Parallel trends assumption?

[Giuseppe Polito]

In the STATA DID intro guide, the parallel trends assumption is written as:

What does it mean?

That the difference post and pre treatment period in the control group, if it was treated, equals the difference post and pre treatment in the control group, given it was not treated?

But if the control group is treated, it would experience a treatment effect, no?

I think it should be written as:

That is, the ATE on the control group if it were treated equals the ATE on the treatment group which was effectively treated?

[Jared Greathouse]

No. Parallel trends says that the average of the control group is a good counterfactual for the treated units. Or, that had the intervention not happened, the expected value of the treated units would move in parallel to that of the control group.

Further implying that there's no confounding between the intervention and the outcome- that's what it means for the expected values of the units given treatment or not to be the same.

I'm sorry I rambled a little, but in English, parallel trends says your control units are good ones for your treated unit. Spoiler alert, in my business, this isn't the case. Not usually. Guaranteed! If you have one treated unit, say, and 88 untreated ones, all 88 of those ain't a good comparison to your treated unit. No chance. A sparse vector of these will be enough... hence, my advocacy for synthetic controls.

[Giuseppe Polito]

Parallel trends says that the average of the control group is a good counterfactual for the treated units.

In the first formula I posted there are no treated units

Or, that had the intervention not happened, the expected value of the treated units would move in parallel to that of the control group

Again, in the first formula there are no treated units

[Øyvind Snilsberg]

the groups are defined by D so the assumption states that the difference between post and pre under control for the treated units are equal to the difference between post and pre under control for the control units.

[Giuseppe Polito]

the groups are defined by D so the assumption states that the difference between post and pre under control for the treated units are equal to the difference between post and pre under control for the control units.

And what is the "g" in $Y_{igt}$?

[Øyvind Snilsberg]

g indicates whether the potential outcomes are stated in terms of treatment (=1) or control (=0). E(Y01|D=1) is the (unobserved) potential outcome under control for the treated units in post.

[George Ford]

The formula says the expected difference for the treated group (D=1) between two periods (00, 01) equals the expected difference for the control group (D=0) between the two periods.