The three main methods for testing for cointegration are Phillips–Ouliaris cointegration test, Johansen test, and Engle-Granger two-step method.

However, the Johansen test is subject to asymptotic properties (large samples). The results will not be reliable if the sample size is too small and in a situation like that, Auto-Regressive Distributed Lags (ARDL) should be used.

See this for more insight;

With correlation, we typically calculate four kinds of correlations in statistics: Pearson correlation, Kendall rank correlation, Spearman correlation, and the Point-Biserial correlation. However, the first 3 are the commonly used ones, Point-Biserial correlation comes in when one variable is dichotomous.

Hope this is helpful, Best regards.

I agree with Caleb Danquah .

As matter of fact, to delve into this topic you can first read the wikepedia page about it and then check the references included there:

https://en.wikipedia.org/wiki/Cointegration

Regards,

Ángel

Doe cointegration you should use Johanssen Test

Johansen cointegration approach can be adopted when all the variables are integrated of the same order whereas ARDL (Bounds Test) approach can be used when the variables have mixed order of integration (i.e. I(0) and I(1) and not I(2).

The bound test within the ARDL framework can be used. Thank you.

Cointegraton techniques are used to test for longrun equilibrium relatioship among series. Appropriate test to use depends on the levels of integration of the series. Assuming we have two series Y and X, if the two are integrated of oder zero, that is stationary at their levels when unit root is conducted, OLS is the appropriate method of analysis as there is no need for short run adjustments. However if the series are integrated of the same order 1, Johansen Cointegraton technique is appropriate, with the ECM to determine the shortrun adjustments. If the series are of mixed integration but not I(2), the appropriate cointegraton test is the ARDL Bound Test.