# How to interpret dummy variables and interactions terms on dummy variables in a regression?

Suppose I have a linear regression form of

$$\log(Y) = \beta_0 + \beta_1X_2 + \beta_2X_3 + \beta_3X_1Z + \beta_4X_2Z + \epsilon$$

where $$X_1, X_2, X_3$$ are binary and $$X_1$$ is omitted as a reference variable. Suppose $$Z$$ is also binary 0-1. I am wondering how we would be interpret $$\beta_1$$ and $$\beta_2$$?

• Note that $\beta_1 = \mathbb{E}[\log Y \mid X_2 = 1, Z = 0] - \mathbb{E}[\log Y \mid X_2 = 0, Z = 0]$ which of course the mean difference in $\log Y$ between a particular treatment ($X_2 = 1$) and control ($X_2 = 0$) when $Z = 0$. If the difference is small then you can interpret it as a percentage change in $Y$ between treatment and control holding Z at 0 ... Feb 14 at 6:22

You can interpret $$\beta_1$$ as the percent change in Y when the treatment effect $$X_2$$ is applied, when you're holding other variables constant, in this case that is Z.
You can scale this interpretation to other $$\beta_s$$ as well.
• This is incorrect in two ways. (1) Assuming "treatment effect $X_2$ is applied" means that $X_2$ is changed from $0$ to $1,$ then $\log(Y)$ changes by adding $\beta_1+\beta_4 Z,$ not just $\beta_1$ alone. (2) Interpreting that as a proportional change in $\log(Y)$ assumes the change is relatively small: less than $0.1$ in absolute value will do; less than $0.25$ is OK; beyond that it's usually a good idea to distinguish changes in the log from proportional changes.