# Interpretation of two dummies interacted with one continuous variable

I am currently looking at the following equation using Stata and unsure of how to interpret the interaction terms when there are three variables interacted together (2 binary variables and one continuous variable).

I ran this equation on Stata:

probit ownland i.employed##i.female##c.eduyrs


where ownland is =1 if the respondent owns agricultural land; employed= 1 if the respondent is currently working, female= 1 if the respondent is a woman, and EDU_YEARS captures years of education.

Own land Coef. Std. Err. z P>z [95% Conf. Interval]
Employed .2022467 .1431765 1.41 0.158 -.078374 .4828674
Female .2049693 .1994331 1.03 0.304 -.1859123 .5958509
Employed#Female -.0853641 .267807 -0.32 0.750 -.6102563 .439528
eduyrs -.0240563 .0113061 -2.13 0.033 -.046216 -.0018967
Employed#c.eduyrs -.0061285 .0191439 -0.32 0.749 -.0436497 .0313928
Female#c.eduyrs -.0248372 .0321575 -0.77 0.440 -.0878647 .0381902
Employed#Female#c.eduyrs .0368203 .0428535 0.86 0.390 -.047171 .1208115

I am pretty clear on the interpretation of "employed", "female", and "eduyrs" on the outcome:

• Employed variable would be the effect of being employed on Y (ownland) for uneducated men
• Female variable would be the effect of being a female on Y for unemployed and uneducated women, compared to men.
• eduyrs variable would be the effect of years of education for unemployed men

But I am confused about the interpretation of other variables.

1. How would you interpret the estimated coefficient for Employed#Female#c.eduyrs? Would it be the effect of being employed women on Y for a unit change in years of education?
2. How can I obtain the net effect of an increase in year of education for employed female versus unemployed male? Would that be nlcom (employed+ female+ i.employed#i.female+eduyrs + Employed#Female#c.eduyrs)-(eduyrs)?
3. How can I obtain the net effect of an increase in year of education for employed female versus unemployed female? Would that be
(employed+ female+ i.employed#i.female+eduyrs + Employed#Female#c.eduyrs)-(female + eduyrs + female#eduyrs)?

If someone could enlighten me on this, that would be very helpful. Thank you so much!

• If you are using Stata, you should find that using ## creates not only the 3-way interaction but also all the 2-way interactions and "main" effects. So, you will have 3 more coefficients in your model than you have included (not including the intercept). These additional terms in the model are essential for interpreting the coefficient on the 3-way interaction. Please clarify this, perhaps including the regression output so we can help interpret specific numbers.
– Noah
Jan 27, 2022 at 20:56

The trick in thinking about this (at least for me) is not to jump immediately to the highest level of interaction. Rather, start at the beginning, the intercept and the individual coefficients, and recognize that each higher level of interaction represents a difference from the previous level.

You seem to grasp the intercept (outcome with all predictors at 0 or reference levels) and the individual coefficients pretty well. If you modeled c.eduyrs coefficient as a continuous linear predictor, the Employed coefficient is the change in outcome from Unemployed at 0 years' education, and the c.eduyrs is the change per year of education beyond 0 for Unemployed for Male, as you say.

So next move up one interaction level to the Employed#c.eduyrs coefficient. That's the difference in outcome per year of education for Employed Male from Unemployed Male.

Then the interpretation of the 3-way interaction is (a bit) easier to think about. You can think of it as the extra difference per year of education for Female beyond what you estimate for an Employed male.

In terms of predictions, it's safest just to write out the entire linear predictor of the model:

$$\beta_0 + \beta_E E + \beta_F F + \beta_C C + \beta_{EF}EF + \beta_{EC}EC + \beta_{FC}FC + \beta_{EFC} EFC$$

where $$E =1$$ is Employed, $$F=1$$ is Female, and $$C$$ is c.eduyrs. The interactions are just products--for each term, multiply the $$\beta$$ coefficient by the 0/1 values for the binary predictors and the continuous value for c.eduyrs. Plug in the values to get the linear predictor for a scenario, then (for a generalized linear model like your probit) pass that through your inverse link function to get the estimated outcome. Extend as needed for multiple scenarios or differences between them.