The casual link between stock returns and trading volume: Some evidence from an emerging market

Thispaper investigates the dynamic association between daily stock index returns andpercentage trading volume changes. Toproceed with this, linear and nonlinear Grangercausality tests are applied to the Karachi Stock Exchange (KSE) data. The analysis covers the span of about 5 years with 1266 daily observations. The same methodology is employed for two non-overlapping sub-periods to examine the robustnessof the results. Unidirectional linear Granger causality from stock returns to trading volume is observedfor the entire sample period and for both the sub-periods as well. The null hypothesisof linear Granger noncausality from percentage volume changes to stock returns is rejected only in optimal lag length for the second sub-period. Regarding nonlinear Granger causality, the modified Baek and Brock's test (l992a) for nonlinear Granger causality provides evidence of significant unidirectional nonlinear Granger causality from percentage volume changes to stock returns in both the sub-periods for all the common lag lengths used but not for vice versa. The analysis exposed that volume has significant nonlinear explanatory power for stock returns,whereas stock returns have linear explanatory power for trading volume. JEL Classification Code: G12; C14


INTRODUCTION
M ostof the empirical work on stock market has focused traditionally on whether future stock prices moments can be projected or not. Through a brief review of the literature, it is found two different groups of studies in this area. The first one considers those studies that tested the Market Efficient Hypothesis (MEH) or/and Random Walk Hypothesis (RWH) for stock markets. A large number of studies including Larson (1960), Alexander (1961Alexander ( , 1964, Granger and Morgenstern (1963), Mandelbort (1963) and Fama (1963Fama ( , 1970 have claimed that stock markets are weak form efficient i.e., past price movements could not be used to predict future prices changes. In contrast, some studies have been rejected the Random Walk Business Review -Volume 2 Number 2 July -December 2007 Hypothesis for stock prices (e.g., Fama (1965), Lo and Mackinlay (1988), Husain (1997), Rashid (2005Rashid ( , 2006».
The second mass of studies on stock market has focused primarily on to identify the financial and socio-economic factors that have significant associations with stock prices. These studies have generally used OLS regression analysis, co integration and causality techniques to explore the relationship between stock prices and the other variables such as profits, book-to-market value, term premium, dividend yields, exchange rates, interest rates, inflation, money supply, output growth, etc.
One common problem with this sort of studies is that stock prices are fluctuated more frequently, while the financial and economic variables are relatively less volatile. Moreover, the factors/news that causes changes in stock prices may be different from the factors that are responsible for the changes in these variables. Under this scenario, these variables may not be able to reflect the available set of relevant information on the stock market.
Trading volume, however, is another most frequent variable like stock prices and perhaps both the fmancial variables are affected by the same sort of risk. Stock prices and trading volume are two prime indicators of stock market performance and jointly determined by the same market fluctuations. As with stock prices, aggregate trading volume mainly reflects investors' expectations on the future performance. As mentioned earlier, most of the empirical studies in existing literature have focused on only stock prices/returns for forecasting with a very little attention on trading volume. The study by Blume et al. (1994) has highlighted that trading volume captures the important information about speculators' trading activities and hence, movements in trading volume may be useful in forecasting of stock price dynamics. Another study by Gallant et al. (1992) claimed that more can be learned about the stock market behavior through studying the jointly dynamics of stock prices and trading volume rather than by focusing only on one of them.
The empirical work on the stock price-volume relation focused on the study of causality between stock prices and trading volume l . Causality tests can provide useful information on whether knowledge of past stock prices (trading volume) movements improves short-run forecasts of current and future movements in trading volume (stock prices). The traditional Granger causality test is only useful to examine the linear relation; however, it is unable to explore the nonlinear relationship between two variables (for details, see Baek and Brock (1992a) and Hiemstra and Jones (1993». Business Review -Volume 2 Number 2

