Causality in quantiles and dynamic stock return–volume relations
Introduction
The relationship between financial asset return and trading volume, henceforth the return–volume relation, is important for understanding operational efficiency and information dynamics in asset markets. Models related to this topic include, e.g., the sequential information arrival model (Copeland, 1976, Jennings et al., 1981, Jennings and Barry, 1983) and mixture of distributions model (Clark, 1973, Epps and Epps, 1976, Tauchen and Pitts, 1983). There are also equilibrium models that emphasize the information content of volume, e.g., Harris and Raviv, 1993, Blume et al., 1994, Wang, 1994, Suominen, 2001. For instance, Blume et al., 1994) stress that volume carries information that is not contained in price statistics and hence is useful for interpreting the price (return) behavior. On the empirical side, there have been numerous studies on contemporaneous return–volume relation since Granger and Morgenstern, 1963, Ying, 1966; see Gallant et al. (1992) and also Karpoff (1987) for a review. Yet, as far as prediction and risk management are concerned, the dynamic (causal) relation between return and volume is more informative.
Causal relations between variables are typically examined by testing Granger non-causality. While Granger non-causality is defined in terms of conditional distribution, it is more common to test non-causality in conditional mean based on a linear model (Granger, 1969, Granger, 1980). Granger et al., 1986, Cheung and Ng, 1996 consider testing non-causality in conditional variance, whereas Hiemstra and Jones (1994) derive a test for nonlinear causal relations. These tests have been widely used in the literature (e.g., Fujihara and Mougoué, 1997, Silvapulle and Choi, 1999, Chen et al., 2001, Ciner, 2002, Lee and Rui, 2002). A serious limitation of this approach is that non-causality in mean (or in variance) need not carry over to other distribution characteristics or different parts of the distribution. Diks and Panchenko (2005) also give examples that the test of Hiemstra and Jones (1994) may not test Granger non-causality. These motivate us to consider characterizing and testing causality differently.
This paper investigates causal relations from the perspective of conditional quantiles. We first define Granger non-causality in a given quantile range and non-causality in all quantiles. The quantile causal effects are then estimated by means of quantile regressions (Koenker and Bassett, 1978, Koenker, 2005). The hypothesis of non-causality in all quantiles is tested by the sup-Wald test of Koenker and Machado (1999). This test checks significance of the entire parameter process in quantile regression models and hence is consistent against any deviation from non-causality in distribution, as opposed to the conventional tests of non-causality in a moment and the tests of Lee and Yang, 2006, Hong et al., forthcoming. The test of Koenker and Machado (1999) is easily extended to evaluate non-causality in different quantile ranges and enables us to identify the quantile range for which causality is relevant. Our approach thus provides a detailed description of the causal relations between return and volume.
In the empirical study we examine the causal relations between return and (log) volume in three stock market indices: New York Stock Exchange (NYSE), Standard & Poor 500 (S&P 500), and Financial Times-Stock Exchange 100 (FTSE 100). Despite that the conventional test may suggest no causality in mean, there are strong evidences of causality in quantiles in these indices. For NYSE and S&P 500, we find two-way Granger causality in quantiles between return and volumes; for FTSE 100, only volume Granger causes return in quantiles. In particular, the causal effects of volume on return are heterogeneous across quantiles, in the sense that they possess opposite signs at lower and upper quantiles and are stronger at more extreme quantiles. On the other hand, the causal effects of return on volume, if exist, are mainly negative and remain stable across quantiles.
With log volume on the vertical axis and return on the horizontal axis, the quantile causal effects of volume on return exhibit a spectrum of symmetric V-shape relations for NYSE and S&P 500.While many existing results (e.g., Karpoff, 1987) find a simple V-shape relation based on a least-squares regression of absolute return on volume, our V-shape results are very different. First, what we find are dynamic rather than contemporaneous relations. Second, these relations hold across quantiles rather than at the mean only. Moreover, the identified V spectrum suggests that distribution dispersion increases with lagged volume. This constitutes an alternative evidence that volume has a positive effect on return volatility and is compatible with the empirical finding based on conditional variance models (e.g., Lamoureux and Lastrapes, 1990, Gallant et al., 1992).
It is interesting to note that the quantile causal relations we find are quite robust to different sample periods and different model specifications. Indeed, the inclusion of the squares of lagged returns in the model may weaken the quantile causal effects of volume on return but does not affect the causality per se. Thus, lagged volumes carry information that is not contained in lagged returns and their squares, as argued by Blume et al. (1994). Our results also confirm that non-causality in mean bears no implication on non-causality in distribution (quantiles). A conventional test may find no causality in mean because the positive and negative quantile causal effects cancel out each other in least-squares estimation, as demonstrated in our study. It is therefore vulnerable to draw a conclusion on causality solely based on a test of non-causality in mean.
