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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

Empirical performance of GARCH, GARCH-M, GJR-

GARCH and log-GARCH models for returns volatility

D B Nugroho*, D Kurniawati, L P Panjaitan, Z Kholil, B Susanto, L R Sasongko

Department of Mathematics
*Study Center for Multidisciplinary Applied Research and Technology (SeMARTy)
Universitas Kristen Satya Wacana
Jl. Diponegoro 52–60 Salatiga 50711, Jawa Tengah, Indonesia

*didit.budinugroho@staff.uksw.edu; didit.budinugroho@uksw.edu

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.

1. Introduction
The theory and practice of economic development led to the empirical findings that financial return time
series exhibit heteroscedasticity, means the volatility (standard deviation) of returns changes over time.
Since Engle [1] introduced Autoregressive Conditional Heteroscedasticity (ARCH) model for modeling
the current variance (squared-volatility) as a linear function of past squared-return and generalized by
Bollerslev [2] to the GARCH (Generalized ARCH) model by adding the past conditional variance, study
in the financial econometrics area is dominated by the modification of ARCH and GARCH-type models
and get serious attention from researchers, practitioners, and policymakers. GARCH-type models
contribute to the ease of predicting the future volatility of financial time series and hence the result in
financial applications can help investors to make investment decisions.
A number of modifications of the standard GARCH models have been conducted for many years
such as the GARCH-in-Mean (GARCH-M), GJR-GARCH, and log-GARCH models. The GARCH-M
model was proposed by Engle et al. [3] which introduces an effect of conditional volatility into the
returns process. Glosten et al. [4] extended the GARCH-M and suggested a model, popularly known as
the GJR-GARCH, allowing the current variance has a different response to the past return. Another class
of the GARCH-type model that appears to have the same characteristics but less attention is the so-
called log-GARCH models introduced by Geweke [5] and Pantula [6] independently. Asai [7] derived

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

this model as a representation of the simple Stochastic Volatility (SV) process (see [8] for a discussion
on SV models). Recently, Sucarrat et al. [9] proposed a generalization of the log-GARCH model by
adding other conditioning variables to the conditional volatility equation. Unfortunately, Asai and
Sucarrat et al. do not provide an empirical study whether the log-GARCH model improves the GARCH
model. Motivated by their studies, it is important to check the performance of the log-GARCH model
for simulated and real data. Meanwhile, some empirical studies showed that the GARCH-M and GJR-
GARCH models work well, e.g., in [10–14]. To the best of authors’ knowledge, there is no literature
that provides an empirical comparison of the above models. Therefore, the main contribution of this
study is to investigate the empirical performance and some characteristics of the GARCH-M, GJR-
GARCH, log-GARCH models compared to the standard GARCH model. Another important
contribution is the use of Excel’s Solver to estimate the model parameters. So, the main difference
between this paper and others is the use of Excel’s Solver and both simulated and real data to check the
modeling performance of GARCH models.
The remainder of this paper is organized as follows. Section 2 discusses the considered models and
the empirical methods employed in estimation. Section 3 conducts the empirical study and its results on
the basis of simulation and real data. Section 4, finally, concludes and offers some possible modifications
to improve the models.

2. Four types of GARCH model and their estimation

2.1. GARCH models

Conditional variance determined through the GARCH model allows the current conditional variance
depends not only on past return but also on the past conditional variance. Let 𝑅𝑡 denote the asset returns
at time t and computed by 𝑅𝑡 = 100 × (log 𝑆𝑡 − log 𝑆𝑡−1 ), where 𝑆𝑡 denote the price of an asset at time
t. A simple specification of GARCH-type models, the so-called GARCH(1,1), is expressed by
𝑅𝑡 = 𝜎𝑡 𝜀𝑡 , 𝜀𝑡 ~𝑁(0,1),
2 2 2
𝜎𝑡 = 𝜔 + 𝛼𝑅𝑡−1 + 𝛽𝜎𝑡−1 ,
where 𝜔 > 0, 𝛼 ≥ 0 and 𝛽 ≥ 0 for the positivity of variances and 0 ≤ 𝛼 + 𝛽 < 1 for the stationarity
of variances. On the basis of empirical study, Hansen and Lunde [15] found no evidence that the
GARCH(1,1) model is inferior to other ARCH-type models.

