Not sure about that! But here's a first attempt. Suppose I have a control group and G treatment levels. The treatment, W, is in {0,1,2,...,G} is unconfounded conditional on X. Assume the overlap condition 0 < p0(x) = P(W=0|X=x) for all x in Support(X).
https://twitter.com/himbonomics/status/1535056821584465920
This isn't a trivial assumption b/c it requires that for and subset of the population as determined by values of x, there are some control units. However, if this isn't true, one can trim the sample -- as in the Crump et al. "Moving the Goalposts" work.
If overlap holds and conditional means are linear, the following regression recovers the ATTs of each group g relative to control:
Y on 1, W1, W2, ... WG, X, W1*(X - Xbar1), W2*(X - Xbar2), ..., WG*(X - XbarG) where Xbarg is the sample average of treatment group g.
Y on 1, W1, W2, ... WG, X, W1*(X - Xbar1), W2*(X - Xbar2), ..., WG*(X - XbarG) where Xbarg is the sample average of treatment group g.
If we don't like linear regression, replace with logit if Y is binary or fractional; multinomial if Y is multinomial or fractional with more than two outcomes (recent paper with Akanksha Negi). If Y is nonnegative (count, corner) use Poisson.
My view is we can estimate these ATTs very generally and these parameters are interesting. This is very similar to what underlies my extended TWFE DiD work. We have to worry about overlap, but having to trim out parts of the population with no controls is not surprising.
The previous regression adjustment is, of course, different from
Y on 1, W1, W2, ..., WG, X
and we know from @TymonSloczynski, the DiD literature, and now @paulgp, @instrumenthull, and Kolesar that this can be very misleading.
Y on 1, W1, W2, ..., WG, X
and we know from @TymonSloczynski, the DiD literature, and now @paulgp, @instrumenthull, and Kolesar that this can be very misleading.
Plus, the full (separate) RA methods extend to doubly robust estimators that combine separate linear/logit/MNL/Poisson means with inverse probability weighting (obtained via MNL).
I certainly hope labor economists don't abandon these methods.
#metricstotheface
I certainly hope labor economists don't abandon these methods.
#metricstotheface
BTW, as a practical matter, I don't think Stata's teffects supports estimation of the ATTs. It provides estimates of the ATEs E[Y(g) - Y(0)], and this requires both stronger unconfoundedness and overlap assumptions. But the regressions are easy to do "by hand."
Standard errors that condition on the covariates follow immediately, and it might be possible to use the vce(uncond) option to account for sampling error in the Xbarg.
BTW, I can recommend the paper by @TymonSloczynski and me in Econometric Theory (2018) for the doubly robust stuff. 😬
• • •
Missing some Tweet in this thread? You can try to
force a refresh
