The study of volatility is always a serious concern for analysts and researchers because high degree of volatility can affect the smooth functioning of any stock market. It may also affect the economic growth and development of the economy through its effect on investor’s confidence and risk taking ability. The researchers worldwide have attempted to identify the major factors affecting the level of volatility in the stock markets. The available theoretical and empirical literature suggests that the main source of volatility in any stock market is the arrival of new information or news.

There is a general agreement on what constitutes stock market volatility and, to a smaller extent, on how to quantify it, there is far less conventionality on the reasons of changes in stock market volatility .A number of researchers investigated the causes of volatility in the arrival of new, unexpected information that affect expected returns on a stock (Engle and Mcfadden, 1994). Thus, changes in market volatility would just reproduce changes in the domestic or global economic environment. Others maintain that volatility is caused largely by changes in trading volume, practices or tends, which in tum are resolute by factors such as changes in macroeconomic policies, shifts in investor’s risk appetite and growing uncertainty.

Conditional Heteroscedasticity (ARCH) became a very popular method in the modeling of stock market volatility. As comparison to traditional time series models, ARCH models allowed the conditional variances to change during time as functions of precedent errors. First approach was to improve the univariate. ARCH model with a different requirement of the variance function. One development was introduced by Bollerslev (1986) where the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) method was presented. Then after, the Integrated GARCH (IGARCH) Engle and Bollerslev (1994) and the exponential GARCH (EGARCH)Nelson (1991) were significant one wherever re-specification of variance equation was considered.


Objective of study

The purpose of this paper is to understand the daily return data volatility of stocks & to develop an asymmetric GARCH models can explain determination of shock and volatility. This study is an attempt to develop models to elucidate the volatility of the stock in Pakistan. To this end, the study includes main indicex of Pakistan stock market i.e KSE 100. This study uses the Autoregressive Conditional Heteroskedasticity (ARCH) models and its extension, the Generalized ARCH, EGARCH and TARCH models was used to find out the presence of the stock market volatility on Pakistan stock market. The objective is to model the phenomena of volatility clustering and persistence of shock using asymmetric GARCH family of models.

Research Methodology

The study spanned the period from Jan 2003 to Dec 2014. This period of study is selected in order to check volatility in Karachi stock exchange returns. The sample population of the study consists of the daily returns of the most prominent domestic index, KSE. The data was collected from official website of respective stock exchange. Daily closing price of the index was considered for the period of study. This market index was fairly representative of the various industry sectors. The daily stock prices were converted to daily returns. Returns are calculated by difference of old and new.

The econometric software package Eviews 5.0 has been used to do the estimations. Arch and Garch Models Conventional econometric models assume a constant one-period forecast variance. To simplify this implausible assumption, Robert Engle presented a set of methods called autoregressive conditional heteroscedasticity (ARCH). These are zero mean, serially uncorrelated methods with non constant variance conditional on the past.

A practical generalization of this model is the GARCH parameterization introduced by Bollerslev (1986). This model is also a weighted average of past squared residuals, but it has waning weights that by no means go entirely to zero.

In the third equation ht= var (€t /CPt-I), cpt-Iit is the information prior to time t-l. Because GARCH (p,q) is an annex of ARCH model, it has all the properties Asia-Pacific Business Review of the original ARCH model. And because in GARCH model the conditional variance is not only the linear function of the square of the lagged residuals, it is also a linear function of the lagged conditional variances, GARCH model is more precise than the original ARCH model and it is easier to compute. The most commonly used GARCH model is GARCH (1,1) model. The (1,1) in parentheses is a standard notation in which the first number refers to how many autoregressive lags, or ARCH terms, come into view in the equation, as the second number refers to how many moving average lags are specified, which here is frequently called the number of GARCH terms. Occasionally models with more than one lag are needed to find better variance forecasts.

GARCH (1,1) is the most extensively used GARCH model because it is correctness and ease. Although GARCH model is very helpful in the predicting of volatility and asset pricing, there are still many problems GARCH model cannot clarify. The main difficulty is that standard GARCH models presume that positive and negative error conditions have a symmetric effect on the volatility.

In other terms, good and bad news have the similar impact on the volatility in this model. In real life this hypothesis is often desecrated, in particular by stock returns, in that the volatility increases more often after a flow of bad news than after good news. According to the challenges in the standard GARCH model, a number of parameterized extensions of the standard GARCH model have been recommended in recent times.

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E-Garch Model Exponential GARCH (EGARCH) model was first developed by Nelson in 1991. The main purpose of EGARCH model is to explain the asymmetrical response of the market under the positive and negative shocks.. If one compared the above equations with the premises of the conventional GARCH model, one can see that there are no constraints for the parameters. This is one of the biggest benefits of EGARCH model as compared to the standard GARCH model.

