The main aim of this study is the empirical exploration for the proper volatility models of some selected pharmaceutical companies listed in the DSE, Bangladesh e.g. Square, Beximco, Beacon, IBN SINA, and Orion Pharmaceuticals Ltd. The data covers the 667 days daily log returns calculated based on closing prices of these five selected companies from 28th January 2019 to 30th December 2021. The beginning portion of the analysis contains the stylized facts of the sampled companies. Afterward, by employing both symmetric along with asymmetric GARCH models different best-fitted models for different pharmaceuticals companies were found. Based on our model selection criteria AIC, SBIC, Log-Likelihood, as well as residual diagnostics GARCH(1,1) is considered to be more appropriate models for both Square Pharmaceuticals Ltd., and Beacon Pharmaceuticals Ltd. The EGARCH (1,1) is deemed to be best for both IBN SINA and Orion Pharmaceuticals Ltd. Whereas, anyone of the GARCH(1,1), and TGARCH(1,) can be applied for the volatility estimation of Beximco Pharmaceuticals Ltd.
Being a thermometer of an economy every capital market reflects the health of any economy. The well-being of every economy largely depends on a vibrant capital market. The growth of new businesses or any economy would not be possible without the avail-ability of equity finance and the advancement of finan-cial markets. The capital markets have a pivotal role in the collection of long-term funds for the deficit unit that wants to invest in the industrial arena.
Since their invention, ARCH, and its generalization GARCH models had been applied widely in measuring and predicting the variance of markets all over the world. We also employed GARCH-type modelsand tried to discover the better models for volatility esti-mation of five selected pharmaceuticals firms regis-tered in DSE, Bangladesh e.g. Square Pharmaceuticals Ltd., Beximco Pharmaceuticals Ltd., Beacon Pharma-ceuticals Ltd., IBN SINA Pharmaceuticals Ltd, and Orion Pharmaceuticals Ltd. We have applied GARCH (1,1), GARCH-M (1,1), EGARCH(1,1), and TGARCH (1,1) models in our investigation. The following section contains a brief explanation.
Modeling Techniques
In our study, we employed four symmetric and asym-metric GARCH approaches. Symmetric GARCH app-roaches consider past positive and negative infor-mation in the same manner. This happens becausetheir variance equations depend onthe extent of lagged resi-duals and not their signs. But a negative blow can have a larger impact on variance than a similar positive one. Hence in our study, we have employed both types of approaches and tried to recognize the appropriate models for each company based on three commonly used criteria e.g. Akaike information criterion (AIC), Schwarz Bayesian information criterion (BIC), and log-likelihood.
GARCH (1,1) Model
An elementary representation for GARCH (1,1)is as follows-
Mean equation: r_t=ω + Є_t
Varianceequation: 〖σ^2〗_t= α_0+α_1 〖Є^2〗_(t-1)+ β_1 〖σ^2〗_(t-1)
The second equation is termed as variance equation and it has three components: The first component isa constant term, next is the lag squared residuals found from its mean equation, and termed as ARCH. The last one is termed GARCH, which represents the fitted variance from the model during the preceding duration. The conditions are:
α0≥ 0, αi ≥ 0, βi ≥ 0, α1 +β1< 1.
So, using this approach it is possible to interpret the currently fitted discrepancy as a weighted function of a long-term average value(constant), information about volatility during the previous period(ARCH term), and the fitted variance from the model during the previous period (GARCH term).
The GARCH-in-Mean (GARCH-M) Model
In the mean equation, it is included either variance or standard deviation as the following expression.
Mean equation: r_t=ω +ϒ〖σ^2〗_(t-1)+ Є_t
Variance equation: 〖σ^2〗_t= α_0+α_1 〖Є^2〗_(t-1)+ β_1 〖σ^2〗_(t-1)
The parameter ϒ is recognized as risk premium para-meter. A positive ϒ specifies that return is positively related to its volatility. Especially, a rise in average return is caused by an increase in conditional variance in the manner that proxy of increased risk.