2.2. GARCH-M models

The GARCH-M model was proposed by Engle et al. [3] by establishing a relationship between return
and conditional variance directly where the current return is expressed as a linear function of the current
variance. In particular, the GARCH(1,1)-M model is defined by
𝑅𝑡 = 𝑘𝜎𝑡2 + 𝜎𝑡 𝜀𝑡 , 𝜀𝑡 ~𝑁(0,1),
2 2
𝜎𝑡2 = 𝜔 + 𝛼𝑅𝑡−1 + 𝛽𝜎𝑡−1 ,
where the constraints for parameters are the same as those in the GARCH(1,1) model.

2.3. GJR-GARCH models

The GJR-GARCH model of Glosten et al. [4], also known as the threshold GARCH (T-GARCH) model,
is proposed to capture an asymmetric behavior by allowing the current conditional variance has a
different response to the past positive and negative returns. In particular, the model GJR-GARCH(1,1)
is expressed as follows:
𝑅𝑡 = 𝜎𝑡 𝜀𝑡 , 𝜀𝑡 ~𝑁(0,1)
2 2
𝜎𝑡2 = 𝜔 + (𝛼 + 𝛾𝐼𝑡−1 )𝑅𝑡−1 + 𝛽𝜎𝑡−1 ,
where the positivity of conditional variances is assured by 𝜔 > 0, 𝛼 ≥ 0, 𝛽 ≥ 0, and 𝛼 + 𝛾 ≥ 0, the
variances stationary is assured by 𝛼 + 𝛽 + 0.5𝛾 < 1, and I is an indicator function that expressed by
0 if 𝑅𝑡−1 ≥ 0, (good news)
𝐼𝑡−1 = {
1 if 𝑅𝑡−1 < 0. (bad news)

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

The following interpretation is given in [16]. When 𝛾 = 0, the model reduces to the standard GARCH
model which treats bad news (𝑅𝑡−1 < 0) and good news (𝑅𝑡−1 > 0) symmetrically: that is, bad news
2
and good news have the same impact (𝛼𝑅𝑡−1 ) on the conditional variance 𝜎𝑡2 . When 𝛾 ≠ 0, the news
impact is asymmetric: that is, bad news and good news have different impacts on the conditional
variance. Bad news has an impact of 𝛼 + 𝛾 on conditional variance, while good news has an impact of
𝛼 on conditional variance. Hence, if 𝛾 > 0, bad news has a larger impact on conditional variance than
good news.

2.4. Log-GARCH models

Motivated by thinking about the positivity of conditional variances, Geweke [5] and Pantula [6] gave a
logarithmic version of the GARCH model by taking the logarithm of the current conditional variance as
a linear function of the logarithm of past squared-return and the logarithm of past conditional variance.
This study focuses on the log-GARCH(1,1) model expressed by
𝑅𝑡 = 𝜎𝑡 𝜀𝑡 , where 𝜀𝑡 ~𝑁(0,1),
2 2
log 𝜎𝑡2 = 𝜔 + 𝛼 log 𝑅𝑡−1 + 𝛽 log 𝜎𝑡−1 ,
where |𝛼 + 𝛽| < 1 for the stationarity of log-variances and (𝛼 + 𝛽)𝛽 > 0 for the positivity of variances,
see [7] for the explanation.
For all the above models, the objective is to estimate parameters that maximize the conditional log-
likelihood function. In general, the conditional log-likelihood function for the above models is as
follows:
1 (𝑅𝑡 − 𝑘𝜎𝑡2 )2
log 𝐿 (𝑅𝑡 |𝜎𝑡 2 ) = − [log(2𝜋𝜎𝑡2 ) + ],
2 𝜎𝑡2
where the above function is for the GARCH-M(1,1) model when 𝑘 ≠ 0 or for the others when 𝑘 = 0.
The models are adjusted to the simulated and real data by the Excel’s Solver tool, which is easy to use
and has been widely used by academics and financial practitioners. Implementation of the Excel’s Solver
for the GARCH(1,1) model was demonstrated by some researchers, e.g., Alexander [17], Christoffersen
[18], and Nugroho et al [19]. Furthermore, the Adaptive Random Walk Metropolis (ARWM) method
as in [20] is implemented in Matlab to verify the Excel’s Solver estimates. The method has been showed
more efficient and much faster than the Hamiltonian Monte Carlo method that was used in [21–23].

