Mike,
I read your paper. Comments to help you out:
1) identification and estimation are separate things. And matching helps with model dependence only for estimation. Comparing results when the conditioning set changes is about identification and there is no reason to think that moves across different identification assumptions will be smooth. You confuse this in your paper, and if I as a little grad student did that, I would be savaged and would have failed my qual exam.
2) be more careful about finite sample versus asymptotic issues with regard to different matching methods.
3) data mining: see Rubin's design versus analysis article. Matching methods have the feature that one can set them up without any outcome data.
You made yourself look bad. But you seem like a smart guy, and I'm sure you will do better in the future.
To PSR: why are we discussing the worst paper at polmeth instead of the good ones?