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Explanations for nonlinearities between stock prices and trading volume is defined as follows. As said by Granger (1986), univariate and multivariate nonlinear models represent the proper way to model a real world that is "almost certainly nonlinear." Hsieh (1991) and Brock (1993) highlighted that the recent focus on nonlinear structure in stock price movements is motivated by the richer types of asset behavior that nonlinear models provide researchers. Large stock price swings and abrupt changes in the stock market volatility can be properly modeled with nonlinear models". Lellaron (1992) used regression models similar to the one used by Campbell et al. (1993) and documented evidence of significant nonlinear interactions between stock returns and trading volume in the New York Stock Exchange. It is now widely accepted that nonlinearity is present in stock price-trading volume associations. The study thus used both linear and nonlinear causality tests to investigate the dynamic interactions between stock prices and trading volume. The nonlinear Granger causality test used here is based on nonparametric estimators of temporal relations within and across time series. It is modified version of Baek and Brock's (1992a) nonlinear Granger causality test.
The core purpose of this study is to examine whether the information on trading volume can be used to predict the changes in stock prices. This analysis has very useful implications for both hedgers and speculators. Most hedgers do not employ continuous hedges. Instead, they select the times at which they want protection against adverse changes. Defmitely, knowledge of stock price-trading volume association can be used for forecasting future movement of stock prices effectively.
The remainder of the article is organized as follows. Section 2 will discuss the theoretical linkages between stock prices and trading volume. This section will also discuss how volume and price move together. Section 3 will briefly review some recent empirical work in this field. Section 4 will present the notation of Granger causality and traditional linear tests in brief. This section will also discuss Baek and Brock's (1992a) nonparametric approach to nonlinear Granger causality testing and the modified version of their test, which is applied here. Estimations and interpretations of the results will follow in Section 5. Finally, Section 6 will conclude witha summary of this document.

THEORETICAL DISCUSSION
The theoretical studies that provided explanations for the presence of a causal link between stock prices and trading volume can be divided into four categories. The Business Review -Volume 2 Number 2 July -December 2007 studies in the first category include Copeland (1976) and Jennings et a1. (1981) that developed the sequential information arrival models. According to these asymmetric models, there is a positive relationship between stock prices and trading volume and causality runs in either direction. Owing to the sequence information flow, lagged trading volume could have predictive power for current stock prices, and vice versa.
The second category includes Lakonishok and Smidt's study (1989) that explained the relation between current trading volume and past stock prices via tax-and nontax-related motives. Their analysis confirmed that the dynamic relation is negative for tax-related trading motives and this relation turns into positive for certain nontax-related trading motives.
The third group of theoretical studies consists of the mixture of distributions models of Clark (1973) and Epps and Epps (1976). These models suggest a positive relation between current stock returns variance and trading volume in different ways. For example, the mixture model of Epps and Epps (1976) used trading volume as a proxy to measure disagreement as traders revise their reservation prices based on the arrival of new information into the market. Their models thus suggested a positive causal relation running from trading volume to absolute stock returns. In contrast, the mixture model of Clark's (1973) used trading volume as a proxy for the speed of information flow and unable to fmd any significant causal link between trading volume and stock prices.
Noise trader models are included in fourth category. These models stated that aggregate stock returns are positively autocorrelated in the short run, but negative in the long run. A positive causal relation from trading volume to stock prices is based on the assumption made in these models that the trading volume strategies pursued by noise traders cause stock prices to move. On the other hand, a positive causal link from stock returns to volume is consistent with the positive-feedback trading strategies of noise traders, for which the decision to trade is conditioned on the past stock price movements (for details, see Delong et a1. (1990)).