The Exponential GARCH (EGARCH) Model
Nelson (1991) was the proponent of this approach. Among various ways of expression, one possible spe-cification is given by-
Ln(〖σ^2〗_t)=ω+ βln(〖σ^2〗_(t-1)) + λµ_(t-1)/√(〖σ^2〗_(t-1 ) ) + α[|µ_(t-1) |/√(〖σ^2〗_(t-1 ) )-√(2/π)]
Where λ is the leverage effect parameter. The signi-ficance along with negative values for this parameter ensures the presence of leverage effect in the return. That specifies that good information has a lower impact than negative information to the same extent.
The Threshold Garch (TGARCH) Model
The TGARCH model was introduced by Zakoian, (1994) and Glosten et al. (1993). The threshold GAR-CH (1,1) model specification follows asThe variance equation of TGARCH (1,1) version is:
〖σ^2〗_t= α_0+α_1 〖Є^2〗_(t-1)+ β_1 〖σ^2〗_(t-1)+ λ〖Є^2〗_(t-1) I_(t-1)
Where I_(t-1) =1
If Є_(t-1)<0; = 0 otherwise.
The significant and positive values of λ signify the pre-sence of the leverage effect in the series. The other conditions are: α_0>0, α_1>0, β ≥ 0, and α1 + λ ≥ 0.
The Data and Basic Statistics
Data for Analysis
The data comprises 667 days daily ending prices of five selected pharmaceutical companies registered in The Dhaka Stock Exchange Ltd (DSE), Bangladesh from 28th January 2019 to 30th December 2021. These data have been gathered from- https://www.dsebd.org. We have transformed our data into logarithmic daily returns using the following formula:
r_t=log (Pt/P_(t-1) )
The r_(t ) represents the logarithmic daily returns of our sampled companies for time t, Pt is the ending prices of sample companies at time t, P_(t-1) is the corres-ponding previous days prices of selected stocks.The descriptive information for all five selected companys returns is exhibited in Table 1.
Table 1: Illustrative features of returns.
Fig. 1: Represents the simple line graphs of return data from 28.01.2019 to 30.12.2021 of selected pharmaceutical companies listed in DSE, Bangladesh.
The Table 1 represents that the distributions of returns of all companies differ markedly from the normality for all companies, observed from watching the skew-ness figures and excess kurtosis (having value more than 3). Jarque-Beras pieces of evidence also confirm that normality is rejected at a p-value of almost 1. The data of five selected pharmaceuticals under review are demonstrated in Fig. 1. A return series that contains ‘volatility clustering, stimulates to parameterize with an ARCH process. Volatility clustering can be exp-lained as a situation where huge changes within the prices of stocks (of either sign) are taken after by expansive changes, and little changes (of either sign) are taken after by little changes. Such scenarios can be spotted in almost all of the selected pharmaceutical companies from Fig. 1.
Testing for Heteroscedasticity
The heteroskedasticity test in the squared residuals can be used to determine whether ARCH-effects are pre-sent in the residuals of a predicted method. This test is contingent on the joint null hypothesis that the co-efficient values of all q lags of the squared residuals are not considerably different from zero. The null hypothesis is rejected if the test statistic is greater than the critical value from the χ2 distribution (Brooks, 2008). By regressing the squared residuals with ran-domly picked different lags, such as 5,10,15,20, and 25, a check depending upon arbitrarily chosen mean ARMA (1,1) for the presence of ARCH in the resi-duals is performed. The outcomes presented in Table 2 showing the probability values with different lags are smaller than even a 1% level of significance for all companies. Hence, there is enough evidence for reject-ing the joint null hypothesis, which supports the appearances of ‘ARCH. So the ARCH/GARCH type models can be employed on these return data for proper estimation.
Table 2: The aftereffects of ARCH-LM tests of selected pharmaceuticals companies.
