Thread by @jmwooldridge

On my shared Dropbox folder, pinned at the top, I posted the latest version of my TWFE/TWMundlak paper. It's essentially complete (and too long ...). I've included the "truly marvelous" proof of equivalence between pooled OLS and imputation.

#metricstotheface

I also fixed some of the material on testing/correcting for heterogeneous trends. A nice result is that the POLS approach with cohort-specific trends is the same as the obvious imputation approach.

This means that using the full regression to correct for non-paralled trends suffers no contamination when testing. It's identical to using only untreated obs to test for pre-trends. But one must allow full heterogen in cohort/time ATTs for the equiv to hold.

I'm very pleased with that result. In a nutshell, add interactions of everything that makes sense and use all of the data. That's identical to imputation. Add heterogen time trends. That's identical to imputation. The test for violation of PT is clean.

In the Stata do files (caution: use only the most recent ones!) I show how the equiv of POLS/TWFE with different imputation methods. The POLS approach actually gives more TEs than did_imputation. Or maybe it's just my limited knowledge.

I should mention there is a remaining gap in the paper, but I'm not going to pursue it fully. I showed that POLS and imputation are the same when using cohort dummies -- not unit fixed effects as in BJS (2021). They are the same, but the proof eludes me.

I don't want to do hard algebra. The equivalence proof I do have uses a simple observation about the FOCs that requires only trivial linear algebra. I suspect the proof might follow from the observation that the POLS and FE residuals differ by a time-constant factor.

I bet someone can show it pretty quickly, but my brains isn't working that way anymore.

Plus, I extended the result I do have to the nonlinear case, where putting in individual FEs should not be done. So I don't have the drive to pursue that result.