TARCH Model Threshold ARCH (TARCH) model was first developed by Zakoian in 1990. It has the conditional variance Where t d is latent variable d, = 1 for fl < 0 and d, = 0 for fl >= 0 because t d is built-in, the rise (0 E) of stock prices will have different impact on conditional variance. When the stock prices increase, cp we say that there is leverage effect.

Analysis and Discussion Arch Test

A descriptive investigation of the plot of daily returns on KSE (Figure 1) reveals that returns incessantly fluctuated about the mean value that was close to zero. The return measures were both in positive and negative area. More fluctuations be tending to cluster together and were alienated by periods of relative calm

This was in agreement with Fama’s (1965) observation of “volatility clustering”. From the time series graph of the returns for KSE market, it is analyzed that high volatilities are followed by high volatilities and low volatilities are followed by low volatilities. That means both time series have important time varying variances. Additionally, it is appropriate to put conditional variance into the function to clarify the impact of risk on the returns. Hence, GARCH class model is the excellent tool for the study




 Mean  0.003691
 Median  0.004623
 Maximum  0.354357
 Minimum -0.482963
 Std. Dev.  0.055540
 Skewness -0.499174
 Kurtosis  8.308197
 Jarque-Bera  3601.731
 Probability  0.000000
 Sum  10.93704
 Sum Sq. Dev.  9.136997
 Observations  2963

Descriptive statistics (Table 1) for KSE returns showed skewness statistic of daily returns different from zero which indicated that the return distribution was asymmetric. In addition, relatively large excess kurtosis recommended that the underlying data was leptokurtic (heavily tailed and sharp peaked).

The Jarque – Bera statistic is calculated to test the null hypothesis of normality rejected the normality assumption. Kse index appeared to have significant strong autocorrelations in one-day lag returns. In addition, the autocorrelation in the squared daily returns suggested incidence of clustering. The results ruled out the independence assumption for the time series of given data set. Stationary of the return series were tested by conducting Dickey-Fuller. The results of the test confirmed that the series is stationary at level (Table 2).

Table 2

Null Hypothesis: R has a unit root  
Exogenous: Constant    
Lag Length: 0 (Automatic – based on SIC, maxlag=27)
      t-Statistic   Prob.*
Augmented Dickey-Fuller test statistic -46.96748  0.0001
Test critical values: 1% level   -3.432368  
  5% level   -2.862317  
  10% level   -2.567228  
*MacKinnon (1996) one-sided p-values.  
Augmented Dickey-Fuller Test Equation  
Dependent Variable: D(R)    
Method: Least Squares    
Date: 06/17/15   Time: 11:29    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Variable Coefficient Std. Error t-Statistic Prob.
R(-1) -0.854238 0.018188 -46.96748 0.0000
C 0.003153 0.001012 3.115317 0.0019
R-squared 0.427017     Mean dependent var 1.78E-05
Adjusted R-squared 0.426823     S.D. dependent var 0.072602
S.E. of regression 0.054966     Akaike info criterion -2.963536
Sum squared resid 8.942867     Schwarz criterion -2.959489
Log likelihood 4390.997     Hannan-Quinn criter. -2.962080
F-statistic 2205.944     Durbin-Watson stat 2.001239
Prob(F-statistic) 0.000000      


Before ARCH-GARCH is used in the study to approximation the model, the study is required to test whether the data has ARCH effect.

The most commonly used method is Lagrange MUltiplier test (LM).as table below shows that he N*R2 & f-statistic is high with significant p-value shows that the ARCH effect is present .

The present work used GARCH, GARCH-M, TARCH, TARCH-M, EGARCH and EGARCH-M to estimate the data.


Heteroskedasticity Test: ARCH    
F-statistic 23.14889     Prob. F(2,2957) 0.0000
Obs*R-squared 45.63032     Prob. Chi-Square(2) 0.0000


Dependent Variable: R    
Method: ML – ARCH (Marquardt) – Normal distribution
Date: 06/17/15   Time: 11:31    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 14 iterations  
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2  
Variable Coefficient Std. Error z-Statistic Prob.
C 0.004657 0.000780 5.970079 0.0000
R(-1) 0.189642 0.014004 13.54192 0.0000
  Variance Equation    
C 0.001822 3.57E-05 50.98481 0.0000
RESID(-1)^2 0.470400 0.025631 18.35312 0.0000
R-squared 0.018415     Mean dependent var 0.003688
Adjusted R-squared 0.018083     S.D. dependent var 0.055550
S.E. of regression 0.055045     Akaike info criterion -3.097226
Sum squared resid 8.968662     Schwarz criterion -3.089132
Log likelihood 4590.992     Hannan-Quinn criter. -3.094313
Durbin-Watson stat 2.086801      





