Difference-in-Differences with a Continuous Treatment

.title[
# Difference-in-Differences with a Continuous Treatment
]
.author[
### Brantly Callaway, University of Georgia Andrew Goodman-Bacon, Federal Reserve Bank of Minneapolis Pedro H.C. Sant’Anna, Emory University 
]
.date[
### September 22, 2023 Virginia Tech
]

---

# What's been happening in the DID Literature?

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A number of papers have diagnosed issues with very commonly used two-way fixed effects (TWFE) regressions to implement DID identification strategies

Summary of Issues:

* The TWFE regression estimates mix together "good" and "bad" comparisons

* Weights on underlying parameters are (non-transparently) driven by estimation method

There have also been a number of papers fixing these issues

---

# This paper

These papers have (largely) focused on the case with a binary, staggered treatment

* In my view, the issues that arise in this setting have largely been solved

* Some exceptions: de Chaisemartin and D'Haultfouille (2020, 2021)

But there is considerable demand for understanding DID with more general treatments

* Unlike TWFE, "treatment effect heterogeneity robust" estimation strategies do not directly handle more complicated treatment regimes

---

# Twitter

---

count:false

# This paper

Current paper: Generalize binary treatment case to multi-valued or continuous treatment ("dose")

`$$Y_{it} = \theta_t + \eta_i + \beta^{twfe} \cdot D_i \cdot Treat_{it} + v_{it}$$`
Setup:

* Treatment "continuous enough" that researcher would estimate above model rather than include a sequence of dummy variables

* Researchers often interpret `$\beta^{twfe}$` as an average causal response

* i.e., (an average over) casual effects of a marginal increase in the dose
  
---

# This paper

Similar issues as in binary treatment literature related to regression (TWFE) estimation strategies when the treatment is multi-valued and/or continuous

As in the case with a staggered, binary treatment, we can fix all of these by (i) carefully making "good" comparisons and (ii) carefully choosing an appropriate weighting scheme

However, there are new issues related to interpreting differences between treatment effects at different doses as causal effects

* At a high-level, these issues arise from a tension between empirical researchers wanting to use a quasi-experimental research design (which delivers "local" treatment effect parameters) but (often) wanting to compare these "local" parameters to each other

* Unlike the staggered, binary treatment case: No easy fixes here!

`$\implies$` (at least in some sense), this is more negative than previous papers

---

# Outline

1. Identification/Issues in a Baseline Case (two periods, no one treated in first period)

2. TWFE in Baseline Case

3. Application on Medicare Spending and Capital/Labor Ratios

4. Bonus Material: Multiple periods, variation in treatment timing

---

# Baseline Case Two periods, no one treated in first period

---

# Notation

Potential outcomes notation

* Two time periods: `$t^*$` and `$t^*-1$`
  
  * No one treated until period `$t^*$`
    
  * Some units remain untreated in period `$t^*$`
  
* Potential outcomes: `$Y_{it^*}(d)$`

* Observed outcomes: `$Y_{it^*}$` and `$Y_{it^*-1}$`

`$$Y_{it^*}=Y_{it^*}(D_i) \quad \textrm{and} \quad Y_{it^*-1}=Y_{it^*-1}(0)$$`

---

# Parameters of Interest (ATT-type)

* Level Effects (Average Treatment Effect on the Treated)

`$$ATT(d|d) := \E[Y_{t^*}(d) - Y_{t^*}(0) | D=d]$$`

* Interpretation: The average effect of dose `$d$` relative to not being treated *local to the group that actually experienced dose `$d$`*
  
  * This is the natural analogue of `$ATT$` in the binary treatment case

* Slope Effect (Average Causal Responses)

`$$ACRT(d|d) := \frac{\partial ATT(l|d)}{\partial l} \Big|_{l=d} \ \ \ \textrm{and} \ \ \ ACRT^O := \E[ACRT(D|D)|D>0]$$`
  
  * Interpretation: `$ACRT(d|d)$` is the causal effect of a marginal increase in dose *local to units that actually experienced dose `$d$`*
  
  * `$ACRT^O$` averages `$ACRT(d|d)$` over the population distribution of the dose.  If you're estimating a TWFE regression, this is probably what you have in mind that you are targeting.