Testing for Stationarity
Before proceeding for further study it is a prerequisite to know whether the series being dealt with is statio-nary or not. In such a case, tests like Augmented Dicky-Fuller (ADF, 1988) test and Phillips-Perron (P-P, 1987) test are adopted to test the stationarity of the daily return of all five sampled pharmaceuticals comp-anies. On the log return data series; an augmented Dickey-Fuller (ADF) test was done allowed to 12 lags with three types of equations: an intercept but no trend, trend & intercept, and none.For the same purpose, Phillips-Perron (P-P) test was performed also. There are two hypotheses of consideration; H_0: the data emb-race unit root versus H_1: they are stationary. Their outcomes are represented by table 3.0. It is observed clearly that at level, all characters of stats are more negative than the critical values in ADF as well asPP tests in all forms of equations. Again all the probability values are less than even 1% level of significance. So, based on these analyses our null hypothesiscan stron-gly be rejected. Hence, all the return data for all five selected pharmaceutical companies are stationary at a level.
Empirical Results
The significant existence of ARCH in residuals of all five sample companies for our study period inspired us to parameterize their returns by applying the following models. All the models performed under the normal Gaussian error distributions. After the model const-ruction, the residual diagnostic check was administered to observe whether there exists any remainder ARCH-effect. The estimated results of all employed models for all the companies i.e. Square Pharmaceuticals, Beximco Pharmaceuticals, Beacon Pharmaceuticals, IBN SINA, and Orion Pharmaceuticals are demons-trated in Table 4, Table 5, Table 6, Table 7, and Table 8 respectively.
Table 3: Results of stationarity tests of daily returns of selected pharmaceutical companies.
A results summary of different model choosing ben-chmarks of all estimated models of our five sample companies is also exposed in table 09. With the exception of Beximco, the α_0 (constant) components inside the volatility formulae are seen as being statis-tically meaningful at the 1%, 5%, and 10% levels in all of our sample companies. For Beximco, this comp-onent is significant only for EGARCH (1,1). However, the α_(1 )(ARCH term), and β_1 (GARCH term) are seen as being statistically relevant at 1%, 5%, and 10% levels in all models of all our sample companies except only for TGARCH(1,1) of Beacon Pharmaceuticals. Where β_1 (GARCH term) isnt observed as statistically significant. The significance of these ARCH, as well as GARCH components imply that their lagged variance together with lagged squared disturbance have impacts on todays conditional variance. So their previous volatilities have explanatory power on present vola-tilities for all companies.
Table 4: Expresses the findings of various GARCH approaches for the Square Pharmaceuticals Ltd.
Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively.
The outcomes of the variance equation shown in the Table 4 justify that the initial three coefficients α_0 (constant), α_(1 )(ARCH term), and β_1 (GARCH term) are significant. The relevance of the ARCH as well as GARCH elements in GARCH (1,1) suggests that the variance is affected by lagged conditional volatility and squared disturbances. The meaning of that is the past volatility has illustrative power for present vola-tility. The score of the two estimated ARCH, & GAR-CH coefficients; α_(1 )+ β_1in our beginning model is close to one, indicating that volatility shocks are quite persistent (Abdalla, 2012). The consequence of shocks seems not to be dried out. The GARCH-M (1,1) model considers variance in its mean equation in case of estimation. From estimation results in Table 4, the estimated coefficient ϒ (risk premium) of σ^(2 )is seen to be positive and not significant. But such inclusion might be helpful for the significance of the GARCH factor in variance equation. Again, to observe the ap-pearance of leverage effect in data of Square Pharma-ceuticals Ltd. we applied EGARCH (1,1), & TGARCH (1,1) models. The〖 λ〗_1(leverage term) for the EGARCH (1,1) shows a positive and statistically not significant value, indicating no vicinity of such effect. In TGA-RCH (1,1) estimation results the leverage term is positive in conjunction with significant. The signi-ficance of this coefficient ensures the existence of such an effect, which implies that negative shocks can substantially affect the volatility of this firm compared to a positive one. The findings of diagnostic tests for Heteroscedasticity (with lag 5) are displayed in the ending area on Table 4. The ARCH-LM test statistics for all models arent found as statistically significant; which assures that these models dont exhibit additional ARCH effects in their residuals. Despite the fact that the asymmetric term is not found to be significant for EGARCH (1,1), our model choosing criterion AIC having the lowest value (-5.7719), and Log-likelihood with its highest value (1929.92) selects this as a better model for Square Pharmaceuticals Ltd. But after the construction of this model still, serial correlation is noticed in residuals. On the contrary, the criterion SBIC with its lower most figure (-5.7395), finds GAR-CH(1,1) as best, and no serial correlation found in the result of residuals diagnostic check up to 36 lags under this model. So between the two above selected approaches GARCH (1,1) can be considered as best for this company.
