Setting

Exploit a data structure where the researcher observes:

1. Multiple periods of data

2. Some pre-treatment data for all units

3. Some units become treated while other units remain untreated

High-Level Thoughts

Panel data gives researchers the opportunity to follow the same person, firm, location, etc. over multiple time periods

Treatment Effect Heterogeneity

Modern approaches also typically allow for treatment effect heterogeneity

• That is, that effects of the treatment can vary across different units in potentially complicated ways

This is going to be a major issue in the discussion below

We’ll consider implications for “traditional” regression approaches and how new approaches are designed to handle this

Notation for Setting with Two Periods

Data:

• 2 periods: \(t=1\), \(t=2\)

  • No one treated until period \(t=2\)
  • Some units remain untreated in period \(t=2\)

• \(D_{i,t}\) treatment indicator in period \(t\)

• 2 groups: \(G_i=1\) or \(G_i=0\) (treated and untreated)

Target Parameter

Average Treatment Effect on the Treated: \[ATT = \E[Y_{i,t=2}(1) - Y_{i,t=2}(0) | G_i=1]\]

Explanation: Mean difference between treated and untreated potential outcomes in the second period among the treated group

How to Use Panel Data to Learn about \(ATT\)

Notice that: \[\begin{align*} ATT = \underbrace{\E[Y_{i,t=2}(1) | G_i=1]}_{\textrm{Easy}} - \underbrace{\E[Y_{i,t=2}(0) | G_i=1]}_{\textrm{Hard}} \end{align*}\]

How to Use Panel Data to Learn about \(ATT\)

Notice that: \[\begin{align*} ATT = \underbrace{\E[Y_{i,t=2}(1) | G_i=1]}_{\textrm{Easy}} - \underbrace{\E[Y_{i,t=2}(0) | G_i=1]}_{\textrm{Hard}} \end{align*}\]

With panel data, we can re-write this as

\[\begin{align*} ATT = \color{green}{\E[Y_{i,t=2}(1) - Y_{i,t=1}(0) | G_i=1]} - \color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} \end{align*}\]

The second term is how outcomes would have changed over time if the treated group had not been treated

• This is not directly observed in the data \(\implies\) we need to make identifying assumptions

How to Use Panel Data to Learn about \(ATT\)

Notice that: \[\begin{align*} ATT = \underbrace{\E[Y_{i,t=2}(1) | G_i=1]}_{\textrm{Easy}} - \underbrace{\E[Y_{i,t=2}(0) | G_i=1]}_{\textrm{Hard}} \end{align*}\]

With panel data, we can re-write this as

\[\begin{align*} ATT = \color{green}{\E[Y_{i,t=2}(1) - Y_{i,t=1}(0) | G_i=1]} - \color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} \end{align*}\]

The second term is how outcomes would have changed over time if the treated group had not been treated

• This is not directly observed in the data \(\implies\) we need to make identifying assumptions

• There are many possibilities here:

  2. Lagged outcome unconfoundedness: \(\color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} = \E\Big[ \E[Y_{i,t=2}(0) | Y_{i,t=1}, G_i=0] - Y_{i,t=1}(0) \Big| G_i=1\Big]\)

How to Use Panel Data to Learn about \(ATT\)

Notice that: \[\begin{align*} ATT = \underbrace{\E[Y_{i,t=2}(1) | G_i=1]}_{\textrm{Easy}} - \underbrace{\E[Y_{i,t=2}(0) | G_i=1]}_{\textrm{Hard}} \end{align*}\]

With panel data, we can re-write this as

\[\begin{align*} ATT = \color{green}{\E[Y_{i,t=2}(1) - Y_{i,t=1}(0) | G_i=1]} - \color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} \end{align*}\]

The second term is how outcomes would have changed over time if the treated group had not been treated

• This is not directly observed in the data \(\implies\) we need to make identifying assumptions

• There are many possibilities here:

  3. Change-in-changes: \(\color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} = \E\Big[ Q_{Y_{i,t=2}(0)|G_i=0}\big(F_{Y_{i,t=1}(0)|G_i=0}(Y_{i,t=1}(0))\big) - Y_{i,t=1}(0) \Big| G_i=1\Big]\)

How to Use Panel Data to Learn about \(ATT\)