---

# Discrete Dose

* Level Effects (Average Treatment Effect on the Treated)

`$$ATT(d|d) := \E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=d]$$`

* This is exactly the same as for continuous dose

--
* Slope Effect (Average Causal Responses)

* Possible doses: `$\{d_1, \ldots, d_J\}$`

`$$ACRT(d_j|d_j) := ATT(d_j|d_j) - ATT(d_{j-1}|d_j)$$`
--

* Interesting side-comment: In the case with a binary treatment, `$ACRT(1|1) = ATT$`
  
    `$\implies$` In binary treatment case, `$ATT$` is (in some sense) both a slope and level effect

---

# Identification
<div class="assumption-box"> "Standard" Parallel Trends Assumption

For all `d`,

`\mathbb{E}[\Delta Y_{t^{\ast}} (0) | D=d] = \mathbb{E}[\Delta Y_{t^{\ast}}(0) | D=0]`

</div>

Then,

$$
`\begin{aligned}
ATT(d|d) &= \E[Y_{t^*}(d) - Y_{t^*}(0) | D=d] \hspace{150pt}
\end{aligned}`
$$

---

count:false
# Identification
<div class="assumption-box"> "Standard" Parallel Trends Assumption

For all `d`,

`\mathbb{E}[\Delta Y_{t^{\ast}} (0) | D=d] = \mathbb{E}[\Delta Y_{t^{\ast}}(0) | D=0]`

</div>

Then,

$$
`\begin{aligned}
ATT(d|d) &= \E[Y_{t^*}(d) - Y_{t^*}(0) | D=d] \hspace{150pt}\\
&= \E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=d] - \E[Y_{t^*}(0) - Y_{t^*-1}(0) | D=d]
\end{aligned}`
$$

---

count:false
# Identification
<div class="assumption-box"> "Standard" Parallel Trends Assumption

For all `d`,

`\mathbb{E}[\Delta Y_{t^{\ast}} (0) | D=d] = \mathbb{E}[\Delta Y_{t^{\ast}}(0) | D=0]`

</div>

Then,

$$
`\begin{aligned}
ATT(d|d) &= \E[Y_{t^*}(d) - Y_{t^*}(0) | D=d] \hspace{150pt}\\
&= \E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=d] - \E[Y_{t^*}(0) - Y_{t^*-1}(0) | D=d]\\
&= \E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=d] - \E[\Delta Y_{t^*}(0) | D=0]
\end{aligned}`
$$

---

count:false
# Identification
<div class="assumption-box"> "Standard" Parallel Trends Assumption

For all `d`,

`\mathbb{E}[\Delta Y_{t^{\ast}} (0) | D=d] = \mathbb{E}[\Delta Y_{t^{\ast}}(0) | D=0]`

</div>

Then,

This is exactly what you would expect
---

# Are we done?

Unfortunately, no

Most empirical work with a multi-valued or continuous treatment wants to think about how causal responses vary across dose

* For example, plot treatment effects as a function of dose

* Does more dose tends to increase/decrease/not effect outcomes?
  
* Average causal response parameters *inherently* involve comparisons across slightly different doses

---

# Interpretation Issues
Consider comparing `$ATT(d|d)$` for two different doses
--

$$
`\begin{aligned}
& ATT(d_h|d_h) - ATT(d_l|d_l) \hspace{350pt}
\end{aligned}`
$$

---

count:false
# Interpretation Issues
Consider comparing `$ATT(d|d)$` for two different doses

---

count:false
# Interpretation Issues
Consider comparing `$ATT(d|d)$` for two different doses

$$
`\begin{aligned}
& ATT(d_h|d_h) - ATT(d_l|d_l) \hspace{350pt}\\
& \hspace{25pt} = \E[Y_{t^*}(d_h)-Y_{t^*}(d_l) | D=d_h] + \E[Y_{t^*}(d_l) - Y_{t^*}(0) | D=d_h] - \E[Y_{t^*}(d_l) - Y_{t^*}(0) | D=d_l]\\
& \hspace{25pt} = \underbrace{\E[Y_{t^*}(d_h) - Y_{t^*}(d_l) | D=d_h]}_{\textrm{Causal Response}} + \underbrace{ATT(d_l|d_h) - ATT(d_l|d_l)}_{\textrm{Selection Bias}}
\end{aligned}`
$$

"Standard" Parallel Trends is not strong enough to rule out the selection bias terms here

* Implication: If you want to interpret differences in treatment effects across different doses, then you will need stronger assumptions than standard parallel trends

* This problem spills over into identifying `$ACRT(d|d)$`

Positive side-comment: `$ATT(d_h|d_h) - ATT(d_l|d_l) = \E[\Delta Y_{t^*} | D=d_h] - \E[\Delta Y_{t^*} | D=d_l]$` (which doesn't involve the untreated group)

---

# Alternative Parameters of Interest (ATE-type)

The underlying reason for the difficulty comparing `$ATT$`-type parameters is that they are "local".  This suggests that "global" parameters could circumvent these issues.