Table 5: shows the findings of various GARCH-type models for Beximco Pharmaceuticals Ltd.
Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively.
Table 5 represents the results for Beximco Pharma-ceuticals Ltd. The figures for the second equation show that α_(1 ) (ARCH) including β_1 (GARCH) are sta-tistically significant. The α_0 (constant) isrecognized as significant only for EGARCH (1,1). The relevance of the ARCH, as well as GARCH elements in GARCH (1,1), suggests that the variance is affected by lagged conditional volatility and squared disturbances. The meaning of that is the past volatility has illustrative power for present volatility. Moreover, the score of the two estimated ARCH, & GARCH coefficients; α_(1 )+ β_1 in our beginning model is is greater than one (1.00065) representing the volatility process as explosive (Ab-dalla & Winker, 2012). So, the consequences of shocks will never dry out. The GARCH-M (1,1) model con-siders variance in its mean equation in the regard of estimation. In estimation results in Table 5, the esti-mated coefficient ϒ (risk premium) of σ^2 is seen to be positive and not significant. But such inclusion might be helpful for the significance of the GAR-CH factor in variance equation. No feedback comes from the variance after the inclusion of its mean equation. The λ_1 (leverage term) for the EGARCH (1,1) shows a negative but statistically not significant value, indicating no leverage effect. In TGARCH (1,1) the coefficient of this component is positive in addition to significant, meaning that bad news will have a bigger impact in its conditional volatility than good news. The diagnostic tests for Heteroscedasticity city (with lag 5) are also reported in Table 5. The LM statistics for all models cannot be located as statistically significant; which assures that these models dont exhibit addi-tional ARCH effects in their residuals. These prove that the variance equations have been properly speci-fied. However, AIC having the lowest value (-4.9529), and Log-likelihood with its highest value (1656.79) select the same approach i.e. TGARCH (1,1) as the best-fitted model for Beximco Pharmaceuticals Ltd. Whereas, SBIC having the lowest value(-4.9213), finds GARCH (1,1) as best. Again, after applying both TGARCH (1,1), and GARCH(1,1) no serial correlation in residuals was detected up to 36 lag.
Table 6: represents the findings of various GARCH-type models for Beacon Pharmaceuticals Ltd.
Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively.