Notice that: \[\begin{align*} ATT = \underbrace{\E[Y_{i,t=2}(1) | G_i=1]}_{\textrm{Easy}} - \underbrace{\E[Y_{i,t=2}(0) | G_i=1]}_{\textrm{Hard}} \end{align*}\]

With panel data, we can re-write this as

\[\begin{align*} ATT = \color{green}{\E[Y_{i,t=2}(1) - Y_{i,t=1}(0) | G_i=1]} - \color{red}{\E[Y_{i,t=2}(0) - Y_{i,t=1}(0) | G_i=1]} \end{align*}\]

The second term is how outcomes would have changed over time if the treated group had not been treated

• This is not directly observed in the data \(\implies\) we need to make identifying assumptions

• There are many possibilities here:

  4. Difference-in-differences: ➡

DID with Two Periods

Explanation: Mean path of untreated potential outcomes is the same for the treated group as for the untreated group

DID with Two Periods

Explanation: Mean path of untreated potential outcomes is the same for the treated group as for the untreated group

Identification: Under PTA, we can identify \(ATT\): \[ \begin{aligned} ATT &= \E[\Delta Y_i | G_i=1] - \E[\Delta Y_i(0) | G_i=1]\\ &= \E[\Delta Y_i | G_i=1] - \E[\Delta Y_i | G_i=0] \end{aligned} \]

Estimation

The most straightforward approach to estimation is plugin:

\[\widehat{ATT} = \frac{1}{n_1} \sum_{i=1}^n G_i \Delta Y_i - \frac{1}{n_0} \sum_{i=1}^n (1-G_i) \Delta Y_i\]

TWFE Regression

It’s easy to make the TWFE regression more complicated:

• Multiple time periods

• Variation in treatment timing

• More complicated treatments

• Introducing additional covariates

Unfortunately, the robustness of TWFE regressions to treatment effect heterogeneity or these more complicated (and empirically relevant) settings does not seem to hold

Setup with Staggered Treatment Adoption

\(\T\) time periods

Staggered treatment adoption: Units can become treated at different points in time, but once a unit becomes treated, it remains treated.

Setup with Staggered Treatment Adoption

Notation (cont’d):

• Potential outcomes: \(Y_{i,t}(g)\) — the outcome that unit \(i\) would experience in time period \(t\) if they became treated in period \(g\).

Target Parameters

Group-time average treatment effects \[\begin{align*} ATT(g,t) = \E[Y_{i,t}(g) - Y_{i,t}(0) | G_i=g] \end{align*}\]

Explanation: \(ATT\) for group \(g\) in time period \(t\)

Target Parameters

Overall ATT

Towards this end: the average treatment effect for unit \(i\) (across its post-treatment time periods) is given by: \[\bar{\tau}_i(g) = \frac{1}{\T - g + 1} \sum_{t=g}^{\T} \Big( Y_{i,t}(g) - Y_{i,t}(0) \Big)\]

Target Parameters

To understand the discussion later, it is also helpful to think of \(ATT(g,t)\) as a building block for the other parameters discussed above. In particular:

DID Identification of \(ATT(g,t)\)

Summary

The previous discussion emphasizes a general purpose identification strategy with staggered treatment adoption:

What Can Go Wrong with TWFE Regression?

With staggered treatments, traditionally DID identification strategies have been implemented with two-way fixed effects (TWFE) regressions: \[\begin{align*} Y_{i,t} = \theta_t + \eta_i + \alpha D_{i,t} + e_{i,t} \end{align*}\]

One main contribution of recent work on DID has been to diagnose and understand the limitations of TWFE regressions for implementing DID

What Can Go Wrong with TWFE Regression?

de Chaisemartin and D’Haultfœuille (2020) intuition: You can write \(\alpha\) as a weighted average of \(ATT(g,t)\)

First, a decomposition: \[\begin{align*} \alpha &= \sum_{g \in \bar{\mathcal{G}}} \sum_{t=g}^{\T} w^{TWFE}(g,t) \Big( \E[(Y_{i,t} - Y_{i,g-1}) | G_i=g] - \E[(Y_{i,t} - Y_{i,g-1}) | U_i=1] \Big) \\ & + \sum_{g \in \bar{\mathcal{G}}} \sum_{t=1}^{g-1} w^{TWFE}(g,t) \Big( \E[(Y_{i,t} - Y_{i,g-1}) | G_i=g] - \E[(Y_{i,t} - Y_{i,g-1}) | U_i=1] \Big) \end{align*}\]