* Level Effects

`$$ATE(d) := \E[Y_{t^*}(d) - Y_{t^*}(0)]$$`
--

* Slope Effects

$$
`\begin{aligned}
  ACR(d) := \frac{\partial ATE(d)}{\partial d} \ \ \ \ & \textrm{or} \ \ \ ACR^O := \E[ACR(D) | D>0]
\end{aligned}`
$$

---

# Comparisons across dose
ATE-type parameters do not suffer from the same issues as ATT-type parameters when making comparisons across dose

$$
`\begin{aligned}
ATE(d_h) - ATE(d_l) &= \E[Y_{t^*}(d_h) - Y_{t^*}(0)] - \E[Y_{t^*}(d_l) - Y_{t^*}(0)]
\end{aligned}`
$$

---

count:false
# Comparisons across dose
ATE-type parameters do not suffer from the same issues as ATT-type parameters when making comparisons across dose

$$
`\begin{aligned}
ATE(d_h) - ATE(d_l) &= \E[Y_{t^*}(d_h) - Y_{t^*}(0)] - \E[Y_{t^*}(d_l) - Y_{t^*}(0)]\\
&= \underbrace{\E[Y_{t^*}(d_h) - Y_{t^*}(d_l)]}_{\textrm{Causal Response}}
\end{aligned}`
$$

Unfortunately, "Standard" Parallel Trends Assumption not strong enough to identify `$ATE(d)$`.

---

# Introduce Stronger Assumptions

<div class="assumption-box">"Strong" Parallel Trends

For all `d`,

`\mathbb{E}[Y_{t^{\ast}}(d) - Y_{t^{\ast}-1}(0)] = \mathbb{E}[Y_{t^{\ast}}(d) - Y_{t^{\ast}-1}(0) | D=d]`

</div>

Under Strong Parallel Trends, it is straightforward to show that

`$$ATE(d) = \E[\Delta Y_{t^*} | D=d] - \E[\Delta Y_{t^*}|D=0]$$`

RHS is exactly the same expression as for `$ATT(d|d)$` under "standard" parallel trends, but here

* assumptions are different

* parameter interpretation is different

---

# Comments on Strong Parallel Trends

* This is notably different from "Standard" Parallel Trends

* It involves potential outcomes for all values of the dose (not just untreated potential outcomes)
  
  * All dose groups would have experienced the same path of outcomes had they been assigned the same dose
  
---

# Comments on Strong Parallel Trends

Strong parallel trends is also closely related to assuming a certain form of treatment effect homogeneity

Consider a stronger version (but similar in spirit) of strong parallel trends: For all `$d$` and `$l$`,
  
`$$\E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=l] = \E[Y_{t^*}(d) - Y_{t^*-1}(0) | D=d]$$`

With a couple of lines of algebra, this implies that

`$$ATT(d|l) = ATT(d|d)$$`

Since this holds for all `$d$` and `$l$`, it also implies that `$ATE(d) = ATT(d|d)$`

This is a form of treatment effect homogeneity, and it has the flavor of a structural assumption as it provides a route for us to extrapolate treatment effects across different values of the dose.

---

# Summarizing

* It is straightforward/familiar to identify ATT-type parameters with a multi-valued or continuous dose

* However, comparison of ATT-type parameters across different doses are hard to interpret

* They include selection bias terms
  
  * This issues extends to identifying ACRT parameters

* This suggests targeting ATE-type parameters

* Comparisons across doses do not contain selection bias terms
  
  * But identifying ATE-type parameters requires stronger assumptions

---

# TWFE in Baseline Case

---

# TWFE

The most common strategy in applied work is to estimate the two-way fixed effects (TWFE) regression:

`$$Y_{it} = \theta_t + \eta_i + \beta^{twfe} \cdot D_i \cdot Post_{t^*} + v_{it}$$`
In baseline case (two periods, no one treated in first period), this is just

