Interpretation of continuous variable in dummy-continuous interaction

Question

Similar questions have been asked before, but all of them focus on the dummy or interaction term.

Say run an OLS regression on the model:

$\ln( housePrice )= \beta_1 \times pollutionLevel + \beta_2 \times D_N + u$

where $D_N$ is a dummy that indicates whether there is a school nearby the house.

The interpretation for $\beta_1$ and $\beta_2$ are simple enough, but in the model:

$ln( housePrice ) = \beta_1 \times pollutionLevel + \beta_2 \times D_N + \beta_3 \times pollutionLevel \times D_N + u$

it's not so clear.

I understand the interpretation of $\beta_3$, but how does the interpretation of $\beta_1$ change? Is $\beta_1$ now just the effect of pollutionLevel when there isn't a nearby school, or is that totally wrong?

Thanks in advance for any help!

Answer 1

Yes, that is correct in your case. A good way to convince yourself of that statement follows.

Say you want to find the impact of the pollution level on the log of house prices.

$$ \dfrac{\partial \ ln(housePrice)} {\partial \ pollutionLevel} = \beta_1 + \beta_3 \times D_N $$

where the impact of the pollution level on the percentage change in house prices when there is no school nearby $(D_N=0)$ is simply $\beta_1$.

Answer 2

One way to generally look at this is via marginal effects as in @Giaco.Metrics' response. Another general technique is a distinction of cases.

For $D_N = 0$ (no school nearby, reference group), your equation simplifies to:

$\ln(housePrice) = \beta_1 \times pollutionLevel + u$,

i.e., you have intercept 0 and slope $\beta_1$ in the reference group.

For $D_N = 1$ (school nearby), you get

$\ln(housePrice) = \beta_1 \times pollutionLevel + \beta_2 + \beta_3 \times pollutionLevel + u\\ = \beta_2 + (\beta_1 + \beta_3) \times pollutionLevel + u$,

i.e., you have intercept $\beta_2$ and slope $\beta_1 + \beta_3$ in the school group. So $\beta_2$ is the difference in intercepts and $\beta_3$ the difference in slopes.