Marginal effect in model with interactions

Question

I am running cross-sectional regressions of the type

$$Y_c = \alpha + \beta X_1 + \gamma X_2 + \delta_1 X_3 + \delta_2 X_1 X_3 + \delta_3 X_2 X_3 + e_c.$$

My theoretical model implies that

• $\delta_2$ should be negative,
• $\delta_3$ should be positive, and
• the marginal effect of $X_3$ should be negative.

My estimates imply that

• $\widehat\delta_2$ is negative and significant,
• $\widehat\delta_3$ is positive and insignificant,
• $\widehat\beta$ is significant, and
• $\widehat\gamma$ is insignificant.

Building on this evidence, can I calculate the marginal effect of $X_3$ as $\delta_1 + \delta_2 E(X_1)$ where $E(X_1)$ is the mean of $X_1$, justifying this procedure with the fact that all the terms incorporating $X_2$ are insignificant?

Answer 1

You seem to be aware that the marginal effect of $X_3$ is $\delta_1 + \delta_2 X_1 + \delta_3 X_2$, which is just the derivative of the response with respect to $X_3$.

Replacing $X_1$ with $E(X_1)$ is a reasonable way to summarize the marginal effect.

However, discarding the final term due to statistical insignificance is nonsense. There are at least two relatively sensible alternatives:

1. If your $n$ is so big that you believe the statistical result that $\delta_3$ is insignificant more than you believe your prior belief that $\delta_3$ should be positive, than you could get rid of the $\delta_3 X_1 X_2$ term in your model and refit the coefficients BEFORE using $\delta_1 + \delta_2 X_1$ as your marginal effect.

2. If you believe that the terms involving $X_2$ need to be in the model, regardless of statistical significance, than you need to keep the $X_2$ term in your marginal effect as well.