Difference-in-Differences: No Anticipation with Parallel Trends (with a simulation and equations)
In a Mixtape Mailbag recently, I talked through what happens if you have parallel trends and SUTVA but not “no anticipation”. What is “no anticipation”? No anticipation simply means that the baseline period is equal to Y(0) not Y(1). In other words, the baseline is not treated.
A reader had written about that, and it was nice because I had actually never just sat down and worked through precisely what happens. And so here’s what I found in short order. When you substitute in all the potential outcome terms, but no longer satisfy the no anticipation assumption, here’s what you find. Let T be the post-treatment period, and D be the treatment group.
The second and third rows are your parallel trends terms, which I’ll just call below the “non PT bias”. It’s biased, in other words, if that’s not zero and it’s unbiased it if is. The fourth term is the mean of Y0-Y1 in the baseline for the treatment group, a reversed ATT. So I move the negative to the exterior and express it as a pro…
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