Thread by @jmwooldridge

Speaking of two-way FE, it's been under fire for the last few years for estimating treatment effects in DID designs -- especially staggered designs. As many on here know. As an older person, I don't let go of my security blankets so easily.

#metricstotheface

Certainly the simple TWFE estimator that estimates a single coefficient can be misleading. We know this thanks to recent work of several talented econometricians (you know who you are). But maybe we're just not being flexible enough with treatment heterogeneity.

Now when I teach panel data interventions, I start with basic TWFE but note that, with multiple treatment periods and different entry times, we can easily include interactions that allow for many different average treatment effects (on the treated).

The ATTs can vary by exposure (cohort) and calendar date. For example, if we have 4 entry times with irreversibility, we estimate 4 + 3 + 2 + 1 = 10 different effects rather than one. These identify the ATTs for the different exposure levels and time periods.

Not surprisingly, identification requires no anticipation and common trends. I dabbled with this a bit in my 2005 REStat paper, but I didn't do a full analysis of what one can identify with different treatment patterns.

When we introduce covariates -- so that CT holds conditional on covariates as in Callaway and Sant'Anna -- we get further flexibility. With four entry periods and one covariate here are 14 additional interactions.

When the covariates are centered about exposure-specific means, the ATTs for each exposure/time period are easily gotten. With 4 control periods and 4 treatment periods and just a single X, the TWFE includes 4 + 10 + 10 regressors (not including FE dummies).

Why am I not abondoning the TWFE framework? I'm getting old and I'm lazy. But also I know FE has resiliency to unbalanced panels. It has bias on the order of 1/T when strict exogeneity is violated. Estimating unit-specific trends, as in my 2005 REStat, is a clear extension.

So I know that, with multiple pre-treatment periods, I can remove unit-specific trends to at least partly relax the common trends assumption. Another reason for studying FE: the equivalence with the Mundlak regression suggests strategies for nonlinear models.

I'm trying to finish a draft of what seems like mostly an expository paper, with the thrilling title "Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-Differences Estimation." Oh, and I'm preparing for an interview with @causalinf.

A sample (and simple) Stata command with T = 4, two treated periods (3 and 4), staggered, one x:

xtreg y c.e3#c.d2013 c.e3#c.d2014 c.e4#c.d2014 c.e3#c.d2013#c.x_dm3 c.e3#c.d2014#c.x_dm3 c.e4#c.d2014#c.x_dm4 d2013 d2014 c.d2013#c.x c.d2014#c.x, fe vce(cluster id)

I expect I'm about to be taught some things. One is never too old for that ....

The coefficients on the first three terms are the estimated TEs. The ATT for cohort first exposed in 2013 during 2013, the effect for that cohort in 2014, and the effect for cohort first exposed in 2014 during 2014.