INTRODUCTION
Pricing of securities depends on volatility of each asset. Therefore, price
changes indicate the average reaction of investors to news. The arrival of new
information makes investors to adapt their expectations and this is the main
cause for price and return changes. Trading volume and volatility are indicators
of the current stock market activity on one hand and a potential source of information
for the future behavior of stock market on the other hand. Numerous papers have
documented the fact that high stock market volume is associated with volatile
returns. However, many theoretical and empirical studies are designed to work
with the conditional variance in developed markets (Dimson
and Marsh, 1990; McMillan et al., 2000).
Various studies reported that there are significant relationships between volume
and stock price movement and volatility. For example, Saatcioglu
and Starks (1998) found that volume led stock prices changes in four out
of the six emerging markets. Chan et al. (2000)
found that trading volume for foreign stocks is strongly associated with NYSE
opening price volatility. Griffin et al. (2007)
investigated the dynamic relation between marketwide trading activity and returns
in 46 markets and reported strong positive relationship between turnover and
past returns. Recently, several authors have investigated the volatility of
stock market by applying econometric models and suggested that, no single model
is superior (Akgiray, 1989; Pagan
and Schwert, 1990). Brailsford and Faff (1996) and
Koutmos (1998) examined the predictive performance of
several statistical methods with GARCH and TGARCH models for Australian stock
exchange. Dimson and Marsh (1990) examined various technical
methods of predicting the volatility of UK stock market returns and find that
exponential smoothening and regression model performed.
The present study reinvestigates the effect of trading volume on volatility
of the Nasdaq index in US stock market using GARCH model to see to what extent
the stock market’s reaction to the arrival of news changed when trading
commenced. Further we also analysis the contemporaneous relationship between
stock price volatility and trading volume. Also a few attempts were made to
model the most prominent features of the time series of Nasdaq index such as
volatility clustering, excess kurtosis and fat tailed by applying the most popular
techniques proposed by Engle (1982). To capture the
above characteristics, ARCH class of models were introduced by Engle
(1982) and GARCH (Generalized Autoregressive Conditional Heteroskedasticity)
by Bollerslev (1986) and Taylor (1986).
Since the intrinsically symmetric GARCH model does not cope with the asymmetry
issues or so called leverage effect, the Exponential Generalized Autoregressive
Conditional Heteroskedasticity process (EGARCH) by Nelson
(1991) is suggested. Finally, to capture asymmetries in terms of negative
and positive shocks TGARCH (Threshold Generalized Autoregressive Conditional
Heteroskedasticity) model was introduced by Zakoian (1994)
and Glosten et al. (1993).
REVIEW OF LITERATURE
Random walk model: The random walk model is the simplest possible models,
where the Ordinary Least Square (OLS) method are constructed on the assumption
of constant variance. As per, efficient market hypothesis the competing market
participants reflect information instantly hence are useless in predicting future
prices. The basic model for estimating stock returns fluctuation by using OLS
in the naïve random walk model is given below:
where, μ is the mean value of the returns, it is expected to be insignificantly differ from zero and ε_{t} is the error term should not be serially correlated over time.
GARCH: Bollerslev (1986) extended Engle’s
ARCH model to the GARCH model and it is based on the assumption that forecasts
of time varying variance depend on the lagged variance of the asset. An unexpected
increase or decrease in returns at time t will generate an increase in the expected
variability in the next period. The basic and most widespread model GARCH can
be expressed as:
where, λ_{j}>0, β_{i} = 0. The GARCH is weekly
stationery Σβ_{i}+Σλ_{j}<1, the latter
two quantifying the persistence of shocks to volatility (Nelson,
1991).
In particular, volatility forecast are increased following a large positive
and negative index return, the GARCH specification that capturing the welldocumented
volatility clustering evident in financial returns date (Engle,
1982).
TGARCH: In TGARCH model, it has been observed that positive and negative
shocks of equal magnitude have a different impact on stock market volatility,
which may be attributed to a leverage effect (Black, 1976).
