Causality in quantiles and dynamic stock return–volume relations

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Abstract

This paper investigates the causal relations between stock return and volume based on quantile regressions. We first define Granger non-causality in all quantiles and propose testing non-causality by a sup-Wald test. Such a test is consistent against any deviation from non-causality in distribution, as opposed to the existing tests that check only non-causality in certain moment. This test is readily extended to test non-causality in different quantile ranges. In the empirical studies of three major stock market indices, we find that the causal effects of volume on return are usually heterogeneous across quantiles and those of return on volume are more stable. In particular, the quantile causal effects of volume on return exhibit a spectrum of (symmetric) V-shape relations so that the dispersion of return distribution increases with lagged volume. This is an alternative evidence that volume has a positive effect on return volatility. Moreover, 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.

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 ifFyt(η|(Y,X)t-1)=Fyt(η|Yt-1),ηR,holds almost surely (a.s.), where Fyt(·|F) is the conditional distribution of yt, and (Y,X)t-1 is the information set generated by yi and xi up to time t-1. That is, Granger non-causality requires that the past information of x does not alter the conditional distribution of yt. 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 Qyt(τ|(Y,X)t-1) 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 yt-1,p=[yt-1,,yt-p],xt-1,q=[xt-1,,xt-q], and zt-1=[1,yt-1,p,xt-1,q], 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 rt=100×(ln(pt)-ln(pt-1)), where pt 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

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