Table 6 represents Beacon Pharmaceuticals Ltd. Un-der all of the methods, the outcomes for the variance equation find the α_0 (constant), as well as theα_(1 ) (AR-CH) as significant. Whereas, the β_1 (GARCH) is signi-ficant except for TGARCH (1,1).The relevance of the ARCH, as well as GARCH factors in GAR-CH(1,1), suggests that the variance is affected by lagged con-ditional volatility and squared disturbances. The meaning of that is the past volatility has illustrative power for present volatility. The sums of ARCH plus GARCH coefficients; α_(1 )+ β_1in all the GARCH-type modelsare in the vicinity of, or higher than one, which implies that shocks are roughly persistent and wouldnt become dry out. The GARCH-M (1,1) model con-siders variance in its mean equation for assessment. The GARCH-M (1,1) model considers variance in its mean equation in the event of estimation. In estimation results in table 06, the estimated coefficient ϒ (risk premium) of σ^2 is seen to be positive and not signi-ficant. So, no feedback comes from the variance after inclusion of it in mean equation for Beacon Pharma-ceuticals Ltd. The λ_1 (leverage term) for the EGARCH (1,1) indicates a non-negative but statistically signi-ficant value, indicating no existence of such effect. In TGARCH (1,1) the digit of the leverage factor is negative as well as significant with only at 10% level, which also supports the non-existence of this effect. The diagnostic tests for Heteroscedasticity (with lag 5), demonstrate that the test statistics for almost all met-hods are not statistically significant. So, no remaining ARCH exists in residuals of the above models. The criteria AIC having the lowest value (-4.3362), and Log-likelihood with its highest value (1451.11) select the same i.e. EGARCH (1,1) as the best-fitted model for Beacon Pharmaceuticals Ltd. Whereas, SBIC having lowest value (-4.3052), finds GARCH(1,1) as best. The residuals of EGARCH(1,1) contain serial correlation. On the contrary, in the event of GARCH (1,1), we didnt find any serial correlation after con-struction up to 36 lags. So, the GARCH(1,1) selected by SBIC appears to be a better model for Beacon Pharmaceuticals Ltd.
Table 7: presents the findings of various GARCH-type models for IBN SINA Pharmaceuticals Ltd.
Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively.
Table 7 represents IBN SINA Pharmaceuticals Ltd. The first three coefficients of the variance equation are acknowledged as significant. The relevance of the ARC, as well as GARCH factors in GARCH (1,1), suggests that the variance is affected by lagged con-ditional volatility and squared disturbances. The meaning of that is the past volatility has illustrative power for present volatility. The sums of ARCH plus GARCH coefficients; α_(1 )+ β_1 in all the GARCH-type models are in the vicinity of, or higher than one, which implies that shocks are roughly persistent and wouldnt become dry out. The GARCH-M (1,1) model con-siders variance in its mean equation for assessment. The GARCH-M (1,1) model considers variance in its mean equation in the event of estimation. In estimation results in Table 7, the estimated coefficient ϒ (risk premium) of σ^2 is seen to be positive and not signi-ficant. So, no feedback comes from the variance after inclusion of it in mean equation for IBN SINA Pharma- ceuticals Ltd. The〖 λ〗_1 (leverage effect) for the EGA-RCH (1,1) shows a negative but statistically not signi-ficant value, indicating the non-existence of such effect in data. In TGARCH (1,1) the asymmetry term is positive together with significant with 5% & 10% levels, which advocates in favor of the existence of such effect. This implies that negative shocks can sub-stantially affect the volatility of this firm compared to a positive one. The diagnostic tests for Heterosce-dasticity (with lag 5), recognize that the test statistics for all methods are not statistically significant. So, no remaining ARCH exists in residuals of the above models. However, each of the two criteria AIC,& SBIC having their respective minimum values (-5.8291),& (-5.7953) select EGARCH(1,1) as the best-fitted model for IBN SINA Pharmaceuticals Ltd. Log-likelihood with its highest value (1949.00) also select the same volatility prediction model for this company. The serial correlation diagnostic also produces the desirable findings for EGARCH (1,1).
Table 8: contains the findings of various GARCH-type models for Orion Pharmaceuticals Ltd.
Remark: ***, **, * represents significant at the 1%, 5% and 10% level respectively.
Table 8 represents Orion Pharmaceuticals Ltd. The first three coefficients of the variance equation were observed as significant. The relevance of the ARCH, as well as GARCH factors in GARCH (1,1), suggests that the variance is affected by lagged conditional volatility and squared disturbances. The meaning of that is that past volatility has illustrative power for present volatility. The sums of ARCH plus GARCH coefficients; α_(1 )+ β_(1 )in all the GARCH-type models are very near to, or higher than one, which implies that shocks are roughly persistent and wouldnt become dry out. The GARCH-M (1,1) model considers variance in its mean equation for assessment.