In the same sense, negative shocks are followed by higher volatility than positive
shocks of the same magnitude (Engle and Ng, 1993). The
threshold GARCH model was introduced by the works of Zakoian
(1994) and Glosten et al. (1993). The main
target of this model is to capture asymmetric in terms of negative and positive
shocks and adds multiplicative dummy variable to check whether there is statistically
significant different when shocks are negative. The conditional variance for
the simple TGARCH model is defined by:
where, d_{t} takes the value of 1 if ε_{t} is negative and 0 otherwise. So good news and bad news have a different impact.
EGARCH: The Exponential GARCH model specifies conditional variance in
logarithmic form, which means that there is no need to impose estimation constraints
in order to avoid negative variance Nelson (1991). The
mean and variance equation for this model is given by:
where, δ captures the asymmetric effect. The exponential nature of EGARCH ensures that the conditional variance is always positive even if the parameter values are negative; thus there is no need for parameter restrictions to impose nonnegativity.
Smirlock and Starks (1985) found that the returnvolume
relation is asymmetric and later, Smirlock and Starks (1988)
found a strong positive lagged relationship between volume and absolute price
changes using individual stock data. Lee and Swaminathan
(2000) used monthly returns and daily trading volume of all the firms listed
on NYSE and American Exchange (AMEX) and find that momentum and trading volume
appear to predict subsequent returns in the US equity market. Bekaert
and Wu (2000) not only support this finding but also suggest that negative
shocks generate a greater response in volatility than positive shocks of an
equal magnitude, evidence of the speed of information transmission in markets.
Thus, the findings of past studies are strong indications of information content
of volatility on the markets, which could be used by investors to earn abnormal
profit. Ratner and Leal (2001) examined the Latin American
and Asian financial markets and find a positive contemporaneous relation between
return and volume in these countries except India. At the same time they observed
that there exists a bidirectional causal relation between return and volume.
In summary, the return and volume are strongly related contemporaneously but
there is little evidence that either can be used to predict the other. De
Medeiros and Doornik (2006) investigated the empirical relationship between
stock returns, return volatility and trading volume using data from the Brazilian
stock market. The study found out there is a contemporaneous and dynamic relationship
between return volatility and trading volume and return volatility contains
information about upcoming trading volumes. Atmeh and Dobbs
(2006) investigated the performance of moving average trading rules in the
Jordanian stock market and found that technical trading rules can help to predict
market movements. AlKhouri and Ajlouni (2007) reported
that the pricelimit technique was effective in reducing the volatility in the
Amman stock exchange. Floros and Vougas (2007) used
GARCH and GMM method to investigate the relationship between trading volume
and returns in Greek stock index futures market and found that trading volume
was used as the indicator of prices.
EMPIRICAL RESULTS AND ANALYSIS
The basic descriptive analysis of the time series of stock returns and trading volume is shown in Table 1. All returns are calculated as the first difference of the log of the daily closing price. Daily trading volume and stock return have positive kurtosis and high JB statistics that implies that the distribution is skewed to the right and they are leptokurtic((heavily tailed and sharp peaked), i.e., the frequency distribution assigns a higher probability to returns around zero as well as very high positive and negative returns. The JarqueBera statistic test indicates that the null hypothesis of normality is rejected and shows that all the series exhibit nonnormality and indicates the presence of Heteroscedasticity. Hence, GARCH (1, 1) model is the suitable for testing of hypothesis.
The study here employs the unit root test to examine the time series properties
of concerned variables. Unit root test describes whether a series is stationary
or nonstationary.
Table 1: 
Diagnostic tests 

For the test of unit root the present study employees the Augmented Dickey
Fuller test and KPSS test (Dickey and Fuller, 1981).
ADF test is used to measure the stationarity of time series data which in turn
tells whether regression can be done on the data or not. The output is presented
in the Table 2. On observing the outputs of ADF and KPSS tests,
it is seen that the ADF test statistic and KPSS test statistics for all is less
than the critical values at 1, 5 and 10% confidence level. Both ADF and KPSS
test statistics confirm that all prices have unit root (nonstationary). So,
the null hypothesis is rejected and the data is found to be stationary.
We investigate that weather trading volume has an explanatory power for Indian
stock market by fitting GARCH (1, 1) model with daily volume included in the
conditional variance equation. It is evident from the Table 3
that parameter β is negative and statistical insignificant indicating that
trading volume does not have GARCH effect in the stock market.