New Approaches

We’ll discuss:

1. Callaway and Sant’Anna (2021), R: did, Stata: csdid, Python: csdid

2. Sun and Abraham (2021), R: fixest, Stata: eventstudyinteract

3. Wooldridge (2021), R: etwfe, Stata: JWDID

4. Gardner et al. (2023) / Borusyak, Jaravel, and Spiess (2023), R: did2s, Stata: did2s and did_imputation

Callaway and Sant’Anna (2021)

Intuition: Directly implement the identification result discussed above

Sun and Abraham (2021)

Intuition: Paper points out limitations of event-study versions of the TWFE regressions discussed above:

\[\begin{align*} Y_{i,t} = \theta_t + \eta_i + \sum_{e=-(\T-1)}^{-2} \beta_e D_{i,t}^e + \sum_{e=0}^{\T} \beta_e D_{i,t}^e + e_{i,t} \end{align*}\]

and points out similar issues. In particular, the event study regression is “underspecified” \(\implies\) heterogeneous effects can “confound” the treatment effect estimates

Wooldridge (2021)

Intuition: Are issues in DID literature due to limitations of TWFE regressions per se or due to misspecification of TWFE regression?

Gardner et al. (2023) / BJS (2023)

Intuition: Parallel trends is closely connected to a TWFE model for untreated potential outcomes \[Y_{i,t}(0) = \theta_t + \eta_i + e_{i,t}\]

Similarities and Differences

In my view, all of the approaches discussed above are fundamentally similar to each other.

Comparison 1: CS and SA

In post-treatment periods, these give numerically identical results: \(\widehat{ATT}^{CS}(g,t) = \hat{\delta}^{SA}_{t,t-g}\)

• This is because a fully interacted regression (SA) is equivalent to taking differences in averages across groups (CS)

In pre-treatment periods, code will give different pre-treatment estimates, but this is due to different default choices

• In SA, all results are relative to a fixed base period (typically the period right before treatment)

• In CS, by default, in pre-treatment periods, estimates are of placebo policy effects on impact (i.e., the base period is always the most recent pre-treatment period)

Comparison 2: SA and Wooldridge

These are clearly closely related, with the difference amounting to whether or not one includes indicators for pre-treatment periods.

Comparison 3: Wooldridge and Gardner/BJS

Wooldridge and Gardner/BJS give numerically the same estimates: \(\hat{\alpha}^W_{gt} = \widehat{ATT}^{G/BJS}(g,t)\)

Intuition: Including full set of interactions is equivalent to estimating separate models by groups

Comments

The above discussion emphasizes the conceptual similarities between different proposed alternatives to TWFE regressions in the literature.

Comments

The above discussion emphasizes the conceptual similarities between different proposed alternatives to TWFE regressions in the literature.

The other major source of differences in estimates across procedures is different default options in software implementations. Examples:

2. Overall average treatment effects
   • CS emphasizes the “overall” treatment effects discussed above
   • Default implementations of imputation run a regression of \(Y_{i,t}-\hat{Y}_{i,t}(0)\) on \(D_{i,t}\) which delivers the “simple” overall average treatment effect which just averages all available treatment effects

Empirical Example: Minimum Wages and Employment

• Use county-level data from 2003-2007 during a period where the federal minimum wage was flat

Code

Full code is available on GitHub.

R packages used in empirical example

library(did)
library(BMisc)
library(twfeweights)
library(fixest)
library(modelsummary)
library(ggplot2)
load(url("https://github.com/bcallaway11/did_chapter/raw/master/mw_data_ch2.RData"))

Setup Data

# drops NE region and a couple of small groups
mw_data_ch2 <- subset(mw_data_ch2, (G %in% c(2004,2006,2007,0)) & (region != "1"))
head(mw_data_ch2[,c("id","year","G","lemp","lpop","lavg_pay","region")])

      id year    G     lemp     lpop lavg_pay region
554 8003 2001 2007 5.556828 9.614137 10.05750      4
555 8003 2002 2007 5.356586 9.623972 10.09712      4
556 8003 2003 2007 5.389072 9.620859 10.10761      4
557 8003 2004 2007 5.356586 9.626548 10.14034      4
558 8003 2005 2007 5.303305 9.637958 10.17550      4
559 8003 2006 2007 5.342334 9.633056 10.21859      4

