Propensity score matching: bias adjustment

Question

I'm using propensity score matching to match similar individuals. I.e., I first estimate a propensity score (the probability of treatment conditional on some set of variables) and then match on the estimated probabilities. Because this matching uses only one covariate (i.e. the propensity score) I had assumed that there's no need to use bias adjustment?!

Apparently I'm wrong about this, but I've not received a convincing argument why. Can anybody out there give me a compelling reason why I'd need bias adjustment in this case?

It was my understanding that one only needs to use bias adjustment when matching on more than one covariate (for example, nearest neighbour matching uses a bias correction term when matching on more than one covariate to speed up convergence).

Thanks (and sorry for the long question).

Answer 1

Bias adjustment after matching is only needed if you have imperfect matches, as typically occurs with more than one covariate. In that case an additional regression adjustment is helpful to reduce residual bias after matching. If you have one covariate only, and conditional on this one covariate the data generating mechanism is unconfounded, and you have overlap in your covariate distribution between treatment and control groups, you might find a perfect match for all cases, and there is no potential bias. This situation is, however, not common in practice. Likewise uncommon in practice is that you find perfect matches for all units on a multivariate covariate distribution. In general, the higher the dimensionality, the less likely it is that you have overlap and can find perfect matches. Hence there is usually bias after matching in practice.