Dependent Variable: R    
Method: ML – ARCH (Marquardt) – Normal distribution
Date: 06/17/15   Time: 11:33    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 45 iterations  
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1)
Variable Coefficient Std. Error z-Statistic Prob.
C 0.004144 0.000625 6.628444 0.0000
R(-1) 0.108704 0.018898 5.752277 0.0000
  Variance Equation    
C 3.91E-05 2.40E-06 16.29193 0.0000
RESID(-1)^2 0.156128 0.010756 14.51604 0.0000
GARCH(-1) 0.845044 0.007936 106.4792 0.0000
R-squared 0.019628     Mean dependent var 0.003688
Adjusted R-squared 0.019297     S.D. dependent var 0.055550
S.E. of regression 0.055011     Akaike info criterion -3.379572
Sum squared resid 8.957575     Schwarz criterion -3.369454
Log likelihood 5010.145     Hannan-Quinn criter. -3.375930
Durbin-Watson stat 1.925959      



Dependent Variable: R    
Method: ML – ARCH (Marquardt) – Normal distribution
Date: 06/17/15   Time: 11:35    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 46 iterations  
Presample variance: backcast (parameter = 0.7)
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0) +
Variable Coefficient Std. Error z-Statistic Prob.
C 0.003833 0.000682 5.623912 0.0000
R(-1) 0.118520 0.019386 6.113554 0.0000
  Variance Equation    
C 4.35E-05 2.83E-06 15.36350 0.0000
RESID(-1)^2 0.133526 0.015151 8.813000 0.0000
RESID(-1)^2*(RESID(-1)<0) 0.059658 0.016272 3.666206 0.0002
GARCH(-1) 0.837064 0.009143 91.54924 0.0000
R-squared 0.020387     Mean dependent var 0.003688
Adjusted R-squared 0.020056     S.D. dependent var 0.055550
S.E. of regression 0.054990     Akaike info criterion -3.381526
Sum squared resid 8.950640     Schwarz criterion -3.369385
Log likelihood 5014.040     Hannan-Quinn criter. -3.377156
Durbin-Watson stat 1.946070      


Dependent Variable: R    
Method: ML – ARCH (Marquardt) – Normal distribution
Date: 06/17/15   Time: 11:37    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 44 iterations  
Presample variance: backcast (parameter = 0.7)
LOG(GARCH) = C(3) + C(4)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(5)
        *RESID(-1)/@SQRT(GARCH(-1)) + C(6)*LOG(GARCH(-1))
Variable Coefficient Std. Error z-Statistic Prob.
C 0.005133 0.000383 13.41781 0.0000
R(-1) 0.111279 0.018531 6.004927 0.0000
  Variance Equation    
C(3) -0.578039 0.035085 -16.47536 0.0000
C(4) 0.346857 0.020478 16.93799 0.0000
C(5) -0.075865 0.010874 -6.976950 0.0000
C(6) 0.948154 0.004057 233.6890 0.0000
R-squared 0.018936     Mean dependent var 0.003688
Adjusted R-squared 0.018605     S.D. dependent var 0.055550
S.E. of regression 0.055030     Akaike info criterion -3.390574
Sum squared resid 8.963899     Schwarz criterion -3.378433
Log likelihood 5027.440     Hannan-Quinn criter. -3.386204
Durbin-Watson stat 1.929444      


Dependent Variable: R    
Method: ML – ARCH (Marquardt) – Normal distribution
Date: 06/17/15   Time: 11:41    
Sample (adjusted): 1/03/2003 5/12/2014  
Included observations: 2962 after adjustments  
Convergence achieved after 74 iterations  
Presample variance: backcast (parameter = 0.7)
GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)
Variable Coefficient Std. Error z-Statistic Prob.
@SQRT(GARCH) 0.027837 0.042554 0.654151 0.5130
C 0.003251 0.001568 2.073169 0.0382
R(-1) 0.109279 0.019097 5.722392 0.0000
  Variance Equation    
C 3.97E-05 2.44E-06 16.27203 0.0000
RESID(-1)^2 0.157702 0.010906 14.45994 0.0000
GARCH(-1) 0.843484 0.008057 104.6846 0.0000
R-squared 0.018986     Mean dependent var 0.003688
Adjusted R-squared 0.018323     S.D. dependent var 0.055550
S.E. of regression 0.055038     Akaike info criterion -3.379067
Sum squared resid 8.963444     Schwarz criterion -3.366926
Log likelihood 5010.398     Hannan-Quinn criter. -3.374697
Durbin-Watson stat 1.924319      

Table 3

Market model rss A-R2 AIC SC
KSE GARCH M 8.963444 0.018323 -3.379 -3.366
  E GARCH 8.963899 0.0186 -3.390 -3.378
  T GARCH 8.950 0.02005 -3.3815 -3.369385
  EGARCH-M 8.952 0.019494 -3.3911 -3.3769


Following is the table with the results estimated from different models. From this table, one can select the best model for the further forecasting of stock market volatility.

