Two-way Fixed Effects Regressions with Several Treatments

TWFE regressions may produce misleading estimates if the policy's effect is heterogeneous between groups or over time.

“It has recently been shown that those regressions may produce misleading estimates, if the policy's effect is heterogeneous between groups or over time, as is often the case.”

The decomposition in (2.2) is the main result in de Chaisemartin and D'Haultfœuille (2020).

“The decomposition in (2.2) is the main result in de Chaisemartin and D'Haultfœuille (2020).”

Similar results apply to beta_fe: more than half of the weights attached to that coefficient are negative, and negative weights sum to -0.53.

“Similar results apply to ̂ β fe : more than half of the weights attached to that coefficient are negative, and negative weights sum to -0.53.”

Under a parallel trends assumption on the potential outcome without treatment, the TWFE coefficient beta_fe equals a weighted sum of treatment effects TE_g,t with weights W_g,t summing to 1.

“de Chaisemartin and D'Haultfœuille (2020) show that under a parallel trends assumption on the potential outcome without treatment Y g,t (0),”

Under parallel trends, beta_fd estimates a weighted sum of the effects of newspapers on turnout in 10,077 county × election cells, where 5,472 effects are weighted positively, 4,605 are weighted negatively, and negative weights sum to -1.43.

“Using the twowayfeweights Stata package, de Chaisemartin and D'Haultfœuille (2020) find that under parallel trends, ̂ β fd estimates a weighted sum of the effects of newspapers on turnout in 10,077 county × election cells, where 5,472 effects are weighted positively while 4,605 are weighted negatively, and where negative weights sum to -1.43.”

de Chaisemartin and D'Haultfœuille (2020) find beta_fd = 0.0026 (s.e. = 0.0009): one more newspaper increases turnout by 0.26 percentage points.

“They find that ̂ β fd = 0 . 0026 (s.e. = 0 . 0009): one more newspaper increases turnout by 0.26 percentage points.”

A decomposition similar to (2.2) can be obtained for TWFE regressions with control variables, and for beta_fd, under common trends and under the assumption that the treatment effect does not change over time.

“A decomposition similar to (2.2) can be obtained for TWFE regressions with control variables, and for ̂ β fd , the treatment's coefficient in a regression of the outcome's first difference on the treatment's first difference and period fixed effects. de Chaisemartin and D'Haultfœuille (2020) also derive decompositions similar to (2.2), for ̂ β fe and ̂ β fd , under common trends and under the”

The weights are negatively correlated with the election year: beta_fd is more likely to upweight newspapers' effects in early elections, and to downweight or weight negatively newspapers' effects in late elections.

“The weights are negatively correlated with the election year: ̂ β fd is more likely to upweight newspapers' effects in early elections, and to downweight or weight negatively newspapers' effects in late elections.”

DID- is unbiased for the treatment effect of the switching-out groups at period 2, under a parallel trends assumption on the treated outcome Y_g,t(1).

“Then, one can show that DID -is unbiased for the treatment effect of the switching-out groups at period 2, under a parallel trends assumption on the treated outcome Y g,t (1).”

DID+ is unbiased for the treatment effect of the switching-in groups at period 2, under a parallel trends assumption on the untreated outcome Y_g,t(0).

“It is similar to the DID estimator in Equation (1.1), and it is unbiased for the treatment effect of the switching-in groups at period 2, under a parallel trends assumption on the untreated outcome Y g,t (0).”

In the Gentzkow et al. (2011) example, de Chaisemartin and D'Haultfœuille (2020) find DID_M = 0.0043 (s.e. = 0.0014), meaning that one more newspaper increases turnout by 0.43 percentage point; DID_M is 66% larger than, and significantly different from, β_fd, the estimator reported by Gentzkow et al. (2011).

“de Chaisemartin and D'Haultfœuille (2020) find that DID M = 0 . 0043 (s.e. = 0 . 0014), meaning that one more newspaper increases turnout by 0.43 percentage point. DID M is 66% larger than, and significantly different from, ̂ β fd , the estimator reported by Gentzkow et al. (2011).”

