Comparative Forecasting Performance of GARCH and GAS Models in the Stock Price Traded on Nigerian Stock Exchange

Subject Areas : International Journal of Mathematical Modelling & Computations

Oluwagbenga Babatunde ¹ , Serifat Folorunso ² , Francisco Saliu ³

1 - Department of Statistics University of Nigeria
2 - Department of Statistics, Faculty of Science, University of Ibadan
3 - Department of Statistics, Faculty of Science, University of Ibadan, Nigeria

Received: 2020-08-31 Accepted : 2021-04-30 Published : 2021-06-20

Keywords: innovations, Probability distribution, GAS Variants, Forecasts Measure,

Abstract :

The forecasting performance of different class of volatility models was compared in this work using the daily adjusted close price of traded stocks of the Nigerian Stock Exchange (NSE) from December 10, 2013 to February 07, 2019. The GARCH and EGARCH models were selected from the GARCH models whereas the GAS and EGAS were selected from the GAS models. Two different distributions were assumed for the innovations of the volatility models and forecasts measure was obtained. Based on the forecasts measure which are Mean Error (ME) and Theil Inequality (TI) obtained, the ability the models to forecast future volatilities was achieved. The outcome of this research showed that the GAS model performed better when compared to the GARCH model under the two distributional assumptions in terms of ability to forecast future volatilities of the close price NSE stocks. However, the EGARCH performed better when student-t distribution was assumed.

Full-Text:



COMPARATIVE FORECASTING PERFORMANCE OF GARCH AND GAS MODELS IN THE STOCK PRICE TRADED ON NIGERIAN STOCK EXCHANGE

 

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Abstract The forecasting performance of different class of volatility models was compared in this work using the daily adjusted close price of traded stocks of the Nigerian Stock Exchange (NSE) from December 10, 2013 to February 07, 2019. The GARCH and EGARCH models were selected from the GARCH models whereas the GAS and EGAS were selected from the GAS models. Two different distributions were assumed for the innovations of the volatility models and forecasts measure was obtained. Based on the forecasts measure which are Mean Error (ME) and Theil Inequality (TI) obtained, the ability the models to forecast future volatilities was achieved. The outcome of this research showed that the GAS model performed better when compared to the GARCH model under the two distributional assumptions in terms of ability to forecast future volatilities of the close price NSE stocks. However, the EGARCH performed better when student-t distribution was assumed.

 

Keywords: GAS Variants, Innovations, Probability Distribution, Forecasts Measure

 

 

1.       Introduction 2.       Materials and Method 3.       Estimation and Forecast Evaluation 4.       Results and Discussion 5.       Conclusion

Index to information contained in this paper

 

 

 

 

1.       Introduction

According to Engle and Patton [1], forecasting of future returns of a series can be done using a volatility model. This implies that a good volatility model should possess the ability to predict volatility in future returns. A lot of models have been developed to describe the pattern and forecast volatility in return series in literature.

Since the introduction of ARCH and GARCH models by Engle [2], and Bollerslev [3] respectively, the models have been applied to study and predict volatility of financial data with the first application to exchange rates by Hsieh [4]. Balaban [5] investigated the out-of-sample forecasting performance of both symmetric and asymmetric type of the GARCH models using the US dollar/Deutsche mark exchange rate. It was concluded that the symmetric GARCH model performs better when compared to the asymmetric GJR-GARCH model based on the obtained forecasts. According to Lee [6], the ability of a volatility model to forecast future volatilities of exchange rate data depends heavily on the performance measure used and using root mean square error (RMSE) as a measure will always make nonlinear GARCH models to outperform linear GARCH models. Brooks and Burke [7] consider new information or selection criteria proposed in Brooks and Burke [8], to determine an appropriate model for forecasting the volatility of US dollar exchange rate from the class of GARCH models and concluded that the mean square error (MSE) performs better than the new information criteria in selecting the appropriate GARCH model. However, the new information criteria outperform the mean absolute error (MAE) in selecting the appropriate GARCH model. The ARCH and GARCH models are symmetric models which do no not capture leverage effects. However, Nelson [9], proposed a volatility model to that effect known as Exponential GARCH (EGARCH).

