M odeling R eturns V olatility of S elected P harmaceutical C ompanies L isted in DSE of B angladesh with GARCH M ethods

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. family modeling and estimating market volatility


INTRODUCTION:
Being a thermometer of an economy every capital market reflects the health of any economy. The wellbeing of every economy largely depends on a vibrant capital market. The growth of new businesses or any economy would not be possible without the availability of equity finance and the advancement of financial 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.
Hence, a high positive link-up between capital market development and industrial development for every economy exists. But greater capital market volatility may hamper the attractions of the potential investors, which ultimately a piece of negative news for industrial upliftment. For Bangladesh, capital market movement is a major concerning issue to the market participants. This market already has witnessed two major debacles e.g. in 1996, and in 2009 within its short history. Hence, capital market movement is a very concerning issue for the market participants in Bangladesh. Individual stock prices or various indices may fall in some situations and maybe constant or rise in another situation. These movements are termed volatility. Such volatility has attracted much concentration of academics, policy-makers, and practitioners during the recent past. In recent years, several models and techniques have been proposed and implemented for volatility analysis. By proper investigation and prediction, the stock markets can be created as an attractive source of investment. There is extensive usage of Generalized and Autoregressive Conditional Hetero skedasticity-(ARCH & GARCH) family models in modeling and estimating the financial market volatility since their advent. Most of the studies on DSE mainly have focused on all inclusive indices. But, investors and other market participants can find their interests in the movement of particular stocks or industries. A big research gap particularly in modeling the variability of return of any single stock or industry was observed in the case of the financial market of Bangladesh.
Hence, this paper has tried to employ GARCH-type models to deduce the best-fitted model for five selected pharmaceutical firms registered in DSE, Bangladesh e.g. Square Pharmaceuticals Ltd., Beximco Pharmaceuticals Ltd., Beacon Pharmaceuticals Ltd., IBN SINA Pharmaceuticals Ltd, and Orion Pharmaceuticals Ltd.

Literature Review
Market participants, researchers, or academicians are very concerned about the measurement and forecast of the inconstancy of equity returns. The swing of equity prices is very common in almost every exchange within the world. Accurate volatility prediction is important for the market participants for their investment strategy and asset selection. Since its inception, the Bangladesh stock market already witnessed two major debacles in its history i.e. in 1996 and in 2009 and it are familiar as very unpredictable market. So to know about the ways of volatility measurement and forecasting techniques is very essential for the market participants. Several studies tried to capture the irregularity of the stock markets of Bangladesh as follows-Huq and Ali, (2018) studied the variability of the DSE Broad Index (DESX) using the GARCH model. Their study period covers DSEX daily closing index from 27/01/2013 to 04/04/2017. They found GJR-GARCH (1,1) to be the most applicable model volatility prediction. The models' performances were also compared under different statistical error measurement tools. Roni et al., (2017) applied GAR-CH-type models to the Bangladesh stock market. They found that the TGARCH model is more accurate in terms of accuracy. They also found according to the error measurement statistic that the GARCH is more efficient and has more forecasting ability also. Hasan and Wadud, (2016) tried to search for an adequate volatility model for DSE. They found that MA (1)-EGARCH (1,1) has a leverage effect in current volatility. Their study also discovers that MA (1)-GARCH (1,1) is the finest model for DSE stock returns in Bangladesh. Huq et al., (2013) also empirically examined and found GARCH (1,1), and GARCH(2,1) along with ARMA (1,1) as better for DSE. Siddikee and Begum, (2016) applied the GARCH (p, q) and ARCH (m) models to DGEN ranging from 1st January 2002 to 30th July 2013. Their results of the GARCH (1, 1) process and standard deviation corroborated the presence of atypical swings in the returns from 2009 to 2012. Parvez et al., (2017) tried to contrast the potentiality of different types of models to forecast the variability of the DSE. They studied 2639 daily observations from 1 st January 2002 to 19th June 2012. After comparing the forecasting performances of the twelve models, they found GARCH (1,1) with conditional distribution student-t model as superior in accordance with the RMSE, MAE, Theil-U, and four asymmetric loss functions (Rayhan et al., 2011). The main focus of their study is the detection of the pattern and reasons behind the monthly variability of DSE, and to search the possible solutions thereto. They observed a random walk phenomenon in the price index, but not in their returns (Sami et al., 2021). Hossain et al., (2012) hunted for a suitable approach for DSE and tried to forecast the future outline of volume data of this market. They selected the ARIMA with the EGARCH model to analyze the volatility as superior. Besides these studies stated above, some studies have tried to correlate the movement of this market along with macroeconomic data. But for individual stocks or specific industries, there has been a big research gap for the Bangladeshi stock market. That's why this paper has tried to employ GARCHtype models in five selected pharmaceuticals limited listed in DSE, Bangladesh.

