How to test if time series are not cointegrated? Is there a test whose null hypothesis is that time series are cointegrated? There are lots of tests whose null is no cointegration. How can I calculate a p-value for the null that two series are cointegrated? I assume a p-value of 0.95 for a null of no-cointegration is not the same as a p-value of 0.05 for a null of cointegration.
 A: The null of cointegration vs. no cointegration can be done in the same way that you have test of stationarity vs. unit roots (KPSS) instead of unit root against stationarity (ADF, PP, etc).
One of the main approach taken is residual based approach: estimating a simple long-run relationship, extract the residual and:


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*test of no cointegration: apply a unit root test on the residuals, this is for example the Philips-Ouliaris test, that applies the Philips-Perron test

*test of cointegration: apply a stationarity test on the residuals. This can be the KPSS test (Shin 1994), or even CUSUM test (Xiao and Philips 2002). 


Now for your question of how to contrast the two results... good question! I have no clear theoretical answer on this, but just to mention that empirically given the low power of unit root tests, and big size issues of stationarity tests, the two approaches most likely will differ in practice. 
Refs:


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*Shin, Y., 1994. A residual based test of the null of cointegration against the alternative of no cointegration. Econometric Theory 10, 91–115.

*Xiao, Z., Phillips, P.C.B., 2002. A CUSUM test for cointegration using regression residuals. Journal of Econometrics 108, 43–61

