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!
Best Answer
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 $\beta_1$.