LITERATURE REVIEW
In 1959, Osborne, first time, hypothesized that secunties prices could be as a lognormal distribution with the variance term dependent on the trading volume. He concluded that the log return process was a Brownian motion process.
Seven years later, another study by Ying (1966) applied a series of statistical tests to a six-year daily series of price and volume. Prices were adjusted by quarterly dividend and similarly trading volume was normalized by the number of shares outstanding to avoid any biases from issues from larger number of outstanding shares. Using the New York Stock Exchange (NYSE) daily percentage volume and Standard and Poor's 500 index returns data covering the period from January 1957 to 70 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 December 1962, he found that a large increase in volume is usually accompanied by a large price change.
He also concluded that if the stock trading volume has decreased (increased) five consecutive trading days, the price will tend to fall (rise) over the next four days. These two studies illustrate the origins of the analysis of stock price-trading volume associations. However, below the study reviews and summarizes some recent studies in this field.
The study by Ganllant et al. (1992) applied nonlinear impulse response functions to examine the nonlinear associations between stock returns and trading volume on the New York Stock Exchange (NYSE). Their study provided evidence that stock returns are playing leading role. They concluded that stock returns might successfully be used to forecast the future movements in trading volume. Hiemstra and Jones (1994) used the traditional linear Granger causality tests along with the modified version of Baek and Brock's (1992a) nonlinear Granger causality test to expose the causal relations between stock prices and trading volume. The study used daily closing prices for Dow Jones Price Index and trading volume on the New York Stock Exchange (NYSE). The sample period is 1915 to 1990. They found evidence of significant unidirectional linear Granger causality from stock returns to percentage volume changes and bidirectional nonlinear Granger causality between stock returns and percentage changes in trading volume. Blume et al. (1994) studied the informational content of trading volume. Rather than describing the simple correlation between stock prices and trading volume, they set out to show how trading volume could affect market behavior. They found that trading volume provides information on the precision and dispersion of information signals, which stock prices alone do not. They suggested the use of trading volume as an additional statistic to observe the stock market movements. Ghysels et al. (2000) investigated the causality between the series of returns and transaction volumes in high frequency data. Their data set consists of 20405 observations on returns and volumes of trades of the Alcatel stock, recorded on the Paris Stock Exchange. The study covers the period from July to August 1996. They found co-movements between volumes and transaction prices and proposed tests for Granger causality. Using standard Granger causality test, they reported that there is a causal relation between stock returns and volume. However, their empirical results indicate that causality directions vary in tinie and depend on the sampling scheme. Chordia and Swaminathan (2000) studied the interaction between trading volume and predictability of short-term stock returns. They found that daily returns of stocks with high trading volume lead daily returns of stocks with low trading volume. They concluded that "trading volume plays a significant role in the dissemination of market wide information". 71 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 Business Review -Volume 2 Number 2 July -December 2007 Chen et al. (2001) used linear Granger causality test to examine the dynamic relation between returns, volume and volatility of stock indexes. The period of the study is 1973 to 2000. Using data for nine national markets viz. New York, Tokyo, London, Paris, Toronto, Milan, Zurich, Amsterdam, and Hong Kong, they found a significant positive correlation between trading volume and the absolute value of the stock prices. Regarding Granger causality, their results provide evidence that for some countries, returns cause volume and volume causes returns. As stated by them, the results are robust across the nine national markets. Fan et al. (2003) examined the relationship between trading volume and stock returns for two Chinese A-share markets and ten individual stocks in the energy sector. Their study also investigated the effects of exogenous government policies on the relationship between trading volume and stock return. Using daily data covering the period from January 1, 1997 to December 31, 2002, they found that the relationship between trading volume and return is asymmetrically V-shaped with the response of trading volume to a rising return being stronger than that to a falling return. Based on Granger causality test, they reported a strong evidence of return causing volume; volume has only a weak effect on future returns but a strong and predictable effect on absolute returns.

