Why use control variables in differences-in-differences?

Question

I have a question on the differences-in-differences approach with the following standard equation: $$ y= a + b_1\text{treat}+ b_2\text{post} + b_3\text{treat}\cdot\text{post} + u $$ where treat is a dummy variable for the treated group and post.

Now, my question is simple: Why do most papers still use additional control variables? I thought that if the parallel trend assumption is correct, then we should not have to worry about additional controls. I could only think of 2 possible reasons for why to use control variables:

1. without them, trends would not be parallel
2. because the DnD specification attributes any differences in trends between treatment and control group at the time of treatment to the intervention (i.e. the interaction term treat*post) - when we don't control for other variables, the coefficient of the interaction may be over-/understated

Could anyone shed some light on this issue? Do my reasons 1) or 2) make sense at all? I don't fully understand the use of control variables in DnD.

Answer 1

without them [i.e., additional variables], trends would not be parallel

Yes, that's right. There may be unit-specific trends that you're not accounting for unless you add time-varying variables to the model.

Even if the parallel trends assumption is satisfied without additional variables, adding additional variables can increase the precision of your estimates, just as in other regressions. I think that this is part of what Michael Chernick has in mind.

Mostly Harmless Econometrics has a nice discussion that may be helpful. See especially pages 236-37.

Answer 2

Sometimes when we look at a treatment effect by computing the difference on response post treatment ot pretreatment we say that the patient act as his own control. The purpose for providing a control group is to account for the so-called placebo effect. Sometimes there can be a positive change even if the treatment is not applied. So the effect we want to determine is the average increase above the "placebo effect."

Answer 3

Yes, both of your points make sense. To see a derivation of two different flavors of diff-in-diff models, you can see my lecture slides on the topic.

Answer 4

Continuing Michael's answer, you want to provide as much evidence as possible that E[u|treat] = 0. That is an assumption and never directly verifiable, but you want to provide as much trust to the readers that you have thought of why it may hold. Adding controls effectively begins to decompose u. And, some controls may not get at everything you would want but may give you a sense of the type of things that you might not need to worry about. For example, if you had a control for IQ then that might help allay concerns of omitted variables on ability.