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Return, volume, and volatility analysis in Indian stock market.



Introduction

The emergence of informationally efficient financial markets is an important facet of any country's economic modernization, with far-reaching implication for its macroeconomic mac·ro·ec·o·nom·ics  
n. (used with a sing. verb)
The study of the overall aspects and workings of a national economy, such as income, output, and the interrelationship among diverse economic sectors.
 stability and performance. Thus, it is in the interest of the economy to achieve efficiency in the dynamics of the stock markets. More can be learned about the market by studying the joint dynamics of prices and trading volume Trading volume

The number of shares transacted every day. As there is a seller for every buyer, one can think of the trading volume as half of the number of shares transacted. That is, if A sells 100 shares to B, the volume is 100 shares.
 than by focusing on the univariate dynamics of prices (Gallant et al. 1992).

In a stock market, return and trading volume are two prime indicators of trading activity, jointly determined by the same market dynamics and may contain valuable information about a security. Prices and trading volume build a market information aggregate out of each new piece of information. Unlike stock price behaviour, which reflects the average change in investors' beliefs due to the arrival of new information, trading volume reflects the sum of investors' reactions. Differences in the price reactions of investors are usually lost by averaging of prices, but they are preserved in trading volume. In this sense, the observation of trading volume is an important supplement of stock price behaviour (Gurgul et al. 2005).

The price--volume relationship depends on the rates of information flow and its diffusion to the market, the extent to which markets convey information, the size of the market, and the existence of short-selling constraints. Trading volume is viewed as the critical piece of information which signals where prices will go next. The trading volume is thought to reflect information which stock prices cannot convey to market participants. Relying on this power of volume and to improve the understanding of the microstructure mi·cro·struc·ture  
n.
The structure of an organism or object as revealed through microscopic examination.


microstructure
Noun

a structure on a microscopic scale, such as that of a metal or a cell
 of stock market, the relationship between return, volume, and volatility has received substantial attention in the market microstructure Market microstructure

The functional setup of a market.
 for a number of years. Furthermore, the stock price--volume relation can be used as the basis of a trading strategy In finance, a trading strategy (see also trading system) is a predefined set of rules to apply.

Usually, this refers to a means used to replicate an option in order to give it an arbitrage free value in the sense that the cost of buying some financial assets to give the same
 and as evidence for or against the efficiency of stock markets.

Financial literature has documented various flavours of the return-volume relationship especially in US stock markets (see survey in Karpoff 1987). By contrast, relatively little attention has been devoted to this relationship in India. Some researchers have made attempts to evaluate return--volume relationship in Indian stock market but these are elementary efforts and moreover, the studies have failed to take the phenomenon of volatility persistence/volatility clustering in return--volume relationship. In most cases, financial time series behave in a way that does not conform to Verb 1. conform to - satisfy a condition or restriction; "Does this paper meet the requirements for the degree?"
fit, meet

coordinate - be co-ordinated; "These activities coordinate well"
 the normality distribution. Hence, the volatility observed in the market is a natural application for the autoregressive conditional heteroscedasticity (ARCH). To observe this phenomenon, ARCH model introduced by Engle (1982) and Bollerslev's (1986) generalized ARCH (GARCH GARCH Generalized Autoregressive Conditional Heteroskedasticity ) model is used in many studies (e.g. Schwert 1990, Lamoureux and Lastrapes 1990; and Kim and Kon 1994). The GARCH specification allows the current conditional variance In statistics, conditional variance is a special form of the variance. If we have a conditional distribution Y|X the conditional variance is defined as



where
 to be a function of past conditional variances. Therefore, the current study investigates return, volume, and volatility relationship in Indian stock market using symmetric and asymmetric GARCH models. The remainder of the paper is as follows. Section I reviews the literature. In Section II, the methodology and data employed are presented. In Section III, the key results from the empirical investigation are reported and in Section IV conclusions are drawn.

Review of Literature

Examination of relationship between return and volume provides significant information regarding the price discovery efficiency of the asset. Based on this logic, return, volume, and volatility relationship has long attracted the attention of many financial economists, which makes contribution not only to a well-established stream of empirical financial studies, but also turn out to be relevant in a broader historical economic perspective.

