Including outcome only related covariates in propensity score estimation When estimating the propensity score in an observational study, there seems to be a relative consensus on the fact that covariates related to the treatment and the outcome (i.e. cofounders) and covariates only linked to the outcome should be included in the model used for estimating the PS. My question is simple: why would one want to do this ?
In search for an answer, I found that many articles on this topic quote an article by Brookhart et al. in which one can read that

one should include in a PS model variables that are thought to be related to the outcome, regardless of whether they are related to the exposure. This result may run counter to intuition for many people. One might wonder why a PS model should include a variable that is unrelated to exposure. The answer is that even if a covariate is theoretically unassociated with exposure, there can be some slight chance relation between the covariate and the exposure for any given realization of a data set. If that covariate is also related to the outcome, then it is a empirical confounder for that particular data set. Including such a covariate in a PS model corrects for small amounts of chance bias or empirical confounding existing within each realization of the data set, thereby improving the precision of the estimator.

I do not understand the idea behind this advice: in my intuition, covariates only associated to the outcome will only add noise to the estimatated PS if included in the estimation model (say for instance a logistic regression). My misunderstanding might come from the fact that I do not understand what Alan Brookhart and his co-authors mean by an "empirical confounder". Anyone willing to share an explanation ? Thanks alot in advance.
 A: The error in an estimate of the treatment effect depends on the difference in covariate means between the treated and control group as well as the coefficient on the covariates in the outcome model. Variables that cause selection into treatment will have high mean differences between the treatment groups, and variables that cause the outcome will have high coefficients in the outcome model. Clearly, if the goal is to reduce error, variables that cause both selection into treatment and the outcome (i.e., confounders) are important to adjust for.
Variables that don't cause selection into treatment (and are otherwise unrelated to treatment) will not have large mean differences, but those differences will not be equal to zero due to random sampling (i.e., in the same way you won't get exactly 5 heads in 10 tosses of a fair coin). So, by reducing the mean differences for these prognostic non-confounders, you can still decrease the error of the effect estimate. When an estimator is unbiased, decreasing the error decreases the variance of the estimate; you get more precision. The larger the coefficient of the coevairate is in the outcome model, the greater gains in error reduciton for balancing the covariate.
There is a tradeoff, though. The propensity score will try to balance any covariate supplied to it. Covariates with large mean differences are the hardest to balance. The more work a propensity score has to do to balance all of the included covariates, the less precise the estimate will be (e.g., due to discarding too many units or having highly variable weights). For variables with only small imbalances, though, the propensity score doesn't have to work very hard, so the tradeoff is generally not realized. So, for these prognostic non-confounders, you can get decreases in error due to improved balance without increasing error due to the effort of the propensity score to balance. This is why these variables should be included in propensity score models, even though they do not cause the treatment.
The opposite is true of instruments, variables that cause the treatment but are unrelated to the outcome. The propensity score uses a lot of effort to balance the instrument, but there are no gains in error reduction because the instrument doesn't have a coefficient in the outcome model. (When some confounders are omitted, including instruments can actually make estimates biased, not just less precise.)
