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Jeffrey Wooldridge Profile picture
@jmwooldridge
Mar 24, 2021 • 13 tweets • 3 min read • Read on X
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.

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More from @jmwooldridge

Jeffrey Wooldridge Profile picture
Jan 24
I wish as a profession we would be more careful about tossing around terms like "endogeneity" -- especially with panel data. For many years, I've been emphasizing that the error consists of two components; I call them c(i) and u(i,t). I always include time dummies, say, f(t).
Endogeneity WRT c(i) and f(t) is handled by TWFE. But that leaves u(i,t), the idiosyncratic, time-varying shocks. For that, we generally need IV along with TWFE.

In terms of DiD, the assignment can be correlated with the level in the control state, y_it(0) -- so it can be endog.
Treatment assignment cannot be correlated with the difference or trend, y_is(0) - y_it(0). This is the parallel trends assumption, and adding controls can help. The unobserved effect, c(i), is part of y_it(0) but it gets removed by differencing or the within transformation.
Read 5 tweets
Jeffrey Wooldridge Profile picture
Jan 1
Nice stuff! Pedro knows I'm competitive, and now he's thrown down the gauntlet. I'll to have to clean up my shared Dropbox (see pinned tweet). For starters, I finally have a new version of my extended TWFE paper -- posted there. It's shorter and hopefully more to the point.
Includes a bunch of equivalences that I've discovered over the past few years -- some recent. And I show that the regression-based "event study" approaches of Sun- Abraham/Callaway-Sant'Anna are the same when S-A includes covariates fully flexibly as with my ETWFE method.
Plus, even the event study ("leads and lags") with full flexibility can be computed by imputation estimation. In previous versions, I only showed this for ETWFE and for estimation with heterogeneous trends.
Read 11 tweets
Jeffrey Wooldridge Profile picture
Nov 3, 2024
There's a good reason the Frisch-Waugh-Lovell Theorem is taught in intro econometrics, at least at the graduate level. It's used to characterize omitted variable bias as well as the plim of OLS estimators under treatment heterogeneity and also diff-in-diffs. And more.
I also teach the 2SLS version of FWL, where exogenous variables, X, are partialled out of the IVs, Z, with endogenous explan vars W. It's important to emphasize that the IV needs to be residualized with respect to X. Let Z" be those residuals. This is the key partialling out.
Then apply 2SLS to any of the equations
Y = W*b + U1
Y" = W*b + U2
Y" = W"*b + U3
Y = W"*b + U4
using IVs Z".

All four deliver the 2SLS estimates of b on the full equation Y = X*a + W*b + U with IVs (X,Z). All " variables have X partialled out from them.
Read 8 tweets
Jeffrey Wooldridge Profile picture
Sep 28, 2024
I think the most commonly used treatment effect estimators when treatment, D, is unconfounded conditional on X, are the following:
1. Regression adjustment.
2. Inverse probability (propensity score) weighting.
3. Augmented IPW.
4. IPWRA
5. Covariate matching.
6. PS matching.
RA, AIPW, and IPWRA all use conditional mean functions; usually linear but can be logit, multinomial logit, exponential, and others.

I like RA because it is straightforward -- even if using logit or Poisson -- and it is easy to obtain moderating effects.
But, technically, RA requires correct specification of the conditional means E[Y(d)|X] for consistency.

IPW uses only specification of the PS. We now know we should use normalized weights. IPW can be sensitive to overlap problems because p^(X) can be close to one or zero.
Read 17 tweets
Jeffrey Wooldridge Profile picture
Sep 28, 2024
It's been too long since I've made a substantive tweet, so here goes. At the following Dropbox link you can access the slides and Stata files for my recent talk at the Stata UK meeting:



It's taken me awhile to see connections among various estimators.dropbox.com/scl/fo/50imn36…
Perhaps even longer to figure out some tricks to make standard error calculation for aggregated, weighted effects easy. I think I've figured out several useful relationships and shortcuts. Ex post, most are not surprising. I didn't have them all in my WP or my nonlinear DiD.
The talk is only about regression-based methods, but includes logit and Poisson regression (and even other nonlinear models). In the linear case, slide 28 shows a "very long regression." I was tempted to call it something like the "grand unified regression."
Read 23 tweets
Jeffrey Wooldridge Profile picture
May 25, 2024
Okay, here goes. T = 2 balanced panel data. D defines treated group, f2_t is the second period dummy, W_t = D*f2_t is the treatment. Y_1 and Y_2 are outcomes in the first and second period. ΔY = Y_2 - Y_1. X are time-constant controls. X_dm = X - Xbar_1 (mean of treated units).
Eight equivalent methods:

1. OLS ΔY on 1, D, X, D*X_dm (cross sec)

2. Pooled OLS of Y_t on 1, W_t, W_t*X_dm, D, X, D*X, f2_t, f2_t*X; ATT is coef on W_t (t = 1,2)

3. Random effects estimation with same variables in (2).

4. FE estimation of (2), where D, X, D*X drop out.
Imputation versions of each:

5. OLS ΔY on 1 X using D = 0. Get residuals TE^_FD. Average TE^_FD over treated units.

6. POLS of Y_t on 1, D, X, D*X, f2_t, f2_t*X using W_t = 0 (control obs). TE_t^_POLS resids. ATT is average of TE_t^_POLS over W_t = 1 (treated observations)
Read 6 tweets

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