Traditional literature on the contemporaneous relationship between volume and price showed that there exits a positive relation between volume and absolute price change (i.e. price volatility) in both equity (e.g. Epps and Epps 1976; and Wood et al. 1985) and futures markets (e.g. Cornell 1981 and Tauchen and Pitts 1983). However, that positive relation between volume and price change (i.e. returns) is found in the stock markets, but not in futures markets (see the surveys in Karpoff 1987). This result was consistent with Karpoff's (1988) costly short-sales hypothesis, indicating that the costs of taking long and short positive positions are asymmetric in stocks markets, but symmetric in futures markets.

Schwert (1989), using monthly aggregates of daily data on Standard and Poor (S&P) composite index Composite Index

A grouping of equities, indexes or other factors combined in a standardized way, providing a useful statistical measure of overall market or sector performance over time. Also known simply as a "composite".
 in NYSE NYSE

See: New York Stock Exchange
, documented the evidence of a positive relationship between estimated volatility and current and lagged volume growth rates Growth Rates

The compounded annualized rate of growth of a company's revenues, earnings, dividends, or other figures.

Notes:
Remember, historically high growth rates don't always mean a high rate of growth looking into the future.
, using linear distributed lag and VAR models. Similar issue was also addressed by Lamoureux and Lastrapes (1990) using individual stocks from the S&P index. They documented a positive conditional volatility--volume relationship in models with Gaussian errors and Generalized Autoregressive Conditional Heteroskedasticity Autoregressive Conditional Heteroskedasticity (ARCH)

A nonlinear stochastic process, where the variance is time-varying, and a function of the past variance. ARCH processes have frequency distributions which have high peaks at the mean and fat-tails, much like fractal distributions.
 (GARCH)-type volatility specifications. However, the finding was cautiously interpreted as it might be biased due to the simultaneity between stock returns and volume. Similar results were also found in Bessembinder and Seguin (1993) for a variety of futures markets. Finally, Gallant et al. (1992), using non-parametric methods, confirmed the positive correlation Noun 1. positive correlation - a correlation in which large values of one variable are associated with large values of the other and small with small; the correlation coefficient is between 0 and +1
direct correlation
 between conditional volatility and volume, when examining daily S&P data from 1928 to 1987.

Kocagail and Shachmurove (1998) examined the contemporaneous relationship between volume and absolute return for sixteen futures markets. They found the relationship to be significantly positive. Daigler and Wiley (1999) examined the effect of different categories of futures traders; and found that the uninformed groups of traders who were distant from the trading floor drove the positive volume--volatility relation.

Gurgul et al. (2005) and Otavio et al. (2006) also documented the evidence of significant contemporaneous interaction between return volatility and trading volume in Polish stock market, Brazilian stock market respectively.

A further analysis of relationship between trading volume and return needs to specify which variable is dependent and which is independent. The studies refared to above primarily focus on the contemporaneous relationship between price change and volume. Although some of these research efforts imply a dynamic relationship between price change and volume using cross-correlation, they do not further pursue causal relationship (Karpoff 1987 and Gallant et. al. 1992). But there are some empirical studies Empirical studies in social sciences are when the research ends are based on evidence and not just theory. This is done to comply with the scientific method that asserts the objective discovery of knowledge based on verifiable facts of evidence.  which have examined the causal relationship between returns and trading volume.

Rogalski (1978), Smirlock and Starks (1985), and Jain and Joh (1986) examined lagged associations and reported the evidence of a unidirectional Granger causality Granger causality is a technique for determining whether one time series is useful in forecasting another. Ordinarily, regressions reflect "mere" correlations, but Clive Granger, who won a Nobel Prize in Economics, argued that there is an interpretation of a set of tests as  from returns to trading volume in US markets. Gallant et. al. (1992) and Hiemstra and Jones (1994) investigated the linkages between volume and returns on the US equity markets. While the former study concluded that volume did not forecast returns, Hiemstra and Jones (1994) found the evidence of unidirectional Granger causality from Dow Jones Dow Jones

the best known of several U.S. indexes of movements in price on Wall Street. [Am. Hist.: Payton, 202]

