Both fixed effect and first difference of some independent variables?

[Christine Green]

Hi, I have a short panel data and after doing unit root ht test, I found that some of the independent variables are nonstationary, therefore I change them into the first difference form. However, I also want to include a lagged term of one explanatory variable, which is stationary and therefore expressed in the normal form. I am using a panel data fixed effect model.

Therefore, my question is
1. does this kind of model make sense (theorectically there is support to add a lagged term)
Y_it= b_0 + b_1X_1it + b_2X_1i(t-1) + b_3D.X_2it + a_i +u_it

2. and if so, how should we interpret b_1, b_2 and b_3?

3. what if b_1 is significant and b_2 is not? however, when we drop the lagged term, b_1 becomes insignificant as well, why is that?

Thank you so much and have a nice day!

[Joro Kolev]

You do not need to worry about non-stationarity in short panels -- and I presume by a "short panel" you mean a short T panel. The asymptotics here are as N grows large, and the time series process within a panel is completely unrestricted.

You can take differences of variables just because you want to, the computer will not break. But this should be based on economic (as opposed to statistical) reasoning.