# drop 2007 as these are right before fed. minimum wage change
data2 <- subset(mw_data_ch2, G!=2007 & year >= 2003)
# keep 2007 => larger sample size
data3 <- subset(mw_data_ch2, year >= 2003)

TWFE Regression

twfe_res2 <- fixest::feols(lemp ~ post | id + year,
                           data=data2,
                           cluster="id")

modelsummary(list(twfe_res2), gof_omit=".*")

\(ATT(g,t)\) (Callaway and Sant’Anna)

attgt <- did::att_gt(yname="lemp",
                     idname="id",
                     gname="G",
                     tname="year",
                     data=data2,
                     control_group="nevertreated",
                     base_period="universal")
tidy(attgt)[,1:5] # print results, drop some extra columns

             term group time    estimate   std.error
1  ATT(2004,2003)  2004 2003  0.00000000          NA
2  ATT(2004,2004)  2004 2004 -0.03266653 0.020267498
3  ATT(2004,2005)  2004 2005 -0.06827991 0.022146742
4  ATT(2004,2006)  2004 2006 -0.12335404 0.020966176
5  ATT(2004,2007)  2004 2007 -0.13109136 0.021773887
6  ATT(2006,2003)  2006 2003 -0.03408910 0.012051925
7  ATT(2006,2004)  2006 2004 -0.01669977 0.008003650
8  ATT(2006,2005)  2006 2005  0.00000000          NA
9  ATT(2006,2006)  2006 2006 -0.01939335 0.008790647
10 ATT(2006,2007)  2006 2007 -0.06607568 0.009371268

Plot \(ATT(g,t)\)’s

Compute \(ATT^o\)

attO <- did::aggte(attgt, type="group")
summary(attO)

Call:
did::aggte(MP = attgt, type = "group")

Reference: Callaway, Brantly and Pedro H.C. Sant'Anna.  "Difference-in-Differences with Multiple Time Periods." Journal of Econometrics, Vol. 225, No. 2, pp. 200-230, 2021. <https://doi.org/10.1016/j.jeconom.2020.12.001>, <https://arxiv.org/abs/1803.09015> 


Overall summary of ATT's based on group/cohort aggregation:  
     ATT    Std. Error     [ 95%  Conf. Int.]  
 -0.0571        0.0084    -0.0735     -0.0406 *


Group Effects:
 Group Estimate Std. Error [95% Simult.  Conf. Band]  
  2004  -0.0888     0.0201       -0.1320     -0.0457 *
  2006  -0.0427     0.0078       -0.0593     -0.0261 *
---
Signif. codes: `*' confidence band does not cover 0

Control Group:  Never Treated,  Anticipation Periods:  0
Estimation Method:  Doubly Robust

Comments

The differences between the CS estimates and the TWFE estimates are fairly large here: the CS estimate is about 50% larger than the TWFE estimate, though results are qualitatively similar.

de Chaisemartin and d’Haultfoeuille weights

\(ATT^o\) weights

Weight Comparison

Discussion

To summarize: \(ATT^o = -0.057\) while \(\alpha^{TWFE} = -0.038\). This difference can be fully accounted for

• Pre-treatment differences in paths of outcomes across groups: explains about 64% of the difference

• Differences in weights applied to the same post-treatment \(ATT(g,t)\): explains about 36% of the difference. [If you apply the post-treatment weights and “zero out” pre-treatment differences, the estimate would be \(-0.050\).]

Additional Comments on Weights

One more comment: there is a lot concern about negative weights (both in econometrics and empirical work).

Discussion

Not a Panacea

• Good properties of new approaches are conditional on parallel trends holding

Advantages:

• Off-the-shelf robust to treatment effect heterogeneity

• Arguably, simpler and more transparent

• Direct implementation of the DID identification strategy

Possible Disadvantages:

• Not yet available for all possible types of treatments where DID identification strategies are used

Introduction

There are a number of complications that can arise for inference in DID settings:

1. Serial correlation

2. Fixed population inference

3. Clustered treatment assignment

4. Multiple hypothesis testing

5. Small number of treated units

Serial Correlation

Probably the most common inference issue in DID settings is serial correlation.

