# Unbiasedness and Consistency of DID estimator - pooled cross sections over time

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?

Thanks!

• You could search for "identification assumption difference in difference". Personally I like the treatment in Lechner's slideshow (unige.ch/ses/dsec/sperlich/L8%20DiD.pdf). Oct 4 '18 at 22:15

You do not need to search the literature about this problem. You fit a linear model, which is classical and common and has solid theoretical background. What you need to do is checking the assumptions for Gauss–Markov theorem with your situation. If there is no violation to these assumptions, your estimate is BLUE (best linear UNBIASED estimate). (here the "unbiased" means $$E(\hat \beta) = \beta$$, which is standard definition in statistics. currently, the (un)biased is used in other meanings, such as biased population, selection bias...)