Are time-variant control variables in DID necessary?

[Li recordyao]

I'm wondering what is the role of time-variant control variables when someone does difference-in-difference regression with two-way fixed effect and the treatment event is 100% exogenous (such as the COVID). The individual and time fixed effects tease out the individual and time heterogeneity. Would the time-variant control variables necessary? Since the treatment event is 100% exogenous, there are no concerns about the reverse causality and missing variable concerns.

[Clyde Schechter]

Only covariates that vary both in time and across panels would be of value in reducing bias. And with a truly exogenous treatment, even those are not needed. I would be somewhat cautious about considering the Covid epidemic exogenous. For most purposes it would be, but the epidemic did not emerge everywhere in an instant: some places were affected later than others, and so some outcomes might have been affected by awareness that the pandemic would be coming. If you can't exclude possibilities like that, then it can't be considered a truly exogenous treatment, though probably close enough for most practical purposes.

Even in the face of a truly exogenous treatment (even a randomization), one might still want to include a covariate that varies both in time and across panels and is associated with the outcome, to reduce residual variance, thereby increasing statistical power.

[Maxence Morlet]

Li recordyao In addition to Dr. Schechter's very enriching response, I recommend you read both following papers, which caution against the use of time-varying covariates in DiD:

- https://pubmed.ncbi.nlm.nih.gov/33978956/

- https://econpapers.repec.org/article...0800000014.htm

[Li recordyao]

Professor Schechter


Thank you for your explanation. Indeed, controlling covariates that vary both in time and across section could reduce residual variance and increase statistical power. But it won't be necessary for the identification under exogenous treatment right? (Let's assume the COVID is 100% exogenous treatment)

To compare two situations, both with two-way fixed effects:
(1). y = β_1a post*treatment
(2). y = β_1b post*treatment + β₂ x_it

where model (1) does not include time-variant control variables and model (2) does.

When the two estimates are different (this is possible if x_it are induced by the COVID and correlated with treatment and y ), which would be true effects of the treatment? β_1a or β_2a ?

My understanding is that β_1a would be the overall treatment effect by COVID, and β_2a would be the direct COVID effect excluding the x_it channel. In a real paper, probably we need to present both results. So having more controls are always better than not having any. But if the dataset lacks such time-variant control variables, would it be a problem for the DID analysis?

[Maxence Morlet]

Are the x_it variables affected by your treatment? If yes, literature is very clear: do NOT include them, notably because you are shutting down a causal channel of the effect of your treatment on the outcome.

[Li recordyao]

Are the x_it variables affected by your treatment? If yes, literature is very clear: do NOT include them, notably because you are shutting down a causal channel of the effect of your treatment on the outcome.

Mr. Morlet


Thank you for your answer and recommendation of reference.
Yes, that makes sense. Then, in the case of COVID, we shouldn't include most time-varying covariates since most socio-economic variables are affected by the big event such as the COVID.
It seems to me, in the case of an 100% exogenous shock and influential event, the dominant strategy for DiD is to exclude any time-varying control variables since the cost of such (potentially shutting down a causal channel) outweighs the benefits (reduce residual variance and increase statistical power), especially when the DID result without these controls is already statistically powerful and robust.

[Maxence Morlet]

If treatment is 100% exogenous, you can theoretically include covariates, however they should not have any effect on the outcome and treatment should not affect them either. If treatment is 100% exogenous (happens rarely), this creates a situation very resemblant to randomised controlled trials.

In this scenario, covariates should also not alter the effect of treatment on the outcome, however they may decrease residual variance, as perfectly explained by Dr. Schechter.

I recommend you also take a look at the community-contributed sdid command.

Also, Kaul et al. (2022) have shown that if you condition the outcome on all pre-treatment lagged outcomes, covariates become redundant, even potentially counter-productive.