# OK to calculate semi-elasticity of interaction term? Can it be less than -1?

I'm running a Stata diff-in-diff regression by using an interaction term. I then try to calculate the semi-elasticity of the interaction term. The Stata code is as follows:

reg interp_tf year year_post_mayo reg_length dem_administration post_mayo is_interpretative interpretative_post_mayo, cluster(agency_num)
margins, eydx(interpretative_post_mayo) post


The interaction term interacts 2 dummy variables. I've seen some suggestions that this might be inappropriate, for example this paper: https://onlinelibrary.wiley.com/doi/pdf/10.1111/boer.12120. But the paper doesn't fully explain, and I don't understand, what the problem might be.

In addition, the margins command produces a result lower than -1, which I don't know how to interpret--e.g., if it is -1.31, does that imply a -131% change? Is that possible?

Any thoughts on either of these questions would be greatly appreciated!

## 1 Answer

I don't think the paper you cited applies since your outcome is neither logged, nor are you are using a non-linear model.

The margins, eydx(x) is calculating the average, calculated across all time periods and observations, in your estimation sample of:

$$\frac{\partial E[y \vert x]}{\partial x} \cdot \frac{1}{E[y \vert x]}=\beta_{x} \cdot \frac{1}{\hat y} \approx \frac{\frac{\Delta \hat y}{\hat y}}{\Delta x}.$$

In your case, $$\Delta x =1$$ since the interaction is dummy, so you have $$\frac{\Delta \hat y}{\hat y}=-1.31.$$

The numerator is change in $$\hat y$$ over $$\hat y$$, so it does make sense to multiply that by 100 to put it into percent like you did. This makes the semi-elasticity -131%. Mathematically, that could be possible, unless you outcome is bounded. For example, profits could become losses. Arrests or wages, on the other hand, are bounded below by zero, so -131% may not make much sense in that setting.

• Thank you! That's good to know about the general validity of using semi-elasticities. On the -131%, my dependent variable (interp_tf) is indeed zero-bounded, so the result is confusing. Is that even mathematically possible, or does this imply that I have some kind of syntax error?
– Jon
Mar 13, 2020 at 20:40
• It's probably not a syntax error. Sometimes linear models struggle with bounded variables. It usually shows up with predictions, but sometimes can spill over to coefficients too. The solution is to try to use a non-linear model, but then the DID gets a bit tricky to interpret. An easier solution would be to try natural logging your outcome variable (and adding a small number to the zeros if necessary), and interpreting like this. An inverse hyperbolic sine in another possible transformation that can handle zeros. Mar 13, 2020 at 20:56