# Table 5 Granger Causality Tests on PMG and PML. Decomposition with High Price Extremes

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Panel A. Monthly Data Observations

Panel B. Quarterly Data Observations

Lags

2

4

6

2

4

6

PMG /â†’ PML

0.009

0.712

0.878

0.036

0.049

0.107

PML /â†’ PMG

0.000

0.000

0.000

0.000

0.000

0.000

P MGF /â†’ P MLF

0.493

0.794

0.869

0.014

0.007

0.029

P MLF /â†’ P MGF

0.000

0.000

0.000

0.000

0.000

0.000

1. Note. X /â†’ Y means the null hypothesis that X does not Granger-causes Y. P MGF and P MLF mean filtered PMG and PML respectively. This table reports the p-values of the F -statistics. When performing Granger causality test, we set m = n in Equation (5) for the sake of being convenient. Di erent lags are used for being robust since the Granger causality tests are sensitive to the lag selection. Panels A and B report respectively the results for monthly and quarterly data observations. Decomposing stock returns with high price extremes are presented as
2. $${\displaystyle \begin{array}{l}{r}_t=\kern0.5em \left[\mathit{\log}\left({H}_t\right)\hbox{-} \mathit{\log}\left({O}_t\right)\right]\hbox{-} \left[\mathit{\log}\left({H}_t\right)\hbox{-} \mathit{\log}\left({C}_t\right)\right]\\ {}\kern1em =P\ {MG}_t\hbox{-} P\ {ML}_t\end{array}}$$