 5 minutes. Adaptive Random Walk Metropolis method was employed in Markov Chain Monte Carlo algorithm to estimate the model parameters by updating the parameters during sampling based on previous samples from the chain. From the results of running the MCMC algorithm 20 times, the mean of the information criteria of competing models is significantly different based on standard deviation and the result suggests that the model with continuous and jump variables can improve the model without jump. The best fit model is provided by RealGJR-CJ with the adoption of 1-minute RV data.

Keywords

Adaptive Random Walk; Continuous and Jump; Realized GJR; TOPIX.

Full Text:

DOWNLOAD [PDF]

References

Atchade, Y. F., & Rosenthal, J. S. (2005). On adaptive Markov chain Monte Carlo algorithms. Bernoulli, 11(5), 815–828. https://doi.org/10.3150/bj/1130077595

Black, F. (1976). Studies of stock market volatility changes. Proceedings of the 1976 Meetings of the Business and Economic Statistics, 177–181. https://www.scirp.org/reference/referencespapers?referenceid=2030459

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327. https://doi.org/10.1016/0304-4076(86)90063-1

Caporin, M., & Costola, M. (2019). Asymmetry and leverage in GARCH models: a News Impact Curve perspective. Applied Economics, 51(3), 3345–3364. https://doi.org/10.1080/00036846.2019.1578853

Chen, H., Zhang, J., Tao, Y., & Tan, F. (2019). Asymmetric GARCH type models for asymmetric volatility characteristics analysis and wind power forecasting. Protection and Control of Modern Power Systems, 4(1), 29. https://doi.org/10.1186/s41601-019-0146-0

Degiannakis, S., Filis, G., Klein, T., & Walther, T. (2022). Forecasting realized volatility of agricultural commodities. International Journal of Forecasting, 38(1), 74–96. https://doi.org/10.1016/J.IJFORECAST.2019.08.011

Dziak, J. J., Coffman, D. L., Lanza, S. T., Li, R., & Jermiin, L. S. (2020). Sensitivity and specificity of information criteria. Briefings in Bioinformatics, 21(2), 553–565. https://doi.org/10.1093/bib/bbz016

Engle, R. (2002). New Frontiers for ARCH Models. Journal of Applied Econometrics, 17(5), 425–446. https://doi.org/10.1002/jae.683

Floros, C., Gkillas, K., Konstantatos, C., & Tsagkanos, A. (2020). Realized measures to explain volatility changes over time. Journal of Risk and Financial Management, 13(6), 125. https://doi.org/10.3390/jrfm13060125

Gerlach, R., & Wang, C. (2016). Forecasting risk via realized GARCH, incorporating the realized range. Quantitative Finance. https://doi.org/10.1080/14697688.2015.1079641

Gkillas, K., Gupta, R., & Pierdzioch, C. (2020). Forecasting realized gold volatility: Is there a role of geopolitical risks? Finance Research Letters, 35, 1–6. https://doi.org/10.1016/j.frl.2019.08.028

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. https://doi.org/10.1111/j.1540-6261.1993.tb05128.x

Hansen, P. R., Huang, Z., & Shek, H. H. (2012). Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility. Journal of Applied Econometrics, 27(6), 877–906. https://doi.org/10.1002/jae.1234

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(9), 2164–2180. https://doi.org/10.1080/03610918.2018.1513141

Mostafa, F., Saha, P., Islam, M. R., & Nguyen, N. (2021). GJR-GARCH Volatility Modeling under NIG and ANN for Predicting Top Cryptocurrencies. Journal of Risk and Financial Management, 14(9), 421. https://doi.org/10.3390/jrfm14090421

Nugroho, D. B. (2018). Comparative analysis of three MCMC methods for estimating GARCH models. IOP Conference Series: Materials Science and Engineering, 403(1), 012061. https://doi.org/10.1088/1757-899X/403/1/012061

Nugroho, D. B., Kurniawati, D., Panjaitan, L. P., Kholil, Z., Susanto, B., & Sasongko, L. R. (2019). Empirical Performance of GARCH, GARCH-M, GJR-GARCH and log-GARCH Models for Returns Volatility. Journal of Physics: Conference Series, 1307(1), 012003. https://doi.org/10.1088/1742-6596/1307/1/012003

Nugroho, D. B., Mahatma, T., & Pratomo, Y. (2021). Applying the non-linear transformation families to the lagged-variance of EGARCH and GJR models. IAENG International Journal of Applied Mathematics, 51(4), 908–919. https://www.iaeng.org/IJAM/issues_v51/issue_4/IJAM_51_4_12.pdf

Nugroho, D. B., & Morimoto, T. (2019). Incorporating realized quarticity into a realized stochastic volatility model. Asia-Pacific Financial Markets, 26(4), 495–528. https://doi.org/10.1007/s10690-019-09276-2

Nugroho, D. B., Panjaitan, L. P., Kurniawati, D., Kholil, Z., Susanto, B., & Sasongko, L. R. (2022). GRG non-linear and ARWM methods for estimating the GARCH-M, GJR, and log-GARCH models. Jurnal Teori Dan Aplikasi Matematika, 6(2), 448–460. https://doi.org/10.31764/jtam.v6i2.7694

Nugroho, D. B., Susanto, B., & Rosely, M. M. M. (2018). Penggunaan MS Excel untuk estimasi model GARCH(1,1). Jurnal Matematika Integratif, 14(2), 71–83. https://doi.org/10.24198/jmi.v14.n2.17680.71-82

Nugroho, D. B., Wibowo, H., & Saragih, A. (2024). Modeling daily return volatility through GJR(1, 1) model and realized volatility measure. Thailand Statistician, 22(1), 50–62. https://ph02.tci-thaijo.org/index.php/thaistat/article/view/252221

Nugroho, D. B., Wicaksono, B. A. A., & Larwuy, L. (2023). GARCH-X(1,1) model allowing a non-linear function of the variance to follow an AR(1) process. Communications for Statistical Applications and Methods, 30(2), 163–178. https://doi.org/10.29220/CSAM.2023.30.2.163

Robert, C. P., Elvira, V., Tawn, N., & Wu, C. (2018). Accelerating MCMC algorithms. Wiley Interdisciplinary Reviews: Computational Statistics, 10(5), 1–14. https://doi.org/10.1002/wics.1435

Roy, V. (2020). Convergence diagnostics for Markov chain Monte Carlo. Annual Review of Statistics and Its Application, 7, 387–412. https://doi.org/10.1146/annurev-statistics-031219-041300

Sheraz, M., & Nasir, I. (2021). Information-theoretic measures and modeling stock market volatility: A comparative approach. Risks, 9(5), 89. https://doi.org/https://doi.org/10.3390/risks9050089

Sinha, A. K. (2021). The reliability of geometric Brownian motion forecasts of S&P500 index values. Journal of Forecasting, 40(8), 1444–1462. https://doi.org/10.1002/for.2775

Wang, Y., Xiang, Y., Lei, X., & Zhou, Y. (2022). Volatility analysis based on GARCH-type models: Evidence from the Chinese stock market. Economic Research-Ekonomska Istrazivanja , 35(1), 2530–2554. https://doi.org/10.1080/1331677X.2021.1967771

Zhang, H., & Lan, Q. (2014). GARCH-type model with continuous and jump variation for stock volatility and its empirical study in China. Mathematical Problems in Engineering, 2014. https://doi.org/10.1155/2014/386721

DOI: https://doi.org/10.31764/jtam.v8i4.24294