Regression with non-cointegrated I(1) series I have 4 variables for a panel of N metropolitan areas over T years:


*

*productivity

*number of patent applications

*population

*share of graduates in the labor force


I wanted to regress log(productivity) over log(patents), log(population) and share of graduates, along with time & metropolitan area fixed effects.
Of course, since all these variables are I(1) I thought about doing an ECM. However, when applying cointegration tests for panel data on these 4 variables (xtwest in Stata), the null of no-cointegration could not be rejected.
What should be the proper procedure in that case ? 
Thank you
 A: If you find that there is no cointegration, your model is likely to be misspecified. This is probably because you are omitting very important variables, which are correlated with other regressors, and that are essential in the long run cointegration relationship. Remember that what cointegration looks for is the set of variables that jointly define a long-run, stable relationship. 
My suggestion is to look for more variables, until you can find a cointegration relationship. The main reference for doing this is, nothing else than the theory. If it happens that you cannot find a cointegration relationship, that is also interesting (publication bias anyone?). You might look to argue why that is the case.
Then, you could follow Richard's suggestion and run the model in differences, but the information you can get is, in my experience, very limited, and you still can have misspecification problems that tests cannot highlight. That is the beauty behind a structural model and the battery of associated tests allowed by, for example, error correction models.  
Finally, if you need to defend your results against serious people (not because of them but because you want a reliable theory), in my experience models in differences are insufficient, specially in modern macroeconomics, where everything is centered about cointegration.
