**3. Empirical Analysis**

First, we estimated the VAR granger causality<sup>8</sup> (both unconditional and conditional on index futures volume) for the purpose of comparisons with the results of causality estimated at the frequency domain. The results are presented in Table 2.


**Table 2.** VAR Granger causality.

\*\*\* indicates significance at 1% level. =<sup>&</sup>gt; refers to "does not granger cause".

No causality was observed from PCRTO to market return. PCROI Granger causes market return. However, this one-shot measure of GC may not hold across frequencies owing to nonlinearities of the data Hosoya (1991). This further justifies the application of Breitung and Candelon's (2006) methodology and the results are discussed in the following section.

Figure 1 presents the frequency domain causality from put–call ratio volume (PCRTO) and open interest (PCROI) to market return9. The blue solid line shows the Granger causality from PCRTO to market return, which is insignificant throughout at both 5% and 10% levels. That means volume PCR does not have predictive power of market return, which is against the popular belief of being a sentiment indicator Open interest put–call ratio (PCROI) significantly (at 5% level) Granger causes market return in long run at a frequency band 0.51 corresponding to 12 days and above. At the 10%

<sup>8</sup> We are thankful to the anonymous referees for this suggestion.

<sup>9</sup> The descriptive statistics of the F-statics of the frequency domain causality results are presented in Appendix A Table A1.

level of significance, it leads the market return between a frequency bands of 0.93–0.51 corresponding to 6–12 days. It implies that open interest PCR is a better predictor of market return than its volume counterpart in the long run. None of these ratios can predict in the short run.

Figures 2 and 3 present the rolling frequency domain causality from PCR volume and open interest to market return. Notably, short term causalities were estimated at frequency of 2.5 corresponding to 2–3 days, as presented in Figure 3, and long-term causality at frequency of 0.50 corresponding to 12–13 days is estimated, aas presented in Figure 2.

The long-term rolling causality in Figure 2 is consistent with the results reported in the in-sample analysis. The predictability of open interest PCR dominates its volume counterparts, as indicated by the dominant and significant peak of its frequency curve from June 2003 to June 2006 and from December 2010 to February 2012. One thing that stands out is that, during the 2008 financial crisis and after the 2012 European sovereign debt crisis, none of the indicators is significant in predicting the market return. Thus, the traders should carefully use these ratios during market crisis.

However, over a short period of 2.5 days, the reported rolling causality in Figure 3 volume PCR dominates its open interest counterpart, which is in stark contrast to the in-sample result. Another interesting thing that stands out from the figures is that, in the short term, volume PCR is a good predictor of market return. Moreover, it is a good predictor during the 2008 financial crisis, as evident from the higher amplitude volume PCR frequency curve, which is significant at 5% level. However, open interest PCR in the short run dominates for a brief period from April 2011 to November 201110. Our results supplement the findings of Pan and Poteshman (2006) and Blau et al. (2014) that the volume PCR is short lived and fleeting, respectively.

<sup>10</sup> We also estimated conditional frequency domain rolling causality analysis after controlling for the futures market activities. The results are quite similar and available upon request.

**Figure 1.** Full sample period frequency domain causality from volume put–call ratio (PCRTO) to market return is represented by blue solid line (FC1 TO) and the yellow solid line (FC1 OI) shows from open interest put–call ratio (PCROI) to market return. The frequencies (omega, ω) are on x-axis, and F-statistics testing the null hypothesis of no Granger causality are on y-axis. The horizontal red solid line and grey solid line indicate the 5% and 10% critical values, respectively.

**Figure 3.** Short-run (ω = 2.5 or 2.5 days) rolling window frequency domain causality. The x-axis represents the date and y-axis shows the F-statistics testing the null hypothesis of no Granger causality from volume Put–call ratio (FC1 TO) and open interest Put–call ratio (FC1 OI) to market return. The horizontal red solid line and grey solid line indicate the 5% and 10% critical values, respectively. The blue solid line (FC1 TO) and yellow solid line (FC1 OI) show short run causality from volume PCR and open interest put ratio, respectively, to market return.
