Correlation with difference-in-difference estimator

[Alex Lampert]

Hello,

Say I'm estimating the impact of a program using an OLS difference-in-difference model, like so:

 regress outcome i.treatment##i.intervention $controls

Is there an issue with controlling for something that is highly correlated with the DiD term (1.treatment*1.after)? My internal monologue is:

(1) You need to control for the correlated variable because without it, you're attributing the effect entirely to the treatment when it could've also been driven by the correlated control variable. BUT
(2) I've learned that the standard errors for highly correlated variables are untrustworthy.

Is this a situation that might be difficult to disentangle? Or am I overthinking this?

Thanks!

[Clyde Schechter]

It depends. If the correlated variable lies on the causal path between the intervention and the outcome, you must not adjust for it. But if it is not on that causal path, then you should adjust for it.

Added: It is not true that standard errors of correlated variables are untrustworthy. What is true, is that correlated variables tend to have larger standard errors, so that estimates of their separate effects have lower precision.