METHODOLOGY:
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 estimation of five selected pharmaceuticals firms' registered in DSE, Bangladesh e.g. Square Pharmaceuticals Ltd., Beximco Pharmaceuticals Ltd., Beacon Pharmaceuticals 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 asymmetric GARCH approaches. Symmetric GARCH approaches consider past positive and negative information in the same manner. This happens becausetheir variance equations depend onthe extent of lagged residuals 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.

Mean equation:
=ω + Є Varianceequation: 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: 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: =ω +ϒ 2 −1 + Є Variance equation: The parameter ϒ is recognized as risk premium parameter. 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 specification is given by- Where λ is the leverage effect parameter. The significance 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.

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 28 th January 2019 to 30 th 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: The represents the logarithmic daily returns of our sampled companies for time t, Pt is the ending prices of sample companies at time t, −1 is the corresponding previous days' prices of selected stocks.The descriptive information for all five selected company's returns is exhibited in Table 1.  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-Bera's 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 explained 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 present in the residuals of a predicted method. This test is contingent on the joint null hypothesis that the coefficient 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 randomly 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 residuals 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 rejecting 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.

Testing for Stationarity
Before proceeding for further study it is a prerequisite to know whether the series being dealt with is stationary 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 companies. 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; 0 : the data embrace unit root versus 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 strongly 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 construction, the residual diagnostic check was administered to observe whether there exists any remainder ARCHeffect. 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 demonstrated in Table 4, Table 5, Table 6, Table 7, and Table 8 respectively. A results summary of different model choosing benchmarks of all estimated models of our five sample companies is also exposed in table 09. With the exception of Beximco, the 0 (constant) components inside the volatility formulae are seen as being statistically meaningful at the 1%, 5%, and 10% levels in all of our sample companies. For Beximco, this component is significant only for EGARCH (1,1). However, the 1 (ARCH term), and 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 1 (GARCH term) isn't 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 today's conditional variance. So their previous volatilities have explanatory power on present volatilities for all companies. The outcomes of the variance equation shown in the Table 4 justify that the initial three coefficients 0 (constant), 1 (ARCH term), and 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 volatility. The score of the two estimated ARCH, & GAR-CH coefficients; 1 + 1 in 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 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 appearance of leverage effect in data of Square Pharmaceuticals Ltd. we applied EGARCH (1,1), & TGARCH (1,1) models. The 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 significance 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 aren't found as statistically significant; which assures that these models don't 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. Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively. Table 5 represents the results for Beximco Pharmaceuticals Ltd. The figures for the second equation show that 1 (ARCH) including 1 (GARCH) are statistically significant. The 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; 1 + 1 in our beginning model is is greater than one (1.00065) representing the volatility process as explosive (Abdalla & Winker, 2012). So, the consequences of shocks will never dry out. The GARCH-M (1,1) model considers variance in its mean equation in the regard of estimation. In estimation results in Table 5, the estimated coefficient ϒ (risk premium) of 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 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 don't exhibit additional ARCH effects in their residuals. These prove that the variance equations have been properly specified. 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. Remark: ***, **, * represents significant at 1%, 5% and 10% level respectively. Table 6 represents Beacon Pharmaceuticals Ltd. Under all of the methods, the outcomes for the variance equation find the 0 (constant), as well as the 1 (AR-CH) as significant. Whereas, the 1 (GARCH) is significant 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 conditional 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; 1 + 1 in all the GARCH-type modelsare in the vicinity of, or higher than one, which implies that shocks are roughly persistent and wouldn't become dry out. The GARCH-M (1,1) model considers variance in its mean equation for assessment.  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 conditional 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; 1 + 1 in all the GARCH-type models are in the vicinity of, or higher than one, which implies that shocks are roughly persistent and wouldn't become dry out. The GARCH-M (1,1) model considers 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  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; 1 + 1 in all the GARCH-type models are very near to, or higher than one, which implies that shocks are roughly persistent and wouldn't 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 coefficient ϒ (risk premium) of 2 is seen to be positive and not significant. So, no feedback comes from the variance after including of it in mean equation for Orion Pharmaceuticals Ltd. The 1 (asymmetry term) for the EGARCH (1,1) indicates a non-negative and statistically significant value, indicating the non-existence 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 nonexistence 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).