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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?

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  • $\begingroup$ Can you post the entire coefficient table? Are all predictors significant? $\endgroup$
    – Mark White
    Commented May 22, 2017 at 20:11
  • $\begingroup$ Could you elaborate on the meaning of "the interpretation of the individual variables changes"? How could that happen? $\endgroup$
    – whuber
    Commented May 22, 2017 at 20:43
  • $\begingroup$ 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. $\endgroup$ Commented May 22, 2017 at 21:36
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    $\begingroup$ I would suggest adding the squares as numeric superscripts rather than text subscripts since that is the more conventional mathematical notation. $\endgroup$
    – dimitriy
    Commented Jun 18, 2020 at 21:20

1 Answer 1

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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 $Z$.

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