Interpreting Interaction Effects in a TWFE model

[Jeffrey Michler]

Hello,

Co-authors and I are going back and forth about the correct way to interpret the interaction effect of a treatment and moderator in a TWFE setting. Our setting is the adoption of a new ag technology (hybrid seed) and how the impact of adoption is moderated by adverse weather effects (drought). Both the treatment and moderator are measured as a continuous variables but could easily be turned into binary indicators. The hybrid seed is supposed to make crops more tolerant to droughts, meaning yields are higher during droughts relative to normal seeds. Additionally, the seeds should have no impact on yields absent a drought (no gain or loss relative to normal seeds). We have panel data, where the unit of observation in farmer over time. The outcome of interest is yield. We are currently estimating:

yield_it = B₁(seed_it*drought_it) + B₂seed_it + B₃drought_it + c_i + c_t + e_it

xtreg             yield c.seed##c.drought i.year, fe cluster(hhid)

The debate we are having is, do we interpret B₁ like we would in a DID framework, where the interaction is the difference (seed/no seed) in the differences (drought/no drought), and so B₁ tells us the effect of seeds on yields during droughts (which is what we want to know)? Or, do we interpret B₁ as measuring heterogeneity in treatment effect by drought, so that we would need to test for the difference between B₁ and B₃ in order to know if seeds performed better than no seeds during a drought? Or is there some third way we should be approach the problem?

Any thoughts are appreciated!

[George Ford]

If dichotomous, the interaction coefficient B1 would measure the difference in seed effect during drought, while B2 measures the effect of seed absent drought.

could run

xtreg yield c.seed#i.drought drought i.year, fe cluster(hhid)

where you'll get the actual coefficient (rather than B1 measuring the difference). or

g nodrought = 1 - drought
xtreg yield c.seed#c.drought c.seed#c.nodrought drought i.year, fe cluster(hhid)

gives you the same. If continuous, then you've got to make computations at chosen values of seed and drought, which margins can do. Dichotomous makes more sense to me.

[Jeffrey Michler]

Thanks!