Solved – Interpreting interaction with quadratic term

$Y$ (scores among black students) $sim X_1 + X_1^2 + X_2 + (X_1 * X_2) + (X_1^2 * X_2)$

X1           0.089626*** (e.g. same ethnic teacher) X1^2         -0.008001*** X1*X2        0.003887*** (e.g. same ethnic teacher * principals' leadership X1^2*X2      -0.000231*** 

In this case,
How can I interpret (X1^2*X2)?

This regression can be written more simply as:

$$Y sim (X_1 + X_1^2)*X_2.$$

This model involves main effect terms plus interaction for the variable $X_2$ and a second-order polynomial in the first variable $X_1$. In such a model, the main effects and interactions are:

$$begin{matrix} text{Main effect of variable } X_1 & & & & X_1+X_1^2 \[6pt] text{Main effect of variable } X_2 & & & & X_2 \[6pt] text{Interaction effect of variables } X_1 text{ and } X_2 & & & & (X_1+X_1^2):X_2 \[6pt] end{matrix}$$

The individual term $X_1^2:X_2$ is not really meaningful in itself, since it is an interaction with only one of the terms in the second-order polynomial for your variable $X_1$. When interpreting the variables you should keep all the parts of your polynomial variable together.

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