Limits to how many control and treatment subjects may be paired in propensity score matching? I'm working on a difference-in-differences project where we're matching up to 5 control subjects to each treatment subject using a combination of techniques to estimate treatment effects (ATT): exact, propensity score, and Mahalanobis matching.
Our 1:5 matching rule is somewhat arbitrary - are there more rigorous rules limiting the number of matches? Conceivably matching one treatment to 50 control subjects, or vice versa, seems extreme - perhaps either increasing bias or variability in ATT estimates?
 A: There is a bias-variance trade-off in deciding how many control units to match to each treated unit. Any units beyond the first match will tend to be further away, yielding more distance pairs and potentially worse balance and therefore more bias. However, increasing the number of control units to pair also increases precision by increasing the sample size. According to Rosenbaum (2020), the variance of the difference-in-means estimator is proportional to $1+\frac{1}{r}$, where $r$ is the number of control units for each treated unit. That means adding more control units does decrease the variance, but going from 1 to 2 controls yields a much greater reduction in variance than going from 9 to 10.
Some simulation studies have been done on this, but I don't think it is worth trusting the results from these studies since the choice depends on the unique features of your dataset that you can assess before estimating the treatment effect. You should try many different matching ratios until you find one that balances covariate balance and sample size.
