# Pearson correlation significance : Issue with $t$-statistic increasing with $N$

I have two assets which seem not correlated (correlation coefficient = 6.3% using monthly frequency and 48 data points).

I want to test the significance of the correlation. Null hypothesis is that correlation is nil, and if the p-value is lower than 0.05 then we can reject the null hypothesis (ie correlation is significantly different than 0, ie there is correlation).

Keeping the correlation constant (at 6.3%) I notice that as I increase $$N$$ (and so I increase $$N-2$$ degrees of freedom) the p-value reduces and will eventually be lower than 0.05.

I am confused, why having more $$N$$ make the correlation becoming statistically significant, if the actual correlation of returns is still low?

• Have you searched on CrossValidated? I imagine they have an answer on this already. – Bob Jansen Sep 4 '20 at 13:58

We can test the hypothesis that the correlation is 0 just as easily as testing that the correlation is 1 or -1. One way to do that is to use Fisher's $$z$$-transformation; for an estimated correlation $$\hat\rho$$, the test statistic $$z$$ is given by: $$z = \frac{1}{2}\log\left(\frac{1+\hat\rho}{1-\hat\rho}\right)$$ and $$z$$ is approximately normal with standard error $$\frac{1}{\sqrt{N-3}}$$.
• A $p$-value below 5% does not say we are 95% sure we can reject the null hypothesis nor that we can accept our estimated value as the correlation. It means that if we had randomly sampled data from the data distribution, the confidence interval would include the true value 95% of the time. This might seem like a subtle point, so I would suggest reading up on the meaning of $p$-values -- and maybe on permutation tests. – kurtosis Sep 7 '20 at 19:43