Interpretation of interaction between binary variable and quadradic variable

[Marlene Waetzold]

Good afternoon!
I have a panel dataset with two survey waves and want to estimate the heterogeneous of effect of negative rainfall deviation on cocoa yields for agroforestry-adopting farmers and non-adopting cocoa farmers.


Negative rainfall deviation is the deviation in rainfall from the historical average and is a continuous variable and measured in absolute terms (i.e. one mm less rainfall than the historical average is 1). Since I expect the relationship between negative rainfall deviation and cocoa yields to be non-linear (there are worse effects on yields at much lower levels of rainfall), I am interested in the quadratic term of negative rainfall deviation.

My hypothesis is that agroforestry (which is an agricultural practice and binary variable) can mitigate the effects of too little rainfall and therefore I want to interact it with the square of negative rainfall deviation.

Agroforestry adoption is time-invariant between our two survey waves.

We use a fixed effects approach with following command:

xtreg cocoa_yield agroforestry##c.negative_rainfall_deviation_sq negative_rainfall_deviation $control_variables_variant [pw=psweight], fe robust

In the stata regression output, agroforestry is dropped (because there are no changes) but I would like to know how to interpret the interaction term between agroforestry and the square of negative rainfall deviation.

Based on my understanding, agroforestry adopters have less negative effects of negative rainfall deviation (since the interaction term is positive, while the negative_rainfall_deviation_sq is negative), however, I would like to know how I can interpret the extract numbers.

Thank you very much in advance for any support!!!

[Clyde Schechter]

I would not use this model. While it is, as you have seen, syntactically legal to interact something with the quadratic term but not the linear one, it makes no sense as a model. Doing so implicitly constrains the location of the vertex of the parabola to a rather bizarre subset of the rainfall deficit-yield plane. I would revise the model so that the interaction also applies to the linear term.

xtreg cocoa_yield agroforestry##c.negative_rainfall_deviation##c.negative_rainfall_deviation $control_variables_variant [pw=psweight], fe robust

Note that the homebrew quadratic variable is not used here--the interaaction is taken as a three way interaction, with two of the terms both being c.negative_rainfall_deviation.

Now, you understandably found the interpretation of your original model difficult, and this one will be even more so. I think that trying to make sense of the regression coefficients you get will be a frustrating effort with a high probability that you (or anybody else doing it) ends up getting it wrong. Instead, use the -margins- and -marginsplot- commands to graph the interaction. Then you will be able to easily see what is going on.