Panel Regression with Region-Specific Dummy. Should I interact the dummy with all control variables?

[Alex Li]

I am investigating the effects of FDI on Economic Growth in a specific region and am running panel regression using data for every country available in the world with a dummy for that specific region, i.e.

(a) Y_it = FDI_it + X_it + Dummy*FDI_it, where X_it are control variables.

Should I also interact my Dummy with the control variables as well (model (b) or should I just use (a)?

(b) Y_it = FDI_it + X_it + Dummy*FDI_it + Dummy*X_it

Hope someone would be able to enlighten me on this issue. Thank you very much for your help!

[Clyde Schechter]

Well, my first response is that both of these models are wrong. It is a mis-specification to have an interaction term in a model without also including both of its constituents. So both models are wrong because Dummy does not appear as a variable in its own right. Now, if you are going to use a fixed-effects regression and if Dummy designates some subset of the levels of the fixed effects variable (e.g. designates one country and the fixed effects variable is country), then that variable is colinear with the fixed effects and will be dropped anyway, so no problem. But otherwise, it needs to be there.

That said, the answer is that it depends. The two models apply to different situations and only you know which one is appropriate for you.

Model a says that the effect of FDI is different when Dummy = 1 than it is when Dummy = 0, but the effects of the X variables are the same, irrespective of the value of dummy. Model b says that not only is the effect of FDI different when Dummy = 1, but the effects of all the X variables are also different when Dummy = 1.

Another way to think about it is, that model b is completely equivalent to doing two separate regressions, one for the Dummy = 1 observations and another for the Dummy = 2 observations. But model a is different: it is equivalent to doing two separate regressions but imposing the constraint that the coefficients of all the X variables must be the same in both regressions.

So that's the statistics underlying the models, and provides the basis for choosing between them. Whether the effects of X are expected to differ between Dummy = 1 and Dummy = 2 is a factual situation, not a statistical question. It depends on what the X's are, what Y is, and how, if at all, Dummy modifies the effects of the X's. That is a content issue, and you need to find the answer yourself, perhaps with the input of colleagues from your own discipline.

[Alex Li]

Well, my first response is that both of these models are wrong. It is a mis-specification to have an interaction term in a model without also including both of its constituents. So both models are wrong because Dummy does not appear as a variable in its own right. Now, if you are going to use a fixed-effects regression and if Dummy designates some subset of the levels of the fixed effects variable (e.g. designates one country and the fixed effects variable is country), then that variable is colinear with the fixed effects and will be dropped anyway, so no problem. But otherwise, it needs to be there.

That said, the answer is that it depends. The two models apply to different situations and only you know which one is appropriate for you.

Model a says that the effect of FDI is different when Dummy = 1 than it is when Dummy = 0, but the effects of the X variables are the same, irrespective of the value of dummy. Model b says that not only is the effect of FDI different when Dummy = 1, but the effects of all the X variables are also different when Dummy = 1.

Another way to think about it is, that model b is completely equivalent to doing two separate regressions, one for the Dummy = 1 observations and another for the Dummy = 2 observations. But model a is different: it is equivalent to doing two separate regressions but imposing the constraint that the coefficients of all the X variables must be the same in both regressions.

So that's the statistics underlying the models, and provides the basis for choosing between them. Whether the effects of X are expected to differ between Dummy = 1 and Dummy = 2 is a factual situation, not a statistical question. It depends on what the X's are, what Y is, and how, if at all, Dummy modifies the effects of the X's. That is a content issue, and you need to find the answer yourself, perhaps with the input of colleagues from your own discipline.

Many thanks for your reply. And yes I will be doing a Fixed-Effect panel regression as the dummy will be dropped and forgot to include it in the regression.

[Zahid Khan]

Clyde Schechter I am doing something similar where I am using a district-level panel over 6 years. I want to include time-specific and region-specific effects but the problem I am facing is that I am getting time-specific effect but when I add region-specific dummies it drops all of these region-specific dummies. Whereas my region-specific dummies are not districts but they are provinces ( which is a bigger unit of the region) and these are only 4 in total.
Adding information to my question I am trying to run a fixed effect regression that drops these province-specific dummies.