Empirical performance of GARCH, GARCH-M, GJR-GARCH and log-GARCH models for returns volatility

Abstract

Volatility plays an important role in the field of financial econometrics as one of the risk indicators. Many various models address the problem of modeling the volatilities of financial asset returns. This study provides a new empirical performance comparison of the four different GARCH-type models, namely GARCH, GARCH-M, GJR-GARCH, and log-GARCH models based on simulated data and real data such as the DJIA, S&P 500, and S&P CNX Nifty indices on a daily period from January 2000 to December 2017. We also investigate the estimation results obtained using Solver’Excel and verify those results against the results obtained using a Markov chain Monte Carlo method. The simulation study showed that the GARCH model is outperformed by other models. Meanwhile, the empirical study provides evidence that the GJR-GARCH model provides the best fitting, followed by the GARCH-M, GARCH, and log-GARCH models. Furthermore, this study recommends the use of Excel’s Solver in practice when the parameter estimates for GARCH-type model do not close to zero.

Skew Normal and Skew Student-T Distributions on Garch(1,1) Model

“…Moreover, the results also show that none of the estimation deviates from the model constraints even though Excel's Solver does not provide strict conditions for inequality. This means that the result of , , > 0 or + = 1, as in Nugroho, Kurniawati, et al (2019) and , is not obtained. Therefore, it can be said that GRG Non-Linear's Excel's Solver method is reliable for estimating GARCH(1,1) models with SN and ST distributions, even though both have complex likelihood functions.…”

“…For example, Solver in Microsoft Excel was studied by Nugroho et al (2018) for GARCH(1,1) model cases with Normal distribution. Recently, Nugroho, Kurniawati, et al (2019, and Kusumawati et al (2020) successfully employed Excel's Solver to estimate GARCH type models, such as GARCH-in-Mean, GJR-GARCH, log-GARCH, and EGARCH, with return errors in the models following Normal and Student-t distributions.…”

“…As a simple framework, they focused on GARCH(1,1) models with Normally distributed return errors, and noted a critical issue in selecting initial values, namely that the initial values should be close to the possible solutions. For more complex models, the GRG Non-Linear's Excel's Solver method was used by Nugroho, Kurniawati, et al (2019) to estimate GARCH-in-Mean, GJR-GARCH, and log-GARCH models with Normally distributed return errors, and by to estimate non-linear GARCH models with return errors that have Normal and Student-t distributions. Both studies found that Excel's Solver produces estimation values for several parameters that violate the model constraints even though it has no effect on other parameter values and objective functions.…”

“…This improvement is done by handling changes in parameter proposal values automatically. To the best of our knowledge, only few researches study the use of ARWM method to estimate GARCH-type models, for example see Salim et al (2016), , Nugroho (2018), Nugroho, Kurniawati, et al (2019), , and Kusumawati et al (2020). In particular, Nugroho (2018) reported that the ARWM method was statistically efficient (in terms of correlation) to estimate GARCH(1,1) models with Normal and Student-t distributions.…”

The Capital Asset Pricing Model

“…The clustering of volatility implies memory in absolute returns over long time horizons [150]. This observation casts doubt on the entire ARCH-GARCH family of volatility models [151][152][153]. Though these approaches recognize autocorrelation in volatility across multiple time steps, they also presume exponential decay [150].…”