Consider two time periods, where In time-period 1: a random sample is collected for group 1 (control) and a random sample is collected for group 2 (treated). In time-period 2: a random sample is collected for group 1 (control) and a random sample is collected for group 2 (treated).
Suppose that in between the two time-periods there was some exogenous event, say, a policy change, causing group 2 to be affected but group 1 not.
A common way of estimating the average treatment effect - the average change in y in the treated group due to the policy change (assuming parallel trends - i.e. both the treated and control group would have had the same path had there not been a policy change) will be given by $\beta_3$ in the model
I have tried google searching the conditions under which we obtain an unbiased (and/or consistent) estimator of beta 3 but I am unable to find it. Can someone help me with this?
My current thinking is
If $x_1=d2$, $x_2=dT$, and $x_3=d2*dT$, estimation of the parameters in the linear projection of $y$ on $[1, x_1, x_2, x_3]$ would be consistently estimated if $x_1$, $x_2$, and $x_3$ are uncorrelated with $u$. But would this condition yield consistent estimators of the coefficient that I want? Particularly $\beta_3$, which equals $(E(y|d2=1,dT=1) - E(y|d2=1,dT=0)) - (E(y|d2=0,dT=1)-E(y|d2=0, dT=0))$.
Do we need $E(u|d2,dT)=0$ for unbiased estimation of the average treatment effect? What are the exact conditions? Is this provable? Is there a relationship between this condition and the parallel trends assumption?