We investigate the out-of-sample forecasting performance of a new class of volatility models proposed in Creal et al [10], and Harvey [11], on the volatility of the stock price traded on Nigerian Stock Exchange. To the best of our knowledge, the forecasting performance of the Generalized Autoregressive Score (GAS) model and its variants for the Nigerian stock prices has not been investigated. A related research, Yaya et al [12] claims that the GAS variants outperform the GARCH models using the Nigeria All Share Index but however the best performing model among the three GAS variants could not be distinguished.

The Generalized Autoregressive Score (GAS) model is considered in Yaya et al [12] as a variant of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model developed by Bollerslev [3], to capture the effects of leverage and jumps in a series. Since the introduction of the GAS model and its variant Exponential Generalized Autoregressive Score (EGAS) in 2013 (see Creal et al [10] and Harvey [11]), several works exist in literature on theoretical framework and application to financial series (see Calvori et al [13], Harvey [14], (Blasques et al [15], [16]), Bernadi and Cantania [17]). However, the GAS model and its variants have not been applied to study the dynamics and forecasting performance of stock prices in Nigeria. Calvori et al [13], proposed a new GAS model for forecasting shares volume. The proposed model was applied to the New York Stock Exchange (NYSE) shares volume and the model was able to capture the dynamics of the shares volume. The misspecification properties of the GAS variants have been considered in Babatunde et al. [18].

This paper tries to investigate the out-of-sample forecasting performance of the GAS model and its variants for the daily adjusted close price of stock traded on the Nigerian Stock Exchange and compare with the standard GARCH models.

2. Materials and Method

The data used for this work is the daily adjusted close price of stock traded on the Nigerian Stock Exchange. It covers a period of 63 months, 10 December 2013 – 07 February 2019, making a total 1289 daily observations. 1000 observations were used for parameter estimation while 289 observations were used for out of sample forecast.

The GARCH (1,1) and GAS (1,1) models are symmetric which evaluate the magnitude of positive and negative  equally.

                                 (1)

     (2)

             (3)

           (4)

Where  and is a function of

Equation (1) is the mean equation where the regressors can be added to the right-hand side . Equation (2) is a GARCH (p,q) process where auto-regression in its squared residuals has an order of p (see Wang [19]). Equation (3) is a GARCH (1,1) model. Equation (4) is a GAS (1,1) model as proposed in Harvey and Chakravarty [20] and Harvey [11].

The EGARCH (1,1) and EGAS (1,1) models consider the magnitude of positive and negative .

          (5)

               (6)

The lag operator is denoted as L,  and  are the asymmetric parameters

We assumed two distributions for the innovations  namely Student-t (t), Skewed-Student-t (skt) so as to capture tail effect of the innovations since the distributions of returns are typically non-normal as noted in Xekelaki and Degiannakis [21].

For the GAS and EGAS models,  is given as:                                                           

;;      (7)                                                                  

;  (8)

,  ,

  and  

3.ESTIMATION AND FORECASTS EVALUATION

The parameters of the GARCH and GAS models were estimated using MaxSA algorithm of Goffe et al. [22] included in the Oxmetrics-GARCH software of Laurent and Peters [23]. The two probability distributions assumed for the innovations of the GARCH and GAS models were Student-t and Skewed Student-t. The log-likelihood functions of these probability distributions were obtained (see Yaya [24, 25]) and optimized with the help of the MaxSA algorithm. The parameters of the models were obtained after achieving strong convergence. The parameters of the models were estimated using 1000 observations out of the 1289 observations available. Out of sample forecast was then carried out after the estimation of the model parameters using the remaining 289 observations. The forecasting performance of the GARCH and GAS models were then evaluated using Mean Error (ME) and Theil Inequality (TI) as obtained in Theil [26, 27].

4. Results and Discussion 

The plots of the daily adjusted close price of stock traded on the Nigerian Stock Exchange and log returns from December 10, 2013 to February 07, 2019 are given in figure 1 with 1289 daily observations. The estimated parameters for each of the volatility models as well as model evaluation assuming student-t distribution for the innovations are presented in table 1. For symmetric models, all the parameters of the GARCH and GAS models were significant except for the constant term . For the asymmetric models, all the parameters of the EGAS were significant except for the constant term  whereas only the constant term was significant for the EGARCH model. On model evaluation for the symmetric models, GARCH (1,1)-t performs better based on log-likelihood and information criteria. The log-likelihood value for GARCH (1,1)-t and GAS (1,1)-t are 3216.564 and 3216.353 respectively. Also, GARCH (1,1)-t has the least information criteria for all the information criteria used in this study. The AIC, SBIC and HQ for GARCH (1,1)-t and GAS (1,1)-t are -6.4231, -6.3986, -6.4138 and -6.4227, -6.3982, -6.4134 respectively.

