# More null rejections than expected at confidence level. Multiple testing

The answer to this question has probably been answered multiple times but I'm lacking the right keywords to find the answer.

I've tested on 24 time series Granger-causality from one series to the other (23 VARs, after checking stationarity and by using AIC minimization). I reject the null hypothesis of no Granger-causality at the 5% level on 4 series out of 23. My understanding is that: I should be finding 5%*23=1 false positive. I am finding 4 times that. Can I conclude there is evidence of Granger-causality? On what? From the series to the whole set?

Edit: to be very clear, I am interested in testing series 1 Granger-causes all other series. So I made 23 VARs of series 1 and each other series:

series 1 and series 2

series 1 and series 3

... Can I conclude there is statistical evidence series 1 Granger-causes the set?

Your null hypothesis is that there is no Gragner-causality. You are finding that you reject the hypothesis on 4 of the 23 series. Note that the probability of rejecting, given the null hypothesis is true, is $$0.05$$. So rejecting 4 or more (incorporating more extreme events) out of the 23 series has probability becomes the value of the binomial distribution with $$n=23$$ and $$p=0.05$$: $$P(N_{rej}\geq 4) = 1- P(N_{rej} \leq 3) = 1- F_{Bin}(3)=1-\sum_{i=0}^3 \left(\begin{array}{c} 23 \\ i \end{array} \right) 0.05^i (1-0.05)^{23-i}$$ This is the probability of this event or a more extreme occurring. Depending on your significance threshold we either reject or do not reject the null hypothesis here.
Assuming you set your confidence level to 5\% we get $$0.025815$$ which means we reject the null hypothesis.