See : Finance
 stock returns to percentage changes in NYSE trading volume. More importantly, they also found a significant bi-directional non-linear causality between returns and volume. Bhanupant (2001), by following the empirical approach of Hiemstra and Jones (1994), examined this relationship in Indian equity market and reported results which are in consistent with the result of Hiemstra and Jones (1994). Kocagail and Shachmurove (1998) investigated the return-volume relationship for US commodity and financial futures financial futures

Obligations to buy or sell particular positions in financial instruments. The features of financial futures are identical to those of any futures contract except that the asset for delivery is of a financial nature.
 contracts and reported that past trading volume did not increase the ability to forecast returns in futures markets.

Chen et al. (2001) examined the dynamic relation between returns, volume, and volatility of stock indices for nine countries and found mixed results. They demonstrated that returns significantly caused volume for US, Japan, UK and France and causal direction from volume to returns was found for Canada only whereas in Switzerland, the Netherlands, and Hong Kong Hong Kong (hŏng kŏng), Mandarin Xianggang, special administrative region of China, formerly a British crown colony (2005 est. pop. 6,899,000), land area 422 sq mi (1,092 sq km), adjacent to Guangdong prov.  they observed bi-directional causality. Lee and Rui (2002) examined the dynamic relation between stock market trading volume and returns for the three large markets (viz., New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, Tokyo, and London). They found that returns caused trading volume in the US and Japanese markets but not in the UK market. However, there was no causality from trading volume to returns in any of these markets.

Griffin et. al. (2004) investigated the dynamic relation between market-wide trading activity and returns in forty-six stock markets and documented the evidence of a stronger relation between return and turnover in countries with restrictions on short sales. Nguyen and Diagler (2005) examined the same relationship for S&P 500, Nasdaq, British pound, Japanese yen, Australian dollar, and Canadian dollar Noun 1. Canadian dollar - the basic unit of money in Canada; "the Canadian dollar has the image of loon on one side of the coin"
loonie

dollar - the basic monetary unit in many countries; equal to 100 cents
 futures. They observed unidirectional causality from returns to volume and volatility, and bi-directional causality between volume and volatility, but returns strongly explained the changes in volatility as compared to volume.

In a nutshell, on the basis of the studies mentioned above it can be stated that significant efforts have been made at the international level to evaluate return, volume, and volatility relationship, whereas in India this relationship has not been well investigated. Therefore, the current study is an attempt to fill this gap and sheds light on the informational efficiency of Indian stock market. This paper examines the relationship between return, return volatility, and volume in a contemporaneous and dynamic context in Indian stock market and contributes to the literature in several respects. Firstly, it deploys the Granger causality test to investigate information flow between the variables instead of ARIMA. In addition, it uses the GARCH models in the study of return-volume ivestigation. This study further checks the information asymmetry Information asymmetry

Condition that information is known to some, but not all, participants.
 with EGARCH (1,1) model. Moreover, the time period considered in the study helps to evaluate the impact of introduction of futures market futures market, a commodity exchange where contracts for the future delivery of grain, livestock, and precious metals are bought and sold. Speculation in futures serves to protect both the developers and the users of the commodities from unfavorable and unpredictable  on stock price-volume linkage. The linkage between reform and information content of volume depends on whether reform increases price efficiency. Thus, the study will enhance the understanding of market asymmetry, market efficiency, and information processing information processing: see data processing.
information processing

Acquisition, recording, organization, retrieval, display, and dissemination of information. Today the term usually refers to computer-based operations.
.

Methodology and Data

Financial time series such as stock prices often exhibit the phenomenon of volatility clustering In finance, volatility clustering refers to the observation, as noted by Mandelbrot, that "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. . To observe this phenomenon, ARCH model introduced by Engle (1982) and Bollerslev's (1986) generalized ARCH (GARCH) model are used.

