How to interpret coefficients in a regression model with two groups

Question

given a linear regression model such as

$ y= \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3D + \beta_4D*X_1 + \beta_5D*X_2 $

where $D$ is a dummy variable, what are the proper interpretations of the coefficients $\beta_1$ and $\beta_4$ ?

My understanding is that the partial base effect of a one unit change in $X_1$ will lead to a change in $y$ by $\beta_1$ units for both groups. (when $D=1 $ or $ D=0$)

and that there is an additional effect on $y$ of $\beta_4$ units only when $D=1$ given a one unit change in $X_1$

a classmate told me that $\beta_1$ only describes changes when $D=0$

Answer 1

You can get the marginal effect on $y$ from $X_1$ increasing by 1 unit by taking the partial derivative of $y$ with respect to $X_1$:

$$ \frac{\partial y}{\partial X_1}= \beta_1 + \beta_4 \cdot D$$

So the marginal effect depends on $D$, which is binary. For people who have $D=0$, the marginal effect is just $\beta_1=\beta_1 + \beta_4 \cdot 0$. For people with $D=1$, it is $\beta_1 + \beta_4 = \beta_1 + \beta_4 \cdot 1$. The effect of $X_1$ is modified by $D$ in this model.

If you are uncomfortable with calculus, you can also calculate a finite difference between the prediction if $X_1$ set to $k+1$ and the prediction with $X_1$ set to $k$.