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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.

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The need for additional control variables can depends on whether the the treatment group was selected at random from a larger group with the remainder becoming controls, or (as is more often the case in post-hoc analysis) because of of some specific features. – Henry Aug 13 '12 at 16:44

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.

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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."

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Hi Michael, thanks for your answer. I think I understand why we need control groups. The control group is incorporated in my regression equation as those who don't have treat=1. So that's not really the question here. The question is why some papers use additional control variables on top of the equation stated above. Would be great if you could answer on that or maybe someone else. Thanks guys! – sachin Aug 12 '12 at 14:22
Why do you call teh additional variables control variables? The only reason I could see including additional variables in the model would be that the variables can account for some of the variation in the response that wasn't explained by the other variables in the model. – Michael Chernick Aug 12 '12 at 14:27
Well, that's basically my question: Why include these variables (i.e. control variables that are included because, as you say, they could explain something we claim the treatment explains) when assuming that the parallel trend assumption holds? I could only assume that including further controls means to relax that assumption - i.e. to see how much the treatment can explain, even when controlling for other variables. This could be a consequence of not being able to fully test the parallel trend assumption and could convince the reader more about the effect of treatment. But not sure about that – sachin Aug 12 '12 at 16:36
The effect on the response does not have to come solely from treatment. I am saying that other variables might be able to explain variation in the response that are independent of treatment. It doesn't have to have anything to do with treatment interacting with anything. – Michael Chernick Aug 12 '12 at 16:44

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.

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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.

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