# Study on The Continuous-Jump Behavior of Asset Return Volatility Through The GJR Model

## Keywords:

ARWM, GJR, jump, volatility## Abstract

Generalized Auto-Regressive Conditional Heteroskeasticity (GARCH) is a model used to predict the volatility of returns. Volatility is a statistical measure of the movement of returns for securities (financial instruments that can only be traded through markets or securities companies) or certain market indices. Then the GARCH model was further developed into an asymmetric form, namely conditional volatility and returns have a relationship, namely the GJR model which is an abbreviation of the name (Glosten- Jagannathan-Runkle). This research focuses on the GJR-X by adding high-frequency exogenous variables in volatility process and on the GARCH-CJ which is a decomposition of the exogenous variable X, namely the continuous component C (Continuous) and the jump J (Jump). TOPIX data (Tokyo Stock Price Index) is the real data used in this study. To estimate the model parameters, the ARWM (Adaptive Random Walk Metropolis) method will be used with the MCMC (Markov Chain Monte Carlo) algorithm. First, it was found that the ARWM method is good at estimating parameters. Second, the AIC value of GJR-CJ was smaller than that of GJR-X, which means that GJR-CJ had better data fitting.

## References

[2] T. Bollerslev, “Generalized autoregressive conditional heteroskedasticity,” J.

Econom., vol. 31, no. 3, pp. 307–327, Apr. 1986, doi: 10.1016/0304-4076(86)90063-1.

[3] V. Mahajan, S. Thakan, and A. Malik, “Modeling and Forecasting the Volatility of

NIFTY 50 Using GARCH and RNN Models,” Economies, vol. 10, no. 5, pp. 1–20,

2022, doi: 10.3390/economies10050102.

[4] L. R. Glosten, R. Jagannathan, and D. E. Runkle, “On the Relation between the

Expected Value and the Volatility of the Nominal Excess Return on Stocks,” J.

Finance, vol. 48, no. 5, pp. 1779–1801, 1993, doi: 10.1111/j.1540-

6261.1993.tb05128.x.

[5] C. Brownlees, R. Engle, and B. Kelly, “A practical guide to volatility forecasting

through calm and storm,” J. Risk, vol. 14, no. 2, pp. 3–22, 2011, doi:

10.21314/JOR.2012.237.

[6] D. B. Nugroho, T. Mahatma, and Y. Pratomo, “Applying the Non-linear

Transformation Families to the Lagged-variance of EGARCH and GJR Models,”

IAENG Int. J. Appl. Math., vol. 51, no. 4, pp. 1–9, 2021.

[7] R. Engle, “New frontiers for ARCH models,” J. Appl. Econom., vol. 17, no. 5, pp. 425–

446, 2002, doi: 10.1002/jae.683.

[8] H. Zhang and Q. Lan, “GARCH-type model with continuous and jump variation for

stock volatility and its empirical study in China,” Math. Probl. Eng., vol. 2014, 2014,

doi: 10.1155/2014/386721.

[9] S. Djamaluddin, R. Ardoni, and A. Herawati, “Stock Price Index (CSPI)In Indonesia

Stock Exchange (IDX) Period 2014-2018,” Dinasti Int. J. Econ. Financ. Account. , vol.

1, no. 1, pp. 40–53, 2020, doi: 10.38035/DIJEFA.

[10] M. Sultonov, “External Shocks and Volatility Overflow among the Exchange Rate of

the Yen , Nikkei , TOPIX and Sectoral Stock Indices,” 2021.

[11] G. Cai, “Markov chain monte carlo,” pp. 1–9, 2020.

[12] D. B. Nugroho, “Comparative analysis of three MCMC methods for estimating

GARCH models,” IOP Conf. Ser. Mater. Sci. Eng. , pp. 0–7, 2018, doi: 10.1088/1757-

899X/403/1/012061.

[13] X.-S. Yang, Introduction to Algorithms for Data Mining and Machine Learning.

Candice Janco, 2019. [Online]. Available: https://www.ptonline.com/articles/how-toget-

better-mfi-results

[14] M. H. Chen and Q. M. Shao, “Monte carlo estimation of bayesian credible and hpd

intervals?,” J. Comput. Graph. Stat., vol. 8, no. 1, pp. 69–92, 1999, doi:

10.1080/10618600.1999.10474802.

[15] H. Akaike, “A New Look at the Statistical Model Identification,” IEEE Trans.

Automat. Contr., vol. 19, no. 6, pp. 716–723, 1974.

[16] S. Portet, “A primer on model selection using the Akaike Information Criterion,”

Infect. Dis. Model., vol. 5, pp. 111–128, 2020, doi: 10.1016/j.idm.2019.12.010.

[17] V. Roy, “Convergence diagnostics for markov chain monte carlo,” Annu. Rev. Stat.

Its Appl., vol. 7, pp. 387–412, 2020, doi: 10.1146/annurev-statistics-031219-041300.

## Downloads

## Published

## How to Cite

*Prosiding University Research Colloquium*, 20–28. Retrieved from http://repository.urecol.org/index.php/proceeding/article/view/2670

## Issue

## Section

## License

Copyright (c) 2023 Yumita Cristin Alfagustina, Didit Budi Nugroho, Faldy Tita

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.