3. Simulation and empirical application

This section investigates the fitting performance of the competing models on the basis of simulated and
real data. In this application, we follow the experimental procedures and algorithms demonstrated by
Nugroho et al [19] and Nugroho [20].

3.1. Simulation study

The simulations are carried out by generating 1,000 returns data from each model, excluding the
GARCH(1,1) model, using the true parameter values presented in tables 1–3. These values are set based
on the empirical studies in the literature. In this case, the Excel’s Solver is initialized by setting the initial
values 𝜔 = 0.03, 𝛼 = 0.04, 𝛽 = 0.89, 𝑘 = 0.5 for the GARCH-M(1,1) model, 𝜔 = 0.06, 𝛼 = 0.38,
𝛾 = −0.31, 𝛽 = 0.4 for the GJR-GARCH(1.1) model, and 𝜔 = 0.04, 𝛼 = 0.04, 𝛽 = 0.89 for the log-
GARCH(1,1) model. The estimation results by the Excel’s Solver are presented in tables 1–3 for the
GARCH(1,1)-M, GJR-GARCH(1,1), and log-GARCH(1,1) models, respectively. The log(L) and LRT
Stat represent log-likelihood and Log-likelihood Ratio Test statistic, respectively. The LRT statistic is
computed by twice the difference in log-likelihoods.
In terms of relative errors, the authors found that Excel’s Solver is reliable to estimate all considered
models. However, the authors noted that Excel’s Solver is very sensitive to the initial guess value of a
parameter. If the parameter estimate cannot be found, the author uses an initial value that is close to the
true value. Furthermore, all cases provide evidence supporting the extended models indicated by the
values of LRT statistic which are larger than the chi-square critical values at any conventional level with

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

degrees of freedom equal to the difference in the number of parameters. It means that all extended
models fit significantly better than the standard model. This result shows that all extended GARCH
models have the potential to outperform the standard GARCH model.

Table 1. Simulation results from the GARCH(1,1)-M models.

True GARCH(1,1)-M GARCH(1,1)

Parameter
value Estimate Relative error Estimate Relative error
𝜔 0.04 0.035 13.6% 0.026 35.4%
𝛼 0.05 0.044 11.4% 0.069 38.0%
𝛽 0.90 0.914 1.6% 0.914 1.6%
𝑘 0.50 0.437 12.7%
Total log(L) –1473.15 –1577.38
LRT Stat 208.46
Source: Authors’ calculations.

Table 2. Simulation results from the GJR-GARCH(1,1) and GARCH(1,1) models.

True GJR-GARCH(1,1) GARCH (1,1)
Parameter
value Estimate Relative error Estimate Relative error
𝜔 0.07 0.071 1.17% 0.082 16.97%
𝛼 0.40 0.399 0.29% 0.280 30.13%
𝛾 –0.30 –0.303 0.94%
𝛽 0.50 0.502 0.40% 0.428 14.35%
Total log(L) –717.13 –726.19
LRT Stat 18.12
Source: Authors’ calculations.

Table 3. Simulation results from the log-GARCH(1,1) and GARCH(1,1) models.