Table 3: 
GARCH model with trading volume 

Systematic variations in trading volume are assumed to be caused only by the
arrival of new information. AIC and SIC criteria used in the study indicating
lower for the regression which is quite reasonable and fit for our model. Further
DurbinWatson value is 2 suggests autocorrelation or specification errors. Since
the DurbinWatson statistic is greater than 2, the error terms are not auto
correlated indicating that the statistical model is fit and appropriate.
LEVERAGE/ASYMMETRIC EFFECT
It is very often observed that downward movement of the markets is followed
by higher volatilities than upward movement of the same magnitude. So it is
important to use TARCH, EGARCH, PGARCH and component ARCH models to test asymmetric
shocks to volatility. Sometimes the simple GARCH models cannot capture some
important features of the data.
Table 4: 
TARCH model with trading volume 

To investigate the leverage effect we have used TARCH (1, 1) model introduced
independently by Zakoian (1994). If the bad news has
a greater impact on volatilities than good news, a leverage effect exists. ARCH
model helps to explain the volatility of spot market when some degree asymmetric
is present in the data.
TARCH model takes the leverage effect into account. The presence of leverage
effect is seen in Table 4 which implies that every price changes
are responding asymmetrically to the positive and negative news in the market.
In the conditional variance equation; α, the coefficient for latest news
which is statistically significant at 1% level indicating that the recent news
has an impact on the volatility of the stock. Similarly β coefficient is
insignificant and suggests that old news is not influencing the stock market
volatility. Coefficient γ (parameter of volume) is positive and greater
than 0 indicating the impact is asymmetric. The analysis shows that trading
volume is associated with an increase in stock return volatility. Good news
therefore induces more trading volume than bad news.
To test the leverage effect, EGARCH model is also used. Table
5 exhibited the existence of leverage effect and news impact is asymmetric
(γ).
Table 5: 
EGARCH model with trading volume 

As Coefficients γ is positive, greater than 0 and significant at 1% level,
the analysis is suggesting that trading volume increases due to good news in
the market. Coefficients γ shows a positive impact of volume on stock return
also generate less impact on volatility of the market.
The empirical evidence in the Table 5 suggests the existence
of leverage effect and news impact is asymmetric (γ). In the models with
a significant power parameter we found δ smaller than 2, in concordance
with Ding et al. (1993) results and the asymmetric
estimated parameter γ is positive. So trading volume increased the stock
return and decrease expected volatility in the market. This supports a positive
correlation between trading volume and predictable volatility of stock returns.
The analysis shows that the PARCH model which exhibits a low power effects but
strong leverage effects in the market.
So far we have used TARCH, EGARCH and PARCH model to find the significance of the asymmetric effects. Alternatively it is also equally important to find the cross correlation between the squared standard residuals and lagged standardized residuals to know the impact of long run/short run movements in volatility.
CONCLUSION
This study examined the relationship between stock returns and trading volume and has used the GARCH (1, 1) model, asymmetric TARCH, EGARCH, PGARCH and CARCH model to empirically examine the persistence of shocks to volatility and to determine the asymmetry in the pattern of volatility. This paper specifically tested the hypothesis of variability in volatility, which implies that volatility is greater when stocks price are moving downwards than upwards. Statistical inferences are drawn from the data by means of significance tests and over all goodness of fit of all the models as reported by the Akaike info criterion &Schwarz criterion. The study found that the recent news has an impact on the volatility of the trading volume. Also, the past news coefficient is statistically insignificant and suggests that old news is not having influencing the trading volume volatility. So it is evident from the study that systematic variations in trading volume are assumed to be caused only by the arrival of new information. To predict volatility, we have used the asymmetric TARCH, EGARCH, PARCH and Component ARCH model and evidence suggests that leverage effects exist and the news impact is asymmetric. This implied that daily new information in market may have significant impact on price volatility. So the study concludes that bad news generate more impact on volatility of the stock return and trading volume. One explanation may be that normally investors have a higher aversion to downside risk, so they react faster to bad news.