The GARCH specification allows the current conditional variance to be a function of past conditional variances, allowing volatility shocks to persist over time. In particular, to test whether the positive contemporaneous relationship between trading volume and returns exists, the following GARCH (1,1) model is estimated where volume is included in mean equation:

[R.sub.t] = [alpha] + [p.summation over (i = 1)] [[beta].sub.t][R.sub.t-i] + [gamma][V.sub.t] + [[epsilon].sub.t] (1)

[h.sub.t] = [omega] + [m.summation over (i = 1)] [[alpha].sub.i][[epsilon].sup.2.sub.t-i] + [n.summation over (j = 1)] [[beta].sub.j] [h.sub.t-j] + [e.sub.t] (2)

Where [h.sup.t] represents the conditional variance term in period t, ai represents the news coefficient and bj represents a persistence coefficient. Parameters w and ai should be higher than 0 and bj should be positive in order to ensure conditional variance ht to be nonnegative non·neg·a·tive  
adj.
Of, relating to, or being a quantity that is either positive or zero.

Adj. 1. nonnegative - either positive or zero
. The sum of parameters ai and bj is a measure of the persistence in the variance of the unexpected return et taking values between 0 and 1. The more this sum tends to unity, the greater the persistence of shocks to volatility, which is known as volatility clustering or hysteresis hysteresis (hĭs'tərē`sĭs), phenomenon in which the response of a physical system to an external influence depends not only on the present magnitude of that influence but also on the previous history of the system. .

GARCH methodology is also instrumental in supporting or refusing the mixture of distribution hypothesis (MDH MDH Minnesota Department of Health
MDH Mälardalens Högskola (Swedish)
MDH Malate Dehydrogenase
MDH Manila Doctors' Hospital
MDH Carbondale, IL, USA - Southern Illinois Airport (Airport Code) 
). According to according to
prep.
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

3.
 the MDH, a serially correlated mixing variable measuring the rate at which information arrives to the market explains the GARCH effect in the returns. This linkage has been documented for the US stock market by Lamoureux and Lastrapes (1990), Andersen (1996), and Gallo and Pacini (2000), and for the UK stock market by Omran and McKenzie (2000). In general, the bulk of empirical studies has found evidence that the inclusion of trading volume in GARCH models for returns results in a decrease of the estimated persistence or even causes it to vanish. This finding, generally interpreted as empirical evidence in favour of the MDH (Sharma, Mougoue, and Kamath 1996; and Brailsford 1996). Thus, to investigate whether trading volume explains the GARCH effects for returns, GARCH (1,1) model with a volume parameter in the variance equation is estimated.

[h.sub.t] = [omega] + [m.summation over (i = 1)] [[alpha].sub.i][[epsilon].sup.2.sub.t-i] + [n.summation over (j = 1)] [[beta].sub.j] [h.sub.t-j] + [[gamma].sub.i][V.sub.t] + [[epsilon].sub.t] (3)

However, the results based upon GARCH (1,1) may again be doubtful because it does not account for asymmetry and non-linearity in the conditional variance. Thus it would be more appropriate to apply asymmetric GARCH model. Engle and Ng (1993) developed an asymmetric GARCH model, which allows for asymmetric shocks to volatility. Thus, among the specifications, which allow for asymmetric shocks to volatility, we estimate the EGARCH (1,1) or exponential GARCH (1,1) model, which was proposed by Nelson (1991) and results are reported and discussed in Section III on Emprirical Results.

[h.sub.t] = [[gamma].sub.1] + [[gamma].sub.2] + [absolute value of [[epsilon].sub.t-1]/ [h.sub.t - 1]] + [[gamma].sub.3] [[epsilon].sub.t-1]/[h.sub.t - 1]] + [[gamma].sub.4][h.sub.t-1] + [[gamma].sub.5][V.sub.t] + [e.sub.t] ... (4)

In this model specification [[gamma].sub.2] is the ARCH term that measures the effect of news about volatility from the previous period on current period volatility and [[gamma].sub.3] measures the leverage effect. Ideally [[gamma].sub.3] is expected to be negative implying that bad news has a bigger impact on volatility than good news of same magnitude. A positive [[gamma].sub.4] indicates volatility clustering implying that positive stock price changes are associated with further positive changes and vice versa VICE VERSA. On the contrary; on opposite sides. . The parameter [[gamma].sub.5] measures the impact of volume on volatility.