Sample consists of:

\[\{Y_{i,1}, Y_{i,2}, \ldots, Y_{i,T}, D_{i,1}, D_{i,2}, \ldots D_{i,T} \}_{i=1}^n\]

which are iid across units, drawn from a “super-population”, and where the number of units in each group is “large”. This is a common sampling scheme in panel data applications (e.g., job displacement).

This sampling scheme allows for outcomes to be arbitrarily correlated across time periods.

\[Y_{i,t}(0) = \theta_t + \eta_i + e_{i,t}\]

• Ignoring serial correlation can lead to incorrect standard errors and confidence intervals (Bertrand, Duflo, and Mullainathan (2004)).

• Instead of modeling the serial correlation, it is most common to cluster at the unit level (i.e., allow for arbitrary serial correlation within units).

• Most (all?) software implementations can accommodate serial correlation (often by default).

Fixed Population Inference

The previous discussion has emphasized traditional sampling uncertainty arising from drawing a sample from an underlying super-population.

In many DID applications, we observe the entire population of interest (e.g., all 50 states).

• Does this means that we “know” the \(ATT\)?…probably not.

Fixed Population Inference

One possibility is to condition on (i.e., treat as fixed/non-random) \((G_i, \theta_t, \eta_i)\) while treating as random \(e_{i,t}\).

• Intuition: repeated sampling thought experiment where we redraw \(e_{i,t} \sim (\mu_i, \sigma^2_i)\) for each unit \(i\) and time period \(t\) while unit fixed effects, time fixed effects, and treatment status are held fixed.

• Adjusted target parameter: mean of finite sample \(ATT\) across repeated samples \[ATT(g,t) := \frac{1}{n_g} \sum_{G_i=g} \E[Y_{i,t} - Y_{i,t}(0) | G_i=g]\]

• Adjusted parallel trends: mean of finite sample parallel trends across repeated samples \[\frac{1}{n_g} \sum_{G_i=g} \E[\Delta Y_{i,t}(0) | G_i=g] - \frac{1}{n_\infty} \sum_{G_i=\infty} \E[\Delta Y_{i,t}(0) | U_i=1]\]

\(\implies\) somewhat different interpretation, but use same \(\widehat{ATT}\) and constructing unconditional standard errors (i.e., same as before) can be conservative in this case. See Borusyak, Jaravel, and Spiess (2023), for example.

Clustered Treatment Assignment

Many DID applications have clustered treatment assignment—all units within the same cluster (e.g., a state) have the same treatment status/timing.

The most common approach in this setting is to cluster at the level of the treatment.

• When we additionally cluster at the unit level (to allow for serial correlation), this is often referred to as two-way clustering. Most software implementations readily allow for this.

To rationalize conducting inference in this way typically requires a large number of both treated and untreated clusters.

• This rules out certain leading applications such as the Card and Krueger (1994) minimum wage study, where the only clusters are New Jersey and Pennsylvania.

Clustered Treatment Assignment

Can we conduct inference in cases where there are only two clusters (but we observe underlying data)?

• Yes, but it may require additional assumptions.

• Let’s start with the setting where there are exactly two clusters. Then (for simplicity just focusing on untreated potential outcomes), \[\begin{align*} Y_{i,j,t}(0) &= \theta_t + \eta_i + e_{i,j,t} \\ &= \theta_t + \eta_i + \underbrace{\nu_{j,t} + \epsilon_{i,j,t}} \end{align*}\] where \(\nu_{j,t}\) is a cluster-specific time-varying error term and \(\epsilon_{i,j,t}\) are idiosyncratic, time-varying unobservables (possibly serially correlated but independent across units).

• \(\nu_{j,t}\) is often viewed as a common cluster-specific shock that affects all units in the cluster (e.g., some other state-level policy).

• For inference with two clusters, we need \(\nu_{j,t}=0\).

  • Without this condition, parallel trends would be violated.
  • If holds, then we can cluster at the unit-level rather than the cluster-level \(\implies\) possible to conduct inference.

Comparing Different Strategies

Suppose that we are studying a policy that is implemented in a few states.

Option 1: Include a large number of states, cluster at state-level

• For clustering: We may only need the weaker condition that \(\E[\nu_{j,t} | G_i] = 0\).

• For identification: less robust to small violations of parallel trends.