From Table 3, one can see that for kse market TGARCH (l,l)  has the lowest RSS and the relative high adjusted 2 R. That means, TGARCH (1,1)  is superior to other models in the~timation.

From the standard of AIC and SC, we can see that GARCH (1,1) has the lowest value. That means GARCH (1,1) is also a relative good model for the estimation.

In addition, when the study use GARCH (1,1) to estimate the data, it is found that the kse market is 0.845. They are very close to 1. This demonstrates that there is high durability of the volatilities in market. That means if there is an expected shock in these markets, the sharp movements will not die out in the short run. That is a sign for high risk. At the same time, the study found that the summation of the parameters is less than 1, which indicates that the GARCH process for the stock return is wide-sense stationary.

When the study used TARCH (1,1) to estimate the model, it is found that the estimate of cps are greater than 0 for stock exchange.

When the study used EGARCH (1,1), it is found the estimates of cps are less than 0 for market. Then one can conclude that there are leverage effects in market. That is to say the volatilities caused by negative shocks are greater than that caused by positive shocks. This is in consistent with most of the existing literature.

The study also used the estimated EGARCH (1,1) to predict the volatilities for KSE. One can see that the model did a great job.


  KSE Return SENSEX Return Nifty Return
 Mean  0.003691 0.000613 0.000561
 Median  0.004623 0.001390 0.001540
 Maximum  0.354357 0.079311 0.079691
 Minimum -0.482963 -0.118092 -0.130539
 Std. Dev.  0.055540 0.015473 0.015388
 Skewness -0.499174 -0.537631 -0.752137
 Kurtosis  8.308197 7.067238 8.508060
 Jarque-Bera  3601.731 1463.086 2695.063
 Probability  0.000000  0.000000  0.000000
 Sum  1.93704 1.215901 1.113573
 Sum Sq. Dev.  0.9136997 0.474761 0.469532
 Observations  2963 1984 1984

By comparing KSE results with base article results (Nifty & SENSEX). At first glance you assume that our market is more volatile then Nifty & SENSEX. While keeping in mind that Indian market is emerging market and Pakistan is also getting a status of Emerging County in 2016 by MSCI. Which show’s that there are a lot of things very common in both markets.

By comparing Mean, Maximum and Minimum values, almost all three markets have same level of fluctuations (Variations). Standard Deviation of KSE 100 Index is higher which shows that market is more volatile as compare to Nifty & SENSEX.

The distribution is negatively skewed for all three markets then the probability density function has a long tail to the left. Kurtosis is positive (It can only be Negative in case of Constant X). Kurtosis > 3 – Leptokurtic distribution, sharper than a normal distribution, with values concentrated around the mean and thicker tails. This means high probability for extreme values.

Sum of Square of Deviations is also high which depict that market is more volatile. The JB statistic is an indication of your distributions deviation of 0 (skewness and Kurtosis if it was truly a normal distribution). P-value = 0 indicates that the null hypothesis: “the distribution is normal” is rejected. Probability is less than 5% which shows that our results are significant. We took more number of observations as compare to Nifty & SENSEX.


The findings of this study suggest that Pakistan stock market has significant ARCH effects and it is appropriate to use ARCH GARCH models to estimate the process.  The research found that both TGARCH (1,1) and EGARCH (l,l) did good jobs in fitting the process for exchange. Research also shows that the investors in stock markets are not grown well and they will be heavily influenced by information (good or bad) very easily.

The results of GARCH (1,1) model indicated the presence of time varying volatility. both α and β coefficients were found positive in variance equation. The magnitude of α is lesser than magnitude of β indicating more impact of past volatility on the current volatility in comparison to impact of past shocks/news on the conditional volatility in equity market in Pakistan.

In addition to this, both α and β coefficients have shown high significance at 1 percent and 5 percent levels of significance indicated that past lagged variance has significant impact on the conditional volatility in Pakistan stock market. It indicated that news shocks in past has noteworthy impact on present returns on equity stocks.


To conclude it can be stated that the historical performance of any market plays a significant role to determine the investment strategy of that market. Therefore, both the market performance in past as well as market sentiments in the past; which have resulted due to various types of news extended by various market participants; have impact on the forecasting strategies of the investors.

A further study can be conducted to examine whether the impact of flow of capital by foreign institutional investors, introduction of derivatives, and impact of trading volume etc. can redefine the volatility of Pakistan stock market in a better way or not.