Interpretation of interaction of a continuous and a dummy variable in linear regression

Question

I have the regression

$$ y = \beta_0 + \beta_1 x_1 + \beta_2 D + \beta_3 x_1 D + \varepsilon $$

where $x_1$ and $y$ are continuous, and $D$ is binary. Say the estimates are as follows:

$$ y = 0.02 + 0.5 x_1 + 2 D + 0.7 x_1 D + \varepsilon $$

Questions:

1. What is the interpretation of $\beta_3$?
2. How do I get the 'total effect' of $x_1$ and $D$? Is it $\beta_1 + \beta_3$ or is it $\beta_1 + \beta_2 + \beta_3$?

Answer 1

1. $\beta_3$ is the difference in slopes of $\mathbb{E}(Y|X_1)$ vs. $X_1$ between observations that have $D=1$ and these that have $D=0$. The slope for the former is $0.5+0.7=1.2$ and for the latter $0.5$.
   On he other hand, consider switching from $D=0$ to $D=1$ and the "effect" of that on $\mathbb{E}(Y|D)$; $\beta_3$ is the difference between two observations that differ by $1$ in terms of $X_1$. ("Effect" need not be causal, it could be just probabilistic.) I.e. $\mathbb{E}(Y|X_1=x_1+1,D=1)-\mathbb{E}(Y|X_1=x_1+1,D=0)=\beta_2+\beta_3 (x_1+1)$, $\mathbb{E}(Y|X_1=x_1,D=1)-\mathbb{E}(Y|X_1=x_1,D=0)=\beta_2+\beta_3 x_1$ and the difference between the two is $\beta_3$.

2. How do you define the "total effect"? E.g. $\mathbb{E}(Y|X_1=x_1+1,D=1)-\mathbb{E}(Y|X_1=x_1,D=0)=\beta_1+\beta_2+\beta_3 (x_1+1)$.