In an earlier substack on Tymon Sloczynski‘s 2022 Restat (hereafter “Tymon”) I went through the algebraic properties of this OLS model:
I noted that Tymon had a theorem he called the “weighted average interpretation of OLS” that showed the OLS estimate of tau was a weighted average of two average partial linear effect quantities — one for the treatment group and one for the control group.
Look closely here at the weights:
Notice how the weight on the first treatment group APLE term (equation 3) is increasing as the share of the control group gets larger. See that? The 1-rho where rho is the share of units in the treatment group? That means as the number of units in the treatment group falls, the weight on that APLE rises and vice versa. You can see the same thing in that second weight — as the share of units in the treatment group, rho, rises, the weight falls.
I will explore this odd feature of OLS weighting today with you with a simulation. Then I want to transition from the alge…
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