In most of the emprical papers using panel data, authors do not seem to "worry" too much aboout the non-stationarity of the individual variables.
Yes, there is asymptotic theory for N and T going to infinity. In my practice, both N and T may be in the range between 5 and 20. If so, what should be my highest concerns regarding the reliability of the results?
Would the following be "acceptable"? Testing for common unit root - if there is no indication for common unit root in the variables used in regressions, I would proceed with the original variables. If there is a common unit root - test for panel cointegration - if there is none, use first differences of the variables? Then, proper regression diagnostics would include what? Check for the presence of autocorrelation? What else? If autocorrelation is found, how to proceed? -> Using lagged values would imply I need to estimate the regression using dynamic panel method, would it not? Or is it "enough" to use some options for covariance adjustment (like period weights or something like that)?