“…Thus, we conclude that, for NYSE and S&P 500, the quantile causal effects are mainly due to the tail quantiles outside the interquartile range (except that of ln v tÀ1 for NYSE). Our results are in contrast with many existing findings of non-causality that are based on a test for linear causality in mean (e.g., Kocagil and Shachmurove, 1998;Chen et al, 2001;Lee and Rui, 2002 Hutson et al (2008). The symmetry of these quantile causal effects helps to explain why the conventional methods, such as correlation coefficient and LS estimation, usually yield an insignificant estimate of the causal effect of volume, as the positive and negative effects at corresponding upper and lower quantiles tend to cancel out each other in ''averaging."…”