In staggered adoption designs with a binary treatment, the DID_M estimator proposed by de Chaisemartin and D'Haultfœuille (2020) also uses the not-yet-treated as controls, and is identical to the DID_0 estimator of the instantaneous treatment effect using the not-yet-treated as controls in Callaway and Sant'Anna (2021).

“Note that in staggered adoption designs with a binary treatment, the DID M estimator proposed by de Chaisemartin and D'Haultfœuille (2020) also uses the not-yet-treated as controls, and is identical to the DID 0 estimator of the instantaneous treatment effect using the not-yet-treated as controls in Callaway and Sant'Anna (2021).”

de Chaisemartin and D'Haultfœuille (2021b) propose both estimators that generalize the DID_M estimator in de Chaisemartin and D'Haultfœuille (2020) and rule out dynamic effects, and estimators that generalize those in Callaway and Sant'Anna (2021) and allow for dynamic effects.

“They propose both estimators that generalize the DID M estimator in de Chaisemartin and D'Haultfœuille (2020) and rule out dynamic effects, and estimators that generalize those in Callaway and Sant'Anna (2021) and allow for dynamic effects.”

Standard TWFE coefficients may not represent a straightforward weighted average of unit-level treatment effects when treatment effects are heterogeneous.

“Multiple authors have noted that the coefficients from standard TWFE models may not represent a straightforward weighted average of unit-level treatment effects when treatment effects are allowed to be heterogeneous.”

de Chaisemartin and D'Haultfœuille (2020) propose reporting the number or fraction of group-time ATTs that receive negative weights, as well as the degree of heterogeneity in treatment effects necessary for the estimated treatment effect to have the wrong sign.

“de Chaisemartin and D'Haultfœuille (2020) propose reporting the number/fraction of group-time ATTs that receive negative weights, as well as the degree of heterogeneity in treatment effects that would be necessary for the estimated treatment effect to have the 'wrong sign.'”

When there is heterogeneity in time since treatment only, βpost corresponds with a potentially non-convex weighted average of the parameters τs, where weights sum to 1 but may be negative.

“In this case, β post corresponds with a potentially non-convex weighted average of the parameters τ s , i.e. β post ' ř s ω s τ s , where the weights ω s sum to 1 but may be negative.”

This approach yields an interpretable causal effect under generalizations of the parallel trends assumption and an additional no carryover assumption that imposes that potential outcomes depend only on current treatment status and not on the full treatment history.

“This approach yields an interpretable causal effect under generalizations of the parallel trends assumption and and an additional 'no carryover' assumption that imposes that the potential outcomes depend only on current treatment status and not on the full treatment history.”

de Chaisemartin and D'Haultfœuille (2020) propose an estimator that can be applied when treatment turns on and off, but in the staggered setting corresponds with the Callaway and Sant'Anna estimator for ATT_w0 and a particular choice of weights.

“de Chaisemartin and D'Haultfœuille (2020) propose an estimator that can be applied when treatment turns on and off (see Section 3.4 below), but in the context of the staggered setting here corresponds with the Callaway and Sant'Anna estimator for ATT w 0 and a particular choice of weights.”

de Chaisemartin and D'Haultfœuille (2020) and Imai and Kim (2021) consider settings where units are treated at different times, but do not necessarily require that treatment is an absorbing state.

“de Chaisemartin and D'Haultfœuille (2020) and Imai and Kim (2021) consider settings where units are treated at different times, but do not necessarily require that treatment is an absorbing state.”

Their estimators compare changes in outcomes for units whose treatment status changed to other units whose treatment status remained constant over the same periods.

“Their estimators intuitively compare changes in outcomes for units whose treatment status changed to other units whose treatment status remained constant over the same periods.”

de Chaisemartin and D'Haultfœuille (2020, 2022) and Borusyak et al. (2021) consider estimators which include covariates in a linear manner.

“de Chaisemartin and D'Haultfœuille (2020); de Chaisemartin and D'Haultfoeuille (2022) and Borusyak et al. (2021) consider estimators which include covariates in a linear manner.”