I hope someone can clarify this for me:
When I am estimating treatment effects with a regression model, I can use interactions between the treatment indicator $W_i$ and covariates $X_i$ to model potential heterogeneity in treatment effects. While researching I came across this quote:
"Second, we can allow for a model with a full set of interactions: $Y_{i}^{\mathrm{obs}}=\alpha+\tau \cdot W_{i}+\beta^{\prime} {X}_{i}+\gamma^{\prime} {X}_{i} \cdot W_{i}+\varepsilon_{i}$
In general the least squares estimates based on these regression functions are not unbiased for the average treatment effects over the randomization distribution given the finite population. There is one exception. If the covariates are all indicators and they partition the population, and we estimate the model with a full set of interactions, Eq. (13), then the least squares estimate of $\tau$ is unbiased for the average treatment effect. " from "The Econometrics of Randomized Experiments, Athey and Imbens 2017, p.97"
Unfortunately, I don't understand how Athey and Imbens get to this conclusion. I know that I can't take $\hat{\tau}$ as the estimated average treatment effect when I have continuous covariates. The estimated ATE would be $\hat{\tau} + \hat{\gamma}*\mu_x$. Is that what they mean with not unbiased? Why would this bias go away with finite population ?