METHODOLOGY FRAMEWORK
This section first presents the definition of Granger causality and then briefly outlines the basic approach to explore the linear Granger causality. Finally, this section provides a detailed discussion of the modified Baek and Brock's (1992a) test as proposed by Hiemstra and Jones (1994) for testing nonlinear Granger causal relations, which cannot be detected by the traditional linear causality tests.
The Granger (1969) definition for causality of two stationary time series Zt and R, is defmed in general form as follows. Let P, (Z, I It-!) be the conditional probability distribution of Zt given the bivariate information set 1 t -1 consisting of a k -length lagged vector of Z, (say Zt~k ), and n -length lagged vector of R, ' say R/'_n. Given lags k and n , the time series t R, } does not strictly Granger where, Pr(.) denotes conditional probability, where k and n are lag orders, and S~and TV, are stock prices and trading volume, respectively. The regression residuals, {u,} and {v, }, are assumed to be mutually independent and individually ii.d, with zero mean and constant variance.
To test for strict Granger causality for all possible pairs of {S~, TV t } in this linear framework, a standard joint test (For X 2 -test) is used to determine whether lagged value of one time series (say S~) has significant linear predictive power for current value of another series (say TV t ). The null hypothesis that TV/ does not 73 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 s~= fJT~-n x S~_K + n, Baek and Brock (1992a) therefore proposed a nonparametric statistical method for uncovering nonlinear causal associations. They suggested the use of correlation integral, an estimator for spatial probabilities across time, to explore causal relationship between two time series. Below briefly the Baek and Brock's (1992a) procedure to testing for non-linear Granger causality is discussed.

THE BAEK AND BROCK APPROACH TO CAUSALITY TESTING
Let consider S~and T~are two time series that assumed strictly stationary and weakly dependent. Denote the m-length lead vector of S~by s~m and the klength and n-length lag vectors of S~and TV t , respectively, by S~~k and TV t : n ,

IIS~~k -SPS~kII< e]
, and the pair of n-histories of T~differ by less than e 2 ' i.e., IITV/: n -TVs~n II< e 2 · The probability on the right-hand side of equation {6} is the conditional probability that two arbitrary m-Iength lead vectors of S~are within a distance e l of each other, given that their corresponding k-Iength lag vectors of S~are at a distance smaller than e] of each other.
As stated by Baek and Brock (1992a), this defmition depends on the number of lags of future prediction which is one for this study (then, S~] = S~), the e], e 2 ; the number ofk-Iength and n-length lags vectors of S~and T~, respectively.
To test the null hypothesis Ho: T~does not nonlinearly Granger cause S~, it is useful to express the conditional probabilities in terms of the corresponding ratios of : The max~~um norm fo~~= (z., Z2, .. :'~k) E Rk is defined as max(Z), i = 1,2, ... , k.
By definition, the conditional probability Pr(AIB) can be expressed as the ratio Pr(AnB)/Pr(B). 75 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 Business Review -Volume 2 Number 2 July -December 2007 C1 (m + k, n; e) = Pr~IS~~:k -SPs:;k II < e1,IITVt~n -TVs~n II < e2 } C2 (k, n; e) = Pr~IS~~k -SPs~k II < e1, IITVt~n-iv; II < e2} C 3 (m + k; e) = Pr{IS~~:k -SPs:;k II < e 1 } C4 (k; e) = Pr~IS~~k -SPS~kII < e1 } For given values of m, k, and n e 1 1 and e > 0, the null hypothesis H 0 can then be expressed as: Let I(Zi> Zz., e) denote a kernel that equals 1 when two conformable vectors Z, and Zz. are within the maximum-norm distance e of each other and 0 otherwise. Correlation-integral estimators of the joint probabilities in equation {7} can then be written as: .JP (C1(m+k,n, Using t-test, it is found that the linear time trend is not statistical significant in both regressions, therefore, regressions are estimated without linear trend. To determine the number of augmentation terms, the study uses .a general-to-specify datadependent procedure in Hall (1993). In this procedure, initially the maximum lag order based on the nature of data is used to estimate the regression and then a tor Ftest is used to eliminate insignificant augmentation terms. https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 Index (except the full-length period) and stock market trading volume change have positive skewness, and significant excess kurtosis is observed for stock returns. Based on the estimated ADF test statistics, the unit root null hypotheses are rejected at the 99 percent confidence level in all cases. It thus can be confirmed that the stock returns and the percentage change in trading volume are stationary at their levels.
A two-step procedure similar to one used in Gallant, Rossi, and Tauchen (1992) is used to remove systematic day-of-the-week and month-of-the-year calendar effects from stock returns and percentage volume changes. Both the mean and variance of the stock returns and trading volume series are adjusted for both of these effects. For the volume series the two-step adjustment procedure involves estimating the following regression equations: Mean Equation: Variance Equation: In Significant positive correlations observed between stock returns and percentage volume changes at level and on average in lead-lengths for all the examined periods. These findings are providing some evidence of the associations between stock returns and trading volume: a theme that is explored by applying linear and nonlinear Granger causality tests.

