If we a have a time series data for four variables and one of the independent variable is stationary at level and rest are stationary at first difference, in this case how can we check the cause effect relationship?
First you would have to transform all your variables on the same basis. In this case, it entails having all your variables transformed at least on a first difference basis. Given that there are time series, most such variables if you fully test them (ADF test) still have a unit root (meaning they are not entirely stationary). It would be better to transform them further and use for instance a % change from one period to the next as a final variable transformation.
To check the cause-and-effect thing you could then next conduct Granger Causality analysis between any two variables. I have shared a presentation on the subject at the attachment below that clearly spells out how to conduct such an analysis.
http://www.slideshare.net/gaetanlion/granger-causality-presentation
When you have one variable integrated of order zero and the three other variables are integrated of order one, in order to study the causal relationship between them, you should use the ARDL bounds testing to cointegration approach and the Granger causality test or the Toda Yamamoto test.
Respected Mounir Belloumi i have appled ARDL, but the longrun coefficients are insignificant
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