True log-GARCH(1,1) GARCH(1,1)
Parameter
value Estimate Relative error Estimate Relative error
𝜔 0.05 0.061 22.22% 0.070 39.35%
𝛼 0.05 0.066 32.05% 0.110 120.82%
𝛽 0.90 0.857 4.74% 0.800 11.11%
Total log(L) –1254.30 –1264.55
LRT Stat 20.50
Source: Authors’ calculations.

3.2. Empirical results for exchange rates

This section provides the empirical analysis of four considered models using daily returns for the Dow
Jones Industrial Average (DJIA), Standard and Poors 500 (S&P 500), and S&P CNX Nifty stock indices.

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

The sample period is from January 2000 to December 2017, excluding the zero returns. These data have
been downloaded from the Oxford-Man Institute of Quantitative Finance (https://realized.oxford-
man.ox.ac.uk).

Table 4. Empirical results.

Stock Model Tool 𝜔 𝛼 𝛽 𝛾 k log(L)
Solver 0.0107 0.1038 0.8890 –5732.74
GARCH(1,1)
Matlab 0.0111 0.1053 0.8873 –5734.12
GARCH(1,1)- Solver 0.0107 0.1037 0.8891 0.0178 –5731.84
M Matlab 0.0116 0.1072 0.8848 0.0589 –5723.68
DJIA
GJR- Solver 0.0143 0.0000 0.8977 0.1825 –5640.75
GARCH(1,1) Matlab 0.0149 0.0053 0.8919 0.1838 –5642.85
log- Solver 0.0900 0.0635 0.9209 –5806.16
GARCH(1,1) Matlab 0.0934 0.0659 0.9175 –5807.93
Solver 0.0103 0.0947 0.8979 –5868.08
GARCH(1,1)
Matlab 0.0109 0.0964 0.8956 –5869.50
GARCH(1,1)- Solver 0.0104 0.0949 0.8975 0.0075 –5862.74
S&P M Matlab 0.0115 0.0989 0.8925 0.0470 –5862.89
500 GJR- Solver 0.0154 0.0000 0.8974 0.1754 –5772.04
GARCH(1,1) Matlab 0.0161 0.0033 0.8923 0.1790 –5779.07
log- Solver 0.0782 0.0544 0.9331 –5948.55
GARCH(1,1) Matlab 0.0806 0.0560 0.9308 –5950.27
Solver 0.0193 0.1039 0.8850 –5422.50
GARCH(1,1)
Matlab 0.0225 0.1142 0.8734 –5424.29
GARCH(1,1)- Solver 0.0193 0.1038 0.8851 0.0157 –5421.80
S&P M Matlab 0.0228 0.1130 0.8739 0.0367 –5419.90
CNX
Nifty GJR- Solver 0.0229 0.0558 0.8788 0.1040 –5395.81
GARCH(1,1) Matlab 0.0244 0.0582 0.8744 0.1071 –5396.74
log- Solver 0.0367 0.0245 0.9730 –5533.61
GARCH(1,1) Matlab 0.0372 0.0248 0.9725 –5535.13
Source: Authors’ calculations.

The estimation results are summarized in table 4. To verify the results of Excel’s Solver in the case
of real data, the Matlab estimates are also provided. In all cases, the results show that Excel’s Solver
and Matlab produce similar estimates. The only exception is the parameter estimate of k for the
GARCH(1,1)-M model and the parameter estimate of 𝛼 for the GJR-GARCH(1,1) model in the case of
DJIA and S&P 500 stocks. In particular, authors found that the Excel’s Solver produce a value of zero
when the estimate tends to zero. However, athe uthors note that the value of zero for 𝛼 does not greatly
affect other estimates. This result indicates the Excel’s Solver is not reliable when the estimate of a