Further, in order to examine the dynamic relationship between variables, linear Granger causality test is applied with the help of E-Views software following the approach of Mestal et. al. (2003) and Otavio and Bernardus (2006). To test for Granger causality, we use a bi-variate VAR model of order p of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The null hypothesis null hypothesis,
n theoretical assumption that a given therapy will have results not statistically different from another treatment.

null hypothesis,
n
 of return not to have any effect of Granger causality on volume and vice versa implies that [[beta].sub.i] (i=1,.... p) are all equal to 0. To test the null hypothesis we calculate F-statistic as used in Mestal et al. (2003):

F = [SSE (1) An earlier full-screen editor in OS/2.

(2) (Streaming SIMD Extensions) A series of additional instructions built into Pentium CPU chips for improved multimedia performance by performing mathematical operations on multiple sets of data at the
.sub.0] - SSE/SSE x N - 2p - 1/p (7)

Where [SSE.sub.0] stands for the sum of squared residuals of the restricted regression (i.e. [[[beta].sub.i] = -[[beta].sub.p] = 0), SSE is the sum of squared residuals of the unrestricted equation, and N denotes the number of observations. Lag length for Granger causality has been determined on the basis of Schwartz criterion.

The series of stock return is computed from daily closing prices for the S&PCNX PCNX PC Annex (website)  NIFTY index for a period of more than five years from June 2000 till March 2006 (i.e. 1455 observations). This has been the period when derivative products were introduced in the Indian stock market. Introduction of futures trading has affected the movement of the index and volume trades in the market in different ways. So the current study attempts to evaluate the return-volume relationship after the introduction of futures trading. The daily stock returns are continuous rates of return, computed as log of ratio of present day's price to previous day's price (i.e. [R.sub.t] = ln ([P.sub.t]/[P.sub.t]-1)). Data are obtained from website of NSE NSE - Network Software Environment: a proprietary CASE framework from Sun Microsystems.  (www.nseindia.com).

The Empirical Results

The efficiency of stock market in general can be measured in terms of its liquidity and price discovery. The examination of relationship between return, return volatility, and volume provides significant information regarding the price discovery efficiency of the asset. Moreover, the market that provides price discovery will have high liquidity (Blume et al. 1994).

This paper begins the empirical analysis by first investigating the descriptive statistics descriptive statistics

see statistics.
 of volume, return, and volatility. Table 1 provides the sample descriptive statistics, which provides important information regarding the behaviour of variables over the period. Mean returns and average volume are higher in the post-futures period. The standard deviation in returns, which is indicative of the unconditional variance, has come down in this phase. Thus there is decline in the daily volatility in the market after the introduction of futures.

Further, the empirical distribution of the trading volume and return volatility series are positively skewed, indicating a right tail of distributions, which shows that they are asymmetrical. On the other side, negative skewness Skewness

A statistical term used to describe a situation's asymmetry in relation to a normal distribution.

Notes:
A positive skew describes a distribution favoring the right tail, whereas a negative skew describes a distribution favoring the left tail.
 is observed for return and magnitude of skewness has significantly increased, which has led the returns to be asymmetric and non-normal and it can be verified from p value of Jarque-Bera test In statistics, the Jarque-Bera test is a goodness-of-fit measure of departure from normality, based on the sample kurtosis and skewness. The test statistic JB is defined as

.

In addition, Table 1 documents that the coefficient of kurtosis Kurtosis

A statistical measure used to describe the distribution of observed data around the mean.

Notes:
Used generally in the statistical field, it describes trends in charts.
 for all variables are significantly greater than 3, which implies that distribution of the variables does not conform to normal distribution, which is the precondition for any market to be efficient in the weak form (Fama 1965; Stevenson and Bear 1970; Reddy 1997; and Kamath 1998).

Thus, in the light of information asymmetry as observed in descriptive ststistics, it will be an interesting venture to test whether contemporaneous relationship between return and volume exists using GARCH (1,1) model with a volume parameter in the mean equation and the results are reported in Table 2.