In estimation results in Table 8, the estimated co-efficient ϒ (risk premium) of σ^2 is seen to be posi-tive and not significant. So, no feedback comes from the variance after including of it in mean equation for Orion Pharmaceuticals Ltd. The λ_1 (asymmetry term) for the EGARCH (1,1) indicates a non-negative and statistically significant value, indicating the non-exis-tence of the leverage issue. For TGARCH (1,1) the leverage factor is negative together with significant with only at 10% level, which also admits the non-existence of such effect in the data. The diagnostic tests for Heteroscedasticity (with lag 5), illustrate that the test statistics for all approaches are not statistically significant. So, no remaining ARCH exists in residuals of the above models.
However, each of the two criteria AIC & SBIC having their respective minimum values (-4.5740), & (-4.5402) select EGAR-CH(1,1) as the best-fitted model for Orion Pharmaceuticals Ltd. Log-likelihood with its highest value (1530.42) also select the same volatility estimation model for this company. The serial correlation diagnostic also furnishes the desirable outcomes for EGARCH(1,1).
Table 9: shows the summary of the best-fitted models for all five sampled companies.
The main thirst of this piece of research is the empi-rical investigation of the approximation of the vola-tility of a few pharmaceutical companies registered in DSE, Bangladesh. The first part of this manuscript contains the summary stats of the sampled companies. Afterward, by employing GARCH-type methods we found different best-fitted models for different pharma-ceuticals companies. Table 9 shows that in reliance on AIC, and Log-Likelihood, EGARCH (1,1) was turned out to be the best-fitted model for every single com-pany except for Beximco Pharmaceuticals Ltd. For this firm TGARCH (1,1) is appeared to be best by them. Conversely, SBIC finds GARCH (1,1) as a superior method in each of the following three companies i.e. Square Pharmaceuticals Ltd., Beximco Pharma-ceuticals Ltd., and Beacon Pharmaceuticals Ltd. Whereas, the EGARCH (1,1) in each of the two firms i.e. IBN SINA and Orion Pharmaceuticals Ltd. have been appeared to be a better approach depending on SBIC criterion. Lastly, the EGARCH (1,1) has been accepted as superior for all the companies except Beximco Pharmaceuticals Ltd. For this firm TGARCH (1,1) has been accepted by this criterion. Despite the fact that EGARCH (1,1) was chosen as the prediction fits for both Square Pharmaceuticals and Beacon Phar-maceuticals relying on their AIC and Log-likelihood values, the residuals diagnostic checks revealed the serial correlations. Hence, GARCH (1,1) selected by SBIC appears to be superior for each of these two companies. However, there are only limitations for all selected methods for the above companies that their residuals werent proved to be normally distributed even after employing the models. But still, the models can be implemented as they have passed their other two residual diagnostics e.g. non-existence of Arch, and serial correlation. Actually, stock market move-ment is a general aspect of any share market. In occu-pancy of high return volatility, potential investors lose interest to invest in such a market. Political unrest and man-made crisis sometimes create the share market more unstable. Hence, investors need to be aware of the disposition of risks involved in a particular stock or in a specific sector. This paper would be helpful to know these, and thus obviously to take their appro-priate decisions.
Firstly, we are grateful to Allah (SWT). The authors of this paper also want to express gratitude to those res-earchers whose previous studies have been cited here, and substantially helped to conduct this study.
The authors want to declare that there is no conflict of interest with any bodyregarding this research work.
Academic Editor
Dr. Doaa Wafik Nada, Associate Professor, School of Business and Economics, Badr University in Cairo (BUC), Cairo, Egypt.
Department of Finance and Banking, Jatiya Kabi Kazi Nazrul Islam University (JKKNIU), Trishal, Mymensingh-2224, Bangladesh
Rahman MH., and Hussain S. (2022). Modeling returns volatility of selected pharmaceutical companies listed in DSE of Bangladesh with GARCH methods, Int. J. Manag. Account. 4(2), 21-32. https://doi.org/10.34104/ijma.022.00210032