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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