As reported in Table 2, coefficient of trading volume is positive and significant (i.e. there exists a positive contemporaneous relationship between trading volume and returns). Further, significant [[alpha].sub.i] and [[beta].sub.j] coefficients clearly indicate that conditional variance is predominantly affected by lagged variance, which implies that previous information shock significantly affects current returns. These evidences imply that Indian stock market is not efficient in weak form. Moreover, Table 2 shows that there is volatility clustering as measured by the sum of [[alpha].sub.i] and [[beta].sub.j] (0.902), which further supports the increase in asymmetry and inefficiency in market after the introduction of futures.

Further, to investigate whether trading volume explains the GARCH effects for returns, GARCH (1,1) model with a volume parameter in the variance equation is estimated and results are shown in Table 3.

The study finds parameters [[alpha].sub.i] and [[beta].sub.j] to be positive and significant in Table 3 where trading volume is included in the variance equation of GARCH model. The coefficient on the volume [[alpha].sub.i] is significant but indicates negative impact on volatility because of asymmetry, which is further checked through EGARCH model. Further, the study shows a decline in the persistence of volatility when trading volume is included in the variance equation, since the sum ([[alpha].sub.i] + [[beta].sub.j]) falls to 0.75 in the Table 3 as compared to the sum of [[alpha].sub.i] and [[beta].sub.j] (0.902) in Table 2 where volume is not included in the variance equation of GARCH model. It means that the degree of persistence is absorbed by the volume series. Therefore, our results for Indian stock market show weak support for the MDH model.

As significant asymmetry is observed in the returns of Nifty index, it would be more informative if we examine the volume--volatility relation through EGARCH (1,1) model to take into account impact of good and bad news on the volatility knowing the fact that both types of news have different kinds of effect on market. The results of EGARCH (1,1) are shown in Table 4.

The presence of leverage effect can be seen in Table 4, which implies that every price change responds asymmetrically to the positive and negative news in the market. A negative impact of lagged volume on volatility is observed. The parameter [[gamma].sub.2] is statistically significant, which supports the previous evidences of asymmetric distribution of returns in descriptive statistics and significant [[gamma].sub.3] indicates mean reverting behaviour of returns because the value of [[gamma].sub.3] is negative, which implies that every price change responds asymmetrically to the positive and negative news in the market. Coefficient [[gamma].sub.4] (which is a parameter of lagged conditional volatility) is significant which implies that Indian market is informationally inefficient. Coefficient [[gamma].sub.5] (which is a parameter of volume) shows a different picture of the role of trading volume on the volatility as compared to that in GARCH (1,1) model. It indicates the significant positive impact of volume on volatility. On the other side, impact of lagged volume on volatility is negative.

Further, in order to verify the robustness of relationship between trading volume and return volatility and to study the direction of information flow between these two, linear Granger causality test has been applied and results are presented in Table 5.

There is strong evidence of bi-directional causality (i.e. reject the null hypothesis of no Granger causality) between return and volume inconsistent with weak-form efficiency Weak-form efficiency

A pricing theory that the price of a security reflects the past price and trading history of the security. Theory implies that security prices follow a random walk. Related: Semistrong-form efficiency, strong-form efficiency.
. Hence, it is concluded that Nifty index may support the sequential arrival of information hypothesis over the MDH, and trading volume helps to predict return and vice versa. This finding is in agreement with Clark (1973) and Bessembinder and Seguin (1993). In addition, Table 5 illustrates that volatility contains information about upcoming trading volume as observed in Bhagat and Bhatia (1999) and Mestal et al. (2003). Preceding return volatility can be seen as some evidence that new information arrival might follow a sequential rather than a simultaneous process. This implies that the strong form of market efficiency does not hold since some private information exists that is not reflected in stock prices.

