Interpreting the coefficient of an interaction term in the presence of two omitted categories

[Sugandha Huria]

Hi,

I am running a linear probability model. My dependent variable is an indicator variable indicating whether a firm is a high-growth firm or not and I am including an interaction term between (time-variant) R&D_status and (Time-invariant) Technology_intensity amongst the RHS variables. Basically, my regression equation is the following:

xtreg highgrowth_status i.Technology_intensity i.Technology_intensity#L3.R&D_statusI L3.l_n1 i.year, fe cluster(co_code)

When I do so, 1.Technology_intensity also gets omitted because of collinearity. While the coefficient of Technology_intensity=0 & L3.R&D_statusI=1 is (-.004), that of Technology_intensity=1 & L3.R&D_statusI=1 is (0.076). Could you please help me in interpreting these two coefficients? Should I interpret the coefficient of the latter relative to that of the former category?

Thank You.

[Clyde Schechter]

I find your description of the output confusing and I don't really grasp what you have here. Please repost showing the actual regression output you got from Stata. Copy it directly from your Results window and paste it here (between code delimiters). If you are not familiar with code delimiters, pleas read FAQ #12 for instructions.

That said,

xtreg highgrowth_status i.Technology_intensity i.Technology_intensity#L3.R&D_statusI L3.l_n1 i.year, fe cluster(co_code)

appears to me to be a mis-specified model that should not be interpreted at all. It is not admissible to include an interaction term without including the constituent ("main") effects as well. You did specify i.Technology_intensity (which Stata then omitted due to colinearity--not a problem), but unless L3.l_n1 is a clone of L3.R&D_statusI, then the effect of the latter is missing from the model and the model is invalid.

By the way, this cannot possibly be the code you ran because the & character cannot appear in a variable name. When posting about code, always show the exact code you ran. Edited, simplified, or otherwise modified versions are a waste of your time and that of those who wish to help you because in coding, there are no unimportant differences: every detail matters.

[Clyde Schechter]

OK. As I suspected, this is a mis-specified model. Without including the L3.rndI term itself, the model is not valid. This is an easy mistake to make, and people often forget it. The way to avoid making this mistake is, when specifying interaction terms in regression commands, to use ## instead of #. That way Stata will automatically put in the "main" effects along with the interaction.

I also note you have a variable called age2. If that is age², then you should not do it that way, because if you try to have Stata do any predictions involving age, the results will be incorrect. So, assuming that age2 does mean age², your model should be:

xtreg hgf_oecd2 L3.le c.L3.age##c.L3.age L3.exporti i.kis##L3.rndI L3.l_n1 i.year, fe cluster(co_code)

Note: if rndI is a continuous variable, you should specify it as c.L3.rndI.

Stata omits the "main" effect of kis because of colinearity, I presume the colinearity is with the country fixed effects. It's not clear to me why 2013.year is also dropped for colinearity: it is not obvious what it might be colinear with, but perhaps if I knew what these variables meant it would be. In any case, assuming that year really is colinear with some other variable (and that this doesn't reflect some kind of data error), the omission by Stata in no way distorts or invalidates the interpretation of the model. You can just go ahead and interpret it exactly the way you would as if everything else were there. Of course, that is best done with the -margins- command.

[Clyde Schechter]

So the coefficient of the interaction term is the difference between the marginal effect of L3.rndI on hgf_oecd2 when kis = 1 and the marginal effect of L3.rndI on hgf_oecd2 when kis = 0. Eauivalently, it is the difference between the marginal effect of kis on hgf_oecd2 when L3.rndI = 1 and the marginal effect of kis on hgf_oecd2 when L3.rndI = 0.

To get a better feel for this, I suggest you run:

margins kis, dydx(L3.rndI)
margins L3.rndI, dydx(kis)

so that Stata will show you the marginal effects of L3.rndI conditional on each value of kis (first -margins- command) and the marginal effects of kis conditional on each value of kis (second -margins- command). You will notice that the interaction coefficient is, in each case, the difference between the two marginal effects.

[Sugandha Huria]

Ok, thank you so much.