Option 2: Only include a few similar states, cluster at unit-level

• For clustering: We need the stronger condition that \(\nu_{j,t}=0\).

• For identification: possibly more robust to small violations of parallel trends.

In my view, there is a fairly strong case for (in many applications) using tighter comparisons with a smaller number of clusters while arguing that \(\nu_{j,t}=0\) by a combination of

• researcher legwork

• pre-testing

Few Treated Clusters with Aggregate Data

The discussion above supposed that we had access to underlying unit-level data. What if we only have access to aggregate data, such as state-level data, and the number of treated clusters is small (e.g., 1 treated cluster)?

In this case, if there is one treated unit, we can often come up with an unbiased (though not consistent) estimator of the \(ATT\).

For inference, there are a number of ideas, often involving some kind of permutation test

• See, for example, T. G. Conley and Taber (2011), Ferman (2019), Hagemann (2019), Roth and Sant’Anna (2023) for examples

• These approaches often require auxiliary assumptions such as restrictions on heteroskedasiticity or differences in cluster sizes

Multiple Hypothesis Testing

If you report multiple hypothesis tests and/or confidence intervals—e.g., an event study—it’s a good idea to make an adjustment for multiple hypothesis testing.

sup-t confidence band—construct confidence intervals of the form \[\Big[\widehat{ATT}^{es}(e) \pm \hat{c}_{1-\alpha/2} \textrm{s.e.}\big(\widehat{ATT}^{es}(e)\big) \Big]\] but instead of choosing the critical value as a quantile of the normal distribution, choose a (slightly) larger critical value that accounts for the fact that you are testing multiple hypotheses.

• Typically, an appropriate choice for the critical value to ensure the correct (uniform) coverage requires resampling methods.

• The reason for this is that the appropriate critical value depends on the joint distribution of \(\sqrt{n}(\widehat{ATT}^{es}(e) - ATT^{es}(e))\) across all \(e\), and these are generally not independent of each other.

• That said, this is not too difficult in practice, and is a default option in the did package.

Further Reading on Clustering

• T. Conley, Gonçalves, and Hansen (2018) - on clustering in general

• Roth et al. (2023) - on clustering in the context of DID

Event Study

attes <- aggte(attgt, type="dynamic")
ggdid(attes)

Back

How do these results work?

Consider a simplified setting where \(\T=2\), but we allow for there to be units that are already treated in the first period.

\(\implies\) 3 groups: \(G_i=1\), \(G_i=2\), \(G_i=\infty\)

TWFE Explanation (cont’d)

Starting from the previous slide: \[\begin{align*} \alpha = \E[\Delta Y_i | \Delta D_{i,t=2} = 1] - \E[\Delta Y_i | \Delta D_{i,t=2}=0] \end{align*}\] and consider the term on the far right, we have that \[\begin{align*} \E[\Delta Y_i | \Delta D_{i,t=2}=0] = \E[\Delta Y_i | G_i=1] \underbrace{\frac{p_1}{p_1 + p_\infty}}_{=: w_1} + \E[\Delta Y_i | G_i=\infty] \underbrace{\frac{p_\infty}{p_1 + p_\infty}}_{=: w_\infty} \end{align*}\]

TWFE Explanation (cont’d)

Let’s keep going: \[\begin{align*} \alpha = \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i | G_i=1]\Big)}_{\textrm{What is this?}} w_1 + \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i|G_i=\infty]\Big)}_{ATT(2,2)} w_\infty \end{align*}\] Working on the first term, we have that \[ \begin{aligned} & \E[\Delta Y_{i2} | G_i=2] - \E[\Delta Y_{i2} | G_i=1] \hspace{300pt} \end{aligned} \]

TWFE Explanation (cont’d)

Let’s keep going: \[\begin{align*} \alpha = \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i | G_i=1]\Big)}_{\textrm{What is this?}} w_1 + \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i|G_i=\infty]\Big)}_{ATT(2,2)} w_\infty \end{align*}\] Working on the first term, we have that \[ \begin{aligned} & \E[\Delta Y_{i2} | G_i=2] - \E[\Delta Y_{i2} | G_i=1] \hspace{300pt}\\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i1}(\infty) | G_i=2] - \E[Y_{i2}(1) - Y_{i1}(1) | G_i=1] \end{aligned} \]

TWFE Explanation (cont’d)

