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?
 A: Your FE model is
$$E[Z_i \vert X_i,Y_i]= a_i + b_1 \cdot X_i + b_2 \cdot Y_i + b_3 \cdot X_i^2 + b_4 \cdot Y_i^2 + b_5 \cdot X_i \cdot Y_i$$
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 $Z$.
