# 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 at 13:58

## 1 Answer

The question of significance is not about the correlation but about the precision of the estimation. If the value estimated with more data is still near the same value estimated with less data, that means you are more sure of that correlation being close to your estimate.

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 $$\frac{1}{\sqrt{N-3}}$$.

Unsurprisingly, you can note here that the standard error decreases as we have more data. If the data were just noisier, more data would give us a correlation estimate converging to 0. However, if the correlation were low but not zero, more data would reveal that.

• I would also add that it's not quite feasible to assume a normal distribution (as is done for t-tests) for such a low number of observations. However, increasing N leads to anasymptotic normality of your test-statistic (the z-value) by the central limit theorem, wich can explain why the estimation is becoming more precise. – Martin Georg Haas Sep 4 at 17:00
• True. For a low number of observations, any distributional assumption is likely to be off. No point going into asymptotic expansions here though.... :-) – kurtosis Sep 4 at 20:55
• Does this mean it is incorrect to say that if the p-value is below 5% we are 95% sure that we can reject the null hypothesis of no correlation? Would it be more correct to say: I cannot say that my result (which is: low correlation of 6.5%) is statistically significant because the p-value is above 5%? – tweedi Sep 5 at 12:57
• How have you increased N? did you interpolate to increase the samples? How have you kept the correlation constant while increasing the samples? – Luigi87 Sep 7 at 6:45
• 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 at 19:43