SELECTION BIAS: Difference-in-Differences 2

Term B\ in (14) arises when Sox\Sx or S\x\Sx is nonempty. In this case we fail to find counterparts to E(Yq | X, D = 1) in the set S0x\Sx and counterparts to E(YQ | X, D = 0) in the set Six\Sx• Term B2 arises from the differential weighting of E(Y0 | X, D = 0) by the two densities for X given D — \ and D — 0 within the overlap set. Term B$ arises from differences in outcomes that remain even after controlling for observable differences. Selection bias, rigorously defined as Вsx, may be of a different magnitude and even a different sign than the conventional measure of bias В.
Matching methods that impose the condition of pointwise common support eliminate two of the three sources of bias in (14). Matching only over the common support necessarily eliminates the bias arising from regions of nonoverlapping support given by term B\ in (14). The bias due to different density weighting is eliminated because matching on participant P values effectively reweights the non-participant data. Thus PxBsx 1S 0П^У component of (14) that is not eliminated by matching.28 Bsx is the bias associated with a matching estimator.

Examining the Validity of Matching on P
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