t-test for correlation Vs t-test for correlation in time series

[hacky]

Usually the t-test for correlation is:
t = r * ((n-2)/(1-r^2))^(1/2)
while when considering time series (autocorrelation of residual errors) is:
t = r / (1/^(1/2))
where
r –> correlation coefficient
n –> number of observations
Why?

 

[Harrogath]

I don’t know exactly why, but what I can see here is that those TESTS are testing different things.
The first one is for 2 series of data, proving if the correlation coefficient is statistically significant or statiscally different from zero. So that is the appropiate test for it.
In the second one you have 1 serie and the 2nd is the same serie but lagged 1 time, for example:
1 -
2 1
3 2
4 3
and so on.
So I think, as you are in a different scenario, the T-test must be adjusted for that. This formulas are statistically proved, and in some way we rely on them. If you want the fundamental of the formula, go to advanced statistics books.
Regards