This paper is organized as follows. We introduce the notion of Granger (non-)causality in quantiles in Section 2 and discuss the sup-Wald test of non-causality in quantiles in Section 3. The empirical results of different causal models are presented in Section 4. Section 5 concludes the paper.
Section snippets
Causality in mean and quantiles
Following Granger, 1969, Granger, 1980, we say that the random variable x does not Granger cause the random variable y ifholds almost surely (a.s.), where is the conditional distribution of , and is the information set generated by and up to time . That is, Granger non-causality requires that the past information of x does not alter the conditional distribution of . The variable x is said to Granger cause y when (1) fails to hold.
Testing non-causality in quantiles
This paper proposes to verify causal relations by testing (3), rather than testing non-causality in a moment (mean or variance) or non-causality in a given quantile. To this end, we postulate a model for and estimate this model by the quantile regression method of Koenker and Bassett (1978); see Koenker (2005) for a comprehensive study of quantile regression.
Letting , and , we assume that the following model is
Empirical study
Our empirical study of return–volume relations focuses on three stock market indices: NYSE, S&P 500 and FTSE 100. The daily data from the beginning of 1990 (Jan. 2 or Jan. 4) to June 30, 2006 are taken from Datastream database, and there are 4135, 4161 and 4166 observations for NYSE, S&P 500 and FTSE 100, respectively. As will be shown in Section 4.4, our results are quite robust to different sample periods.
Returns are calculated as , where is index at time t; volumes
Concluding remarks
In this paper we estimate quantile causal effects and test Granger non-causality in different quantile ranges based on the quantile regressions of return (log volume). We find that there are quantile causal relations between return and log volume. More importantly, our results indicate that the causal relations may be far more complicated than what can be described using least-squares regression. Indeed, the causal effects may be heterogeneous across quantiles and that the causal effects at
Acknowledgements
We would like to thank a referee and the managing editor for very useful comments and suggestions. We also benefit from the comments by Zongwu Cai, Yongmiao Hong, Po-Hsuan Hsu, Mike McAleer, Essie Maasoumi, Shouyang Wang, and Arnold Zellner. This paper is part of the project “Advancement of Research on Econometric Methods and Applications” (AREMA) and was completed while C.-M. Kuan was visiting USC in the Spring of 2008. Kuan wishes to express his sincere gratitude to Cheng Hsiao and USC for
References (41)
- et al.
A causality-in-variance test and its application to financial market prices
Journal of Econometrics
(1996) Testing for causality: a personal viewpoint
Journal of Economic Dynamic and Control
(1980)- et al.
Volume and skewness in international equity markets
Journal of Banking and Finance
(2008) - et al.
The dynamic relationship between stock return and trading volume: domestic and cross-country evidence
Journal of Banking and Finance
(2002) - et al.
Testing the price-volume relation in emerging Asian stock markets
Journal of Asian Economics
(1995) - et al.
Testing for linear and nonlinear Granger causality in the stock price-volume relation: Korean evidence
Quarterly Review of Economics and Finance
(1999) - et al.
A consistent characteristic function-based test for conditional independence
Journal of Econometrics
(2007) Tests for parameter instability and structural change with unknown change point
Econometrica
(1993)- et al.
Market statistics and technical analysis: the role of volume
Journal of Finance XLIX
(1994) Recent advances in quantile regression models: a practical guideline for empirical research
Journal of Human Resources
(1998)
Trading volume and serial correlation in stock returns
Quarterly Journal of Economics
The dynamic relation between stock returns, trading volume and volatility
The Financial Review
The stock price-volume linkage on the Toronto stock exchange: before and after automation
Review of Quantitative Finance and Accounting
A subordinated stochastic process model with finite variance for speculative prices
Econometrica
A model of asset trading under the assumption of sequential information arrival
Journal of Finance
Crossing probabilities for a square root boundary by a Bessel process
Communications in Statistics – Theory and Methods
A note on the Hiemstra–Jones test for Granger causality
Studies in Nonlinear Dynamic & Econometrics
The stochastic dependence of security price changes and transaction volumes: implication for the mixture-of-distributions hypothesis
Econometrica
An examination of linear and nonlinear causal relationships between price variability and volume in petroleum futures markets
Journal of Futures Markets
Stock price and volume
The Review of Financial Studies
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