For the asymmetric models, EGAS (1,1)-t performs better based on log-likelihood and information or selection criteria. The value of the log-likelihood function for EGAS (1,1)-t and EGARCH (1,1)-t is 3216.893 and 3184.462 respectively. The AIC, SBIC and HQ for EGAS (1,1)-t and EGARCH(1,1)-t are -6.4238, -6.3992, -6.4144and -6.3549, -6.3206 and -6.3419 respectively.

The ARCH LM test statistic for both the symmetric and asymmetric models indicates no serial correlation among the innovations since the test were not significant at 0.05.

 

Figure 1: Original and log return series

 

Table 1. Volatility models with student-t distributional assumption for the innovations

***Significsnt at 5%

The estimated parameters for each of the volatility models as well as model evaluation assuming skew-student-t distribution for the innovations are presented in table 2. All the estimated parameters for all the GARCH, GAS and EGAS models were significant except the constant and asymmetry terms. However, none of the term in the EGARCH model was significant. On model evaluation for the symmetric models, GAS (1,1)-skt perform better based on log-likelihood and information or selection criteria. The value of the log-likelihood function for GAS (1,1)-skt and GARCH (1,1)-skt is 3216.857and 3216.805 respectively. Also, GAS (1,1)-skt has the least information criteria for all the information criteria used in this study. The AIC, SBIC and HQ for GAS (1,1)-skt and GARCH (1,1)-skt are -6.4217, -6.3923, -6.4105 and -6.4216, -6.3922,  -6.4104 respectively.

For the asymmetric models, EGAS (1,1)-skt perform better based on log-likelihood and information or selection criteria. The value of the log-likelihood function for EGAS (1,1)-skt and EGARCH (1,1)-skt were 3217.395 and 3186.642 respectively. The AIC, SBIC and HQ for EGAS (1,1)-skt and EGARCH (1,1)-skt are -6.4228, -6.3933, -6.4116 and -6.3573, -6.3180, -6.3424 respectively.

The ARCH LM test statistic for both the symmetric and asymmetric models indicates no serial correlation among the innovations since the test were not significant at 0.05.

 

Table 2. Volatility models with skew-student-t distributional assumption for the innovations

Parameters	GARCH (1,1)-t	GAS (1,1)-t	EGARCH (1,1)-t	EGAS (1,1)-t
	1.63E-04	1.91E-04	4.79E-04***	1.9E-04
	0.0567***	0.0724***	223.3060	0.0713***
	0.9379***	-	0.9964	-
	-	-	-0.0298	-
	-	-	0.0099	-
	-	0.9894***	-	0.9866***
Student-t df	5.4141***	6.0821***	2.0001	6.0356***
Model Evaluation
Log –Lik.	3216.564	3216.353	3184.462	3216.893
AIC	-6.4231	-6.4227	-6.3549	-6.4238
SBIC	-6.3986	-6.3982	-6.3206	-6.3992
HQ	-6.4138	-6.4134	-6.3419	-6.4144
ARCH LM Test
Lag 2	1.0114	0.9315	0.1914	0.8685
Lag 5	1.0317	0.9357	0.3631	0.9476
Lag 10	0.9346	0.9619	0.7876	0.9729

***Significant at 5%

Table 3 presents the estimates of forecast measure used for model comparison. For the symmetric models, the estimates of the ME and TI indicated that the GAS model outperformed the GARCH model under the two distributional assumptions of the innovations in terms of ability to forecast future volatilities of the stock prices. In a related research, Yaya et al [12] claims that the GAS variants outperform the GARCH models using the Nigeria All Share Index. For the asymmetric models, the estimates of ME and TI indicated that the EGARCH model outperformed the EGAS model when student-t distribution was assumed for the innovations but when skewed student-t distribution was assumed for the innovations, the estimates of ME and TI were inconsistent.