Conclusion

This paper examines the empirical relationship In science, an empirical relationship is one based solely on observation rather than theory. An empirical relationship requires only confirmatory data irrespective of theoretical basis.  between return, volume, and volatility using symmetric and asymmetric GARCH techniques and Granger causality test. The study provides evidence of positive impact of volume on return using GARCH (1,1) model. In addition GARCH (1,1) documents that the persistence of variance over time partly declines if one includes trading volume as a proxy for information arrivals in the equation of conditional volatility but GARCH effects remain significant, which highlights the inefficiency in the market. It also shows the negative impact of volume on conditional volatility because of asymmetry that is observed in significant Jarque-Bera. Next, in the light of Information asymmetry, the study has used the EGARCH (1,1) model, which allows for asymmetric shocks to volatility. It indicates the presence of leverage effect and positive impact of volume on volatility. The differential cost Noun 1. differential cost - the increase or decrease in costs as a result of one more or one less unit of output
incremental cost, marginal cost

monetary value, price, cost - the property of having material worth (often indicated by the amount of money
 of taking long and short positions is the main reason for information asymmetry (leverage effect). In addition, linear Granger causality results support the sequential arrival of information hypothesis, which implies that new information is not simultaneously available to all traders and it takes time to absorb, hampering the price discovery efficiency of the market.

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Notes:
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EMH Eastern Maine Healthcare
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Sarika Mahajan Mahajan is an Indian surname, found among the Vaishya castes (business communities). In India surname Mahajan is used by two communities: - one residing in North of India(mainly on the Amritsar to Jammu belt) and another belonging to North Maharashtra. , Junior Research Fellow, Department of Commerce and Business Management, Guru Nanak Dev University Guru Nanak Dev University, or G.N.D.U., was established at Amritsar, India on November 24, 1969 to commemorate Guru Nanak Dev's birth quincentenary celebrations. Introduction
It is a both residential and an affiliating university.
, Amritsar143005, India.

Balwinder Singh, Reader, Department of Commerce and Business Management, Guru Nanak Dev University, Amritsar-143005, India.
Table 1 : Descriptive Statistics

                Return     Volatility    Volume

Mean            0.000591    0.000195    2.23E+08
Median          0.001566    6.58E-05    1.99E+08
Std. Dev.       0.013940    0.000604    1.17E+08
Skewness       -0.980780    17.39004    0.763804
Kurtosis       10.835230    436.7367    3.016435
JarqueBera      3952.372    11470678    141.3928
Probability     0.000000    0.000000    0.000000

Table 2 : GARCH (1,1) estimates for Nifty returns with
volume in mean equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Volume--Return Relationship

                                    Coefficient   P-value

[gamma]                             8.43E-01      0.0027
[omega]                             1.77E-05      0.0000
[[alpha].sub.i]                     0.18017       0.0000
[[beta].sub.i]                      0.72152       0.0000
[[alpha].sub.i] + [[beta].sub.i]    0.90169         --

Table 3: GARCH (1,1) estimates for Nifty returns with
volume in variance equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Volume-Volatility Relationship

Parameter                          Coefficient   P-value

[omega]                             0.00019      0.0000
[[alpha].sub.i]                     0.15000      0.0000
[[beta].sub.i]                      0.60000      0.0000
[[gamma].sub.i]                    -3.25E-13     0.0000
[[alpha].sub.i] + [[beta].sub.i]    0.75000      --

Note: * [[gamma].sub.i], is a parameter of volume, which is included
in variance equation.

Table 4 : EGARCH (1,1) estimates with volume in
variance equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Volume-Volatility Relationship

Parameter         Coefficient   P-value

[[gamma].sub.1]   -1.469986     0.0000
[[gamma].sub.2]    0.258060     0.0000
[[gamma].sub.3]   -0.215193     0.0000
[[gamma].sub.4]    0.861218     0.0000
[[gamma].sub.5]    4.91E-09     0.0000
[[gamma].sub.6]   -4.73E-09     0.0000

Table 5: Linear Granger Causality Tests (Lags-5)

Null Hypothesis        F-Statistics   P-value

Returns does not       11.4185        8.00E-11
cause Volume

Volume does not        2.27167 *      0.04529
cause Return

Volatility does not    2.25728 **     0.04657
cause Volume

Volume does not        1.27678        0.27129
cause Volatility

Note: * and ** indicate significant at the level of 1% and 5%
respectively.
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Author:Mahajan, Sarika; Singh, Balwinder
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