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

parameter is too close to zero. For this point of view, the next analysis only focuses on the results of the
Matlab.
In the case of DJIA data, the values of LRT statistic for the GARCH(1,1)-M, GJR-GARCH(1,1), and
log-GARCH(1,1) models against the GARCH(1,1) model are 20.88 (significant), 182.54 (significant),
–147.62 (not significant), respectively. In the case of S&P 500, the values of LRT statistic for the
GARCH(1,1)-M, GJR-GARCH(1,1), and log-GARCH(1,1) models against the GARCH(1,1) model are
13.22 (significant), 180.86 (significant), –161.54 (not significant), respectively. In the case of S&P 500
CNX Nifty data, the values of LRT statistic for the GARCH(1,1)-M, GJR-GARCH(1,1), and log-
GARCH(1,1) models against the GARCH(1,1) model are 8.78 (significant), 55.10 (significant), –221.68
(not significant), respectively. These results indicate that the GJR-GARCH(1,1) model provide the best
fit for all considered data, followed by the GARCH(1,1)-M, GARCH(1,1), and log-GARCH (1,1)
models.
Considering the GJR-GARCH(1,1) model, the estimate of 𝛾 in all data cases is positive. It indicates
the presence of the asymmetric effect of the past returns on the current conditional variance. Comparing
the variance persistence, which is denoted by 𝜙 and given by 𝜙 = 𝛼 + 𝛽 for the GARCH(1,1) and
GARCH(1,1)-M models, by 𝜙 = 𝛼 + 𝛽 + 0.5𝛾 for the GJR-GARCH(1,1) model, and by 𝜙 = |𝛼 + 𝛽|
for the log-GARCH(1,1) model, the persistences implied by the GARCH(1,1) and GARCH(1,1)-M
models are very close to each other and greater than those implied by the GJR-GARCH(1,1) model.
These results show that adding the conditional variance in the returns process does not greatly affect on
the variance persistence. Meanwhile, the presence of asymmetric effect causes the conditional variance
is less persistent and more volatile. Furthermore, according to the variance half-life defined by
log(0.5)/ log(𝜙) in [24], the conditional variance implied by the GJR-GARCH(1,1) model is faster to
move halfway back towards its unconditional mean than those implied by the GARCH(1,1) and
GARCH(1,1)-M models. For example, in the case of adopting the S&P 500 returns, the conditional
variance takes about 47 days implied by the GJR-GARCH(1,1) model, which is shorter than 81 days
implied by the GARCH(1,1)-M model and than 87 days implied by the GARCH(1,1) model. Finally,
considering the log-GARCH(1,1) model, the estimation results show that the logarithmic transformation
does not greatly affect the parameter estimates.

4. Conclusion and future works

This study compared empirical performance of the four different GARCH(1,1)-type models, including
the standard GARCH(1,1), GARCH(1,1)-M, GJR-GARCH(1,1), and log-GARCH(1,1) models, on the
basis of simulated data and real data, namely DJIA, S&P 500, and S&P 500 CNX Nifty stock indices.
The results are as follows. First, on the basis of the simulation and empirical results, the Excel’s Solver
is reliable to estimate the GARCH(1,1)-type models when the estimate of 𝛼 does not too close to zero.
It confirms the result of Nugroho et al. [25]. Second, the simulation results show that all extended
GARCH(1,1) models have the potential to better fit than the standard GARCH(1,1) model. In the case
of real application, the first and second best fitting specification are respectively provided by the GJR-
GARCH(1,1) and GARCH(1,1)-M models, meanwhile, the log-GARCH(1,1) model is outperformed by
the GARCH(1,1) model.
There are several possible modifications of the above models that deserve further study. The future
study could focus on the models’ comparison along with two distributions (normal and Student-t). It
would be interesting to investigate the application of Box-Cox transformations for the return series and
the lagged variance as in [26,27].

Acknowledgments
Sincere thanks to: (1) Higher Education Excellent Fundamental Research Grant of The Ministry of
Research, Technology and Higher Education of the Republic of Indonesia for the 2019-2020 fiscal year,
and (2) Universitas Kristen Satya Wacana via Internal Funding for the 2018-2019 fiscal year. Because
of their financial supports, we were able to finish this research work and final publication.

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International Conference on Science and Science Education IOP Publishing
IOP Conf. Series: Journal of Physics: Conf. Series 1307 (2019) 012003 doi:10.1088/1742-6596/1307/1/012003

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