# Interpretation of interaction in presence of squared terms

I am running a fixed effects model with two independent variables, their quadratic terms, and the interaction of the two variables.

$$IV = b_0 + b_1X + b_2Y + b_3X_{sq} + b_4Y_{sq} + b_5XY$$

How do I interpret the interaction? Does it even make sense to interpret it, given that when I include the quadratic terms, the interpretation of the individual variables changes?

• Can you post the entire coefficient table? Are all predictors significant? Commented May 22, 2017 at 20:11
• Could you elaborate on the meaning of "the interpretation of the individual variables changes"? How could that happen?
– whuber
Commented May 22, 2017 at 20:43
• b1 and b2 are not significant, b3, b4 and b5 are significant. What I meant by "interpretation of the individual variables changes" was that the interpretation of b1, for example, differs between a model that includes X_sq in it, compared to one that does not include this squared term. Commented May 22, 2017 at 21:36
• I would suggest adding the squares as numeric superscripts rather than text subscripts since that is the more conventional mathematical notation. Commented Jun 18, 2020 at 21:20

$$E[Z_i \vert X_i,Y_i]= a_i + b_1 \cdot X_i + b_2 \cdot Y_i + b_3 \cdot X_i^2 + b_4 \cdot Y_i^2 + b_5 \cdot X_i \cdot Y_i$$

The intercept $$b_0$$ is not really an ordinary intercept that comes out of the model (since that is eliminated by the demeaning), so I replaced it with the fixed effect $$a_i$$. You can think $$b_0$$ as the average value of the fixed effects (which is what software packages will frequently report).

Let's say you care about the effect of $$X$$ on $$Z$$. Take the derivative of the expected value with respect to $$X:$$

$$\frac{\partial E[Z_i \vert X,Y]}{\partial X}= b_1 + b_3 \cdot 2 \cdot X_i + b_5 \cdot Y_i$$

The marginal effect tells you how the expected value of $$Z$$ changes with an additional unit of $$X$$. Note that it is a function that depends on how much $$X_i$$ is already there, but also on what $$Y_i$$ is. If all three coefficients are positive, for example, adding that extra unit of $$X$$ will be more impactful when there are lots more $$X$$ and $$Y$$ to begin with.

But why stop here? Now we can take this a step further and ask how this marginal effect itself depends on $$Y$$ by taking a derivative of the derivative:

$$\frac{\partial E[Z_i \vert X,Y]}{\partial Y \partial X}= b_5.$$

This means the marginal effect of $$X$$ on $$Z$$ increase by $$b_5$$ Zs for each additional unit of $$Y$$. The significance of the interaction tells you something about whether you can distinguish that effect-on-the-effect from zero.

When your model leaves out interactions, you can think of that as effectively setting their coefficients to zero. This means $$b_1$$ will be comparable and have the same meaning across models with and without those interactions. It's the direct effect of additional $$X$$ on $$Z$$.