Let’s keep going: \[\begin{align*} \alpha = \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i | G_i=1]\Big)}_{\textrm{What is this?}} w_1 + \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i|G_i=\infty]\Big)}_{ATT(2,2)} w_\infty \end{align*}\] Working on the first term, we have that \[ \begin{aligned} & \E[\Delta Y_{i2} | G_i=2] - \E[\Delta Y_{i2} | G_i=1] \hspace{300pt}\\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i1}(\infty) | G_i=2] - \E[Y_{i2}(1) - Y_{i1}(1) | G_i=1] \\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i2}(\infty) | G_i=2] + \underline{\E[Y_{i2}(\infty) - Y_{i1}(\infty) | G_i=2]} \end{aligned} \]

TWFE Explanation (cont’d)

Let’s keep going: \[\begin{align*} \alpha = \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i | G_i=1]\Big)}_{\textrm{What is this?}} w_1 + \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i|G_i=\infty]\Big)}_{ATT(2,2)} w_\infty \end{align*}\] Working on the first term, we have that \[ \begin{aligned} & \E[\Delta Y_{i2} | G_i=2] - \E[\Delta Y_{i2} | G_i=1] \hspace{300pt}\\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i1}(\infty) | G_i=2] - \E[Y_{i2}(1) - Y_{i1}(1) | G_i=1] \\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i2}(\infty) | G_i=2] + \underline{\E[Y_{i2}(\infty) - Y_{i1}(\infty) | G_i=2]}\\ &\hspace{20pt} - \Big( \E[Y_{i2}(1) - Y_{i2}(\infty) | G_i=1] - \E[Y_{i1}(1) - Y_{i1}(\infty) | G_i=1] + \underline{\E[Y_{i2}(\infty) - Y_{i1}(\infty) | G_i=1]} \Big) \end{aligned} \]

TWFE Explanation (cont’d)

Let’s keep going: \[\begin{align*} \alpha = \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i | G_i=1]\Big)}_{\textrm{What is this?}} w_1 + \underbrace{\Big( \E[\Delta Y_i | G_i=2] - \E[\Delta Y_i|G_i=\infty]\Big)}_{ATT(2,2)} w_\infty \end{align*}\] Working on the first term, we have that \[ \begin{aligned} & \E[\Delta Y_{i2} | G_i=2] - \E[\Delta Y_{i2} | G_i=1] \hspace{300pt}\\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i1}(\infty) | G_i=2] - \E[Y_{i2}(1) - Y_{i1}(1) | G_i=1] \\ &\hspace{10pt} = \E[Y_{i2}(2) - Y_{i2}(\infty) | G_i=2] + \underline{\E[Y_{i2}(\infty) - Y_{i1}(\infty) | G_i=2]}\\ &\hspace{20pt} - \Big( \E[Y_{i2}(1) - Y_{i2}(\infty) | G_i=1] - \E[Y_{i1}(1) - Y_{i1}(\infty) | G_i=1] + \underline{\E[Y_{i2}(\infty) - Y_{i1}(\infty) | G_i=1]} \Big)\\ &\hspace{10pt} = \underbrace{ATT(2,2)}_{\textrm{causal effect}} - \underbrace{\Big(ATT(1,2) - ATT(1,1)\Big)}_{\textrm{treatment effect dynamics}} \end{aligned} \]

Plug this expression back in \(\rightarrow\)

TWFE Explanation (cont’d)

Plugging the previous expression back in, we have that \[\begin{align*} \alpha = ATT(2,2) + ATT(1,1) w_1 + ATT(1,2)(-w_1) \end{align*}\]

“Simple” Aggregation

Consider the following alternative aggregated treatment effect parameter \[\begin{align*} ATT^{simple} := \sum_{t=g}^\T ATT(g,t) \frac{\P(G_i=g | G \in \bar{\mathcal{G}})}{\sum_{t=g}^{\T} \P(G_i=g| G \in \bar{\mathcal{G}})} \end{align*}\] Consider imputation so that you have \(Y_{i,t}-\hat{Y}_{i,t}(0)\) available in all periods. This is the \(ATT\) parameter that you get by averaging all of those.

“Simple” Aggregation

“Simple” Aggregation

Besides the violations of parallel trends in pre-treatment periods, these weights are further away from \(ATT^o\) than the TWFE regression weights are!