# Partial derivatives in interaction term interpretation

I have estimated the following regression:

$$logWages = -0.24 +0.046*Schooling + 0.01*SchoolingxBlack -0.29*Black$$

My question is with respect to the interpretation of the interaction term. If I take partial derivative I would get that:

1.$$\frac{\partial logWages}{\partial Schooling}= 0.046 +0.015Black$$ So if you are black you would have an increase of 0.046 +0.015 given a marginal increase in schooling in contrast to 0.046 if you are not black.

1. $$\frac{\partial logWages}{\partial Black}= 0.015*Schooling -0.29$$ So if you are black, one extra year of schooling is 0.015-0.29 decrease in your logwage whereas you would get 0.015 increase o.w.

I think the latter interpretation makes no sense but I am looking to a clearer ilumination here because I always get confused when interpreting the interaction terms . Thanks!

• I hope you're not literally interpreting this regression, because, you need a lot of controls to link wages to the salaries, many more than just schooling. Also, partial derivative on categorical variable is weird to interpret, it's not how people do it Jan 14, 2021 at 18:00
• thank you! this is just a subset of the real regression. " Also, partial derivative on categorical variable is weird to interpret, it's not how people do it" How do people do it? I'm learning, maybe drop me an answer and show me? Jan 14, 2021 at 18:11
• similarly to answer you accepted. you set Black to 0 or 1, then you get -0.29 in salary for Black vs not Black, plus 0.015 for a year of schooling Jan 14, 2021 at 18:16
• On @Aksakal's second point, this is one way to deal with categorical dummies with a logged outcome. Calculating finite differences is another. Jan 14, 2021 at 18:32
• @DimitriyV.Masterov Finite difference makes sense for integer variable like schooling years. If you have 0 and 1 only what’s the point in finite difference? Jan 14, 2021 at 18:35

The two derivatives answer different questions.

The first derivative implies that a year of schooling is associated with 4.6% increase in wages if White, and 6.1% if Black.

The second derivative can be re-written as

$$\frac{\partial \ln Wages}{\partial Black}= -0.29 + 0.015 \cdot Schooling$$

This says the expected wage is ~1/3 lower if Black, reflecting discrimination or other omitted factors that determine wages and are correlated with race, like location. However, additional education starts to eat into that, albeit very slowly. It would take almost 20 years of schooling to offset the "penalty" of the lower intercept.

In other words, the first tells you how the schooling is associated with wages, depending on race. The second tells about how race and wages are associated, as a function of schooling.

• not Black doesn't imply White. not Black is simply not Black. also the race category these days is not disjointed, you can be both White and Black, like BHO Jan 14, 2021 at 18:01
• of course, dont worry, toy example. Jan 14, 2021 at 18:11