Thread by @jmwooldridge

I finally got my TWFE/Mundlak/DID paper in good enough shape to make it an official working paper. I'll put it in other places but it's currently here:

researchgate.net/publication/35…

Also, the Stata stuff is still with the Dropbox link:

dropbox.com/sh/zj91darudf2…

#metricstotheface

I changed the title a bit to better reflect it's contents. I'm really happy with the results, less happy that the paper got a bit unwieldy. It's intended to be a "low hanging fruit" DID paper.

Now I've more formally shown that the estimator I was proposing -- either pooled OLS or TWFE or RE (they're all the same, properly done) identifies every dynamic treatment one is interested in (on means) in a staggered design.

I formalized the no anticipation and common trend assumptions to make them simple and comparable to the literature. The method involves lots of regressors but it's linear. Plus, it is both BLUE and asymp. effic. under a stand no ser correl/homosk assumption.

Combining efffects and testing restrictions on them is as simple as the Stata "test" and "lincom" commands. If you by NA and conditional parallel trends, linearity of the cond means in the covariates is the only shortcoming.

Unlike other previous work, I show what is being identified by the POLS/ETWFE estimator when there is no never treated group. It's quite neat under a natural extension of the PT assumption. In the final period, you get the ATTs of being treated before the final period.

In other approaches, such as Callaway and Sant'Anna, no effects are estimated in the final period. Using POLS/ETWFE, only one parameter is lost: there is not control group for the cohort treated in the final unit.

In preparing a DID course, I discovered some other neat stuff. When you impose PT and allow flexible interactions with covariates the way I do, the POLS estimator is numerically identical to the comparable imputation estimator in Borusyak, Jaravel, Spiess (2021). @borusyak

“Revisiting Event Study Designs: Robust
and Efficient Estimation." I've only partly proven this algebraically. Some may remember my reference to a "proof by Stata" a couple of weeks ago .... POLS makes it trivial to obtain valid inference when N is large, T not too large.

Since BJS showed efficiency of their estimator (albeit under diff assumps), it's ex post not too surprising the methods coincide in the PT case. Could we have two BLUEs?

The POLS approach extends naturally to nonlinear models, but I'm writing a separate paper on that.

Another important point is that, once we know POLS uses the NA/PT assumptions efficiently, we can look for better ways to do doubly robust estimation. Methods that use long differences are inefficient.

Love the Callaway/Sant'Anna work, but long differences throw out useful information, especially in later periods when there are many potential controls or with many pre-treatment periods. @pedrohcgs

POLS uses all valid control units for each cohort/year combo.

A first simulation shows Callaway/Sant'Anna SDs 30% larger than POLS. The bias of each is negligible. Yes, it was set up with linear cond means. And I used csdid in Stata.

There's more: I show how PT can be easily tested while allowing for flexible covariates.

Simply robust Wald tests using the Stata "test" command. It does not get around the pre-testing problem (@jondr44) but it makes it transparent: it's the age old problem of pre-testing a set of regressors.

If the controls are supposed to induce PT, condition on them in testing.

As I suspected would happen, things have come full circle for me. Take the disgraced TWFE estimator, add full heterogeneous effects, allow hetero in covariates across cohort, time, and treatment. Combine and test as you wish. It's incredibly simple, flexible; can be effic.

With decent overlap I'm not too worried about the linear conditional expectation. Don't mess with OLS.

Oh, wait, I am worried about linearity -- that's why I'm working on nonlinear extensions (binary, fractional, nonnegative).