LINEAR GRANGER CAUSALITY TEST RESULTS
As mentioned earlier, the results of the ADF test provided evidence that returns in KSE-IOO Index and percentage volume changes on the Karachi Stock Exchange are stationary at their level for the entire sample period and for the two sub-periods either. Thus, to look at the nonlinear Granger causality, the study estimated the VAR models specified in equations {3} and {4} with calendar-adjusted stock returns and percentage volume change. The models can be expressed as follows: 80 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 BusinessReview -Volume 2 Number 2

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TheAkaike's (1974) information criterion (AlC) is used to determine the appropriate laglengths for VAR models.  Bold lag lengths are optimal that set with the Akaike (1974) information criterion. Theresults of Granger causality are very sensitive to lag order. The study therefore usedsome other lag intervals besides the optimal lag lengths to check the robustness of the results. Focusing on the rejections of the null hypothesis of Granger noncausality at the five percent nominal significance level, the Granger test is able to reject the null hypothesis that stock returns do not cause volume changes for the entire sample period from 1 January 2001 to 31 Mach 2006 and for both the nonoverlapping sub-periods alike. It means that there is unidirectional Granger causality fromstock returns to percentage volume changes. This piece of evidence is robust to otherexamined lag lengths.
Onthe other hand, the null hypothesis of Granger noncausality from volume changes to stock returns cannot be rejected at 5 percent significance in the full-length period . and in either sub-period. The null can be rejected, however, at 8 percent significance level in only optimal lag length selected by AlC for the second sub-period from 1 January 2003 to 31 Mach 2006. Overall, the null hypothesis of Granger noncausality 81 https://ir.iba.edu.pk/businessreview/vol2/iss2/5 DOI: https://doi.org/10. 54784/1990-6587.1390 Published by iRepository, June 2021 Business Review -Volume 2 Number 2 July -December 2007 from stock returns to percentage volume changes is rejected and from percentage volume changes to stock returns is accepted. Thus, it can be concluded that there is a unidirectional linear Granger causality from stock returns to percentage volume changes.

NONLINEAR GRANGER CAUSALITY TEST RESULTS
Using the traditional linear Granger causality tests, it is found that there is unidirectional linear causality from stock returns to change in trading volume. As reported earlier, the traditional Granger causality test is not able to explore the nonlinear relations; the study therefore uses the modified version of Baek and Brock's (1992a) nonlinear Granger causality test to expose the nonlinear interactions between stock returns and percentage volume changes. The study follows Hiemstra and Jones (1994) in order to select values for the lead lengthm, the lag length k and n, and the scale parameter e. Hence, for all cases, the lead length ofm = 1, common lag lengths of 1 to 8 lags and a common scale parameter of e = 1.5aare used to apply the test on standardized series".  Table 4. From the table, it can be seen some fascinating evidence. There is evidence of unidirectional nonlinear Granger causality between stock returns and the percentage trading volume changes that runs from volume to stock returns for both the examined sub-periods. However, the analysis is not able to find any reverse causation for these two variables. * significant at the 1 percent level of significance for a one-sided test. ** significant at the 10 percent level of significance for a one-sided test.
This piece of evidence is robust for about all the common lag lengths used in conducting the test. For the entire sample period, there is rio significant evidence of nonlinear Granger causality between stock returns and trading volume in either direction, except lag length 1 and 4. At these lag lengths; .the modified Baek and Brock's test rejects the null hypothesis of nonlinear Granger noncausality from tradingvolume to stock returns.
It is interesting to compare the results of the modified Baek and Brock's (1992a) test for nonlinear Granger causality with linear Granger causality test results. Note in Table 3 that the test statistics only provide strong evidence of significant linear unidirectional causality from stock returns to percentage volume changes. Quite the opposite,