# Interpretation of the coefficent on an interaction effect in a linear-log model

Hi I have the following regression model:

\begin{align}y=\beta_0 +\beta_1x_1+\beta_2x_2 +\beta_3 \ln(x_3)+\beta_4[\ln(x_3)\times x_1] + \mu_{ti} \end{align}

How would you interpret $$\beta_4$$?

My own thinking is that it is the $$\frac{\beta_4}{100}$$ is the ceteris paribus partial effect of a 1% increase in $$x_3$$ on the partial effect of $$x_1$$ on $$y$$. Is this in any way correct?

Differentiating the expected value of $$y$$ with respect to $$x_1$$ yields $$\frac{\partial y}{\partial x_1} = \beta_1 + \beta_4 \cdot \ln x_3$$ This is the change in expected $$y$$ (in units of $$y$$) associated with a one-unit change in $$x_1$$. It's a function of $$x_3$$.

How does this function behave when you change its argument? Differentiate it with respect to $$x_3$$:

$$\frac{\partial y}{\partial x_3 \partial x_1} =\beta_4 \cdot \frac{1}{x_3}$$

Bring $$x_3$$ over to the left to get:

$$x_3 \cdot \frac{\partial y}{\partial x_3 \partial x_1} =\beta_4$$

You can (approximately) rewrite that as

$$\frac{\frac{\Delta y}{\Delta x_1}}{100\cdot \frac{\Delta x_3}{x_3}} =\beta_4$$

The numerator is the change in expected $$y$$ for 1 unit change in $$x_1$$. The denominator is $$1\%$$ change in $$x_3$$.

If I can be picky, this is for expected $$y$$, since regression models the conditional mean of $$y$$, not $$y$$ itself.

• Amazing, thank you for your response! Dec 12, 2022 at 23:40