Table 3. Estimates of Model Comparison Forecast Measure

Parameters	GARCH (1,1)-skt	GAS(1,1)-skt	EGARCH(1,1)-skt	EGAS (1,1)-skt
	8.5E-05	9.0E-05	1.7E-04	8.9E-05
	0.0561***	0.0721***	206946.8988	0.07089***
	0.9386***	-	0.9917	-
	-	-	-3.4E-05	-
	-	-	5.2E-05	-
	-	0.9897***	-	0.9871***
Student-t df	5.4230***	6.0955***	2.00004	6.0511***
Asymmetry	-0.0319	-0.0457	-0.0126	-0.0457
Tail	5.4230***	6.0955***	2.00004	6.0511***
Model Evaluation
Log –Lik.	3216.805	3216.857	3186.642	3217.395
AIC	-6.4216	-6.4217	-6.3573	-6.4228
SBIC	-6.3922	-6.3923	-6.3180	-6.3933
HQ	-6.4104	-6.4105	-6.3424	-6.4116
ARCH LM Test
Lag 2	1.0074	0.9138	0.1215	0.8523
Lag 5	1.0182	0.9203	0.4655	0.9316
Lag 10	0.9253	0.9474	0.7234	0.9585

Model	Distribution	ME	TI
GARCH (1,1)	Student-t	-1.15E-04	0.9825
	Skew-Student-t	-3.73E-05	0.9908
GAS (1,1)	Student-t	-1.43E-04	0.9796
	Skew-Student-t	-4.24E-05	0.9902
EGARCH (1,1)	Student-t	-4.31E-04	0.9511
	Skew-Student-t	-1.23E-04	0.9817
EGAS (1,1)	Student-t	-1.43E-04	0.9797
	Skew-Student-t	-4.16E-05	0.9903

 

5. Conclusion

This paper investigated the forecasting performance of the GARCH and GAS models using the daily adjusted close price of stock traded on the Nigerian Stock Exchange from December 10, 2013 to February 07, 2019. The series contains 1289 daily observations in which 1000 observations was used for the estimation of the model parameters while the remaining 289 observations was used for the out-of-sample forecast. The GARCH and EGARCH were selected from the GARCH models whereas the GAS and EGAS were selected from the GAS models. The GARCH and GAS models are symmetric while the EGARCH and EGAS models are asymmetric. Two different probability distributions were assumed for each model. Based on the different information criteria used, the fitness of the models was assessed. For the symmetric models, the GARCH (1,1) model fits better compared to GAS (1,1) model under the Student-t distributional assumption. However, GAS (1,1) model fits better compared to GARCH (1,1) model under the Skewed-Student-t distributional assumption. For the asymmetric models, the EGAS (1,1) model fits better compared to EGARCH (1,1) model under the Student-t distributional assumption. Also, EGAS (1,1) model fits better compared to EGARCH (1,1) model under the Skewed-Student-t distributional assumption.

We went ahead to evaluate the forecasting performance of both the symmetric and asymmetric models in forecasting future volatilities of the daily adjusted close price of stock traded on the Nigerian Stock Exchange. The performance of the out-of-sample forecasts was evaluated using ME and TI as forecast measures. Based on different forecast measures which are ME and TI, the forecast performance of the models were compared and summarized in Table 4. The result showed that GAS (1,1) performed better when compare to GARCH (1,1) under the two distributional assumption in terms of ability to forecast future volatility of stock prices and EGARCH (1,1) performed better when compared to EGAS (1,1) when student-t distribution was assumed.

The implication of this result is that the GAS (1,1) model performs better compared to GARCH (1,1) in forecasting the future volatilities of the stock prices of the Nigerian Stock Exchange taking only the magnitude of the returns into consideration. However, if both the magnitude and sign (positive and negative) are of interest, the EGARCH (1,1) model is considered to perform better compared to EGAS (1,1) model in forecasting the future volatilities of the stock prices of the Nigerian Stock Exchange.

Table 4. Summary of Model Forecasting Performance

Distribution	Symmetric Models		Asymmetric Models
	ME	TI	ME	TI
Student-t	GAS	GAS	EGARCH	EGARCH
Skew-Student-t	GAS	GAS	EGAS	EGARCH

 

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