When only half of the independent variables are non stationary, does it make sense to run a cointegration test? I have the following regression equation (panel data):
$Y = f(X_1, X_2, X_3, X_4)$

After obtaining CIPS and CADF statistics, $X_1$ results to be stationary for both intercept and intercept + trend, whereas $X_2$ is stationary only for intercept + trend. $X_3$ and $X_4$ are contrariwise non stationary in neither case.

Because of these results, I was wondering whether running a co-integration test makes sense (since if I'm not wrong, all the variables should be $I(1)$).

Also, I was wondering whether to proceed with using a first difference for $X_2$, $X_3$ and $X_4$, or only for $X_3$ and $X_4$.

Thank you very much.

Kodi
For cointegration analysis all variables should be of the same order of integration. However this being said it depends on how much you trust your stationarity tests. Most unit root tests are riddled with low power problems especially in small samples (see some textbook on econometrics like Stock &amp; Watson introductory econometrics for more details and sources).

Moreover, if the broader literature suggest that all the variables are let’s say I(1) I would even disregard the unit root tests results as they may be just false positives/negatives.

Also, if you want you can use the Pesaran bounds approach to cointegration which allows for both stationary and nonstationary variables being estimated within one model, although cointegration can only happen between stationary variables. This approach is usually not in textbooks yet so you might wanna check the original paper “Bounds testing approaches to the analysis of level relationships”