`$$\Delta Y_i = \beta_0 + \beta^{twfe} \cdot D_i + \Delta v_i$$`

`$\beta^{twfe}$` often loosely interpreted as Average Causal Response

---
# Interpreting `$\beta^{twfe}$`

In the paper, we show that

* Under Standard Parallel Trends:

`$$\beta^{tfwe} = \int_{\mathcal{D}_+} w_1(l) \left[ ACRT(l|l) + \frac{\partial ATT(l|h)}{\partial h} \Big|_{h=l} \right] \, dl + w_0 \frac{ATT(d_L|d_L)}{d_L}$$`

* `$w_1(l)$` and `$w_0$` are positive weights that integrate to 1
  
  * `$ACRT(l|l)$` is average causal response conditional on `$D=l$`
  
  * `$\frac{\partial ATT(l|h)}{\partial h} \Big|_{h=l}$` is a local selection bias term
  
  * `$\frac{ATT(d_L|d_L)}{d_L}$` is the causal effect of going from no dose to the smallest possible dose (conditional on `$D=d_L$`)
  
---

# Interpreting `$\beta^{twfe}$`

* Under Strong Parallel Trends:

`$$\beta^{tfwe} = \int_{\mathcal{D}_+} w_1(l) ACR(l) \, dl + w_0 \frac{ATE(d_L)}{d_L}$$`

* `$w_1(l)$` and `$w_0$` are same weights as before
  
  * `$ACR(l)$` is average causal response to dose `$l$` across entire population
  
  * there is no selection bias term
  
  * `$\frac{ATE(d_L)}{d_L}$` is the causal effect of going from no dose to the smallest possible dose (across entire population)

---

# What does this mean?

* Issue \#1: Selection bias terms that show up under standard parallel trends

`$\implies$` to interpret as a weighted average of any kind of causal responses, need to invoke (likely substantially) stronger assumptions
  
--
  
* Issue \#2: Weights

* They are all positive
 
 * But this is a very minimal requirement for weights being "reasonable"
 
 * These weights have "strange" properties (i) affected by the size of the untreated group, (ii) that they are maximized at `$d=\E[D]$`.
 
 * [[[Example 1 - Mixture of Normals Dose](#weights-example-mixture)]] &nbsp; &nbsp; [[[Example 2: Exponential Dose](#weights-example-exponential)]]
 
---

# What does this mean?

* Issue \#3: Pre-testing

* That the expressions for `$ATE(d)$` and `$ATT(d|d)$` are exactly the same also means that we cannot use pre-treatment periods to try to distinguish between "standard" and "strong" parallel trends 
  
--
  
* Issue \#4: Alternative Interpretations

* The above decompositions of `$\beta^{TWFE}$` are not unique.  For example, you can show that `$\beta^{TWFE}$` is also equal to a weighted combination of `$ATT(d|d)$`
  
  * However, the weights on `$ATT(d|d)$` integrate to 0 `$\implies$` incorrect to think of `$\beta^{TWFE}$` as "approximating" `$ATT(d|d)$`.
  
---

# What should you do?

1. Either (i) report `$ATT(d|d)$` directly and interpret carefully, or (ii) be aware (and think through) that `$\beta^{twfe}$`, comparisons across `$d$`, or average causal response parameters all require imposing stronger assumptions

2. With regard to weights, there are likely better options for estimating causal effect parameters

* Step 1: Nonparametrically estimate `$ACR(d) = \frac{\partial \E[\Delta Y | D=d]}{\partial d}$`
 
 * Side-comment: This is not actually too hard to estimate. No curse-of-dimensionality, etc.
 
 * Step 2: Estimate `$ACR^0 = \E[ACR(D)|D>0]$`.
 
 * These do not get around the issue of requiring a stronger assumption

---

# Empirical Application

---

# Empirical Application

This is a simplified version of Acemoglu and Finkelstein (2008)

1983 Medicare reform that eliminated labor subsidies for hospitals

* Medicare moved to the Prospective Payment System (PPS) which replaced "full cost reimbursement" with "partial cost reimbursement" which eliminated reimbursements for labor (while maintaining reimbursements for capital expenses)

* Rough idea: This changes relative factor prices which suggests hospitals may adjust by changing their input mix.  Could also have implications for technology adoption, etc.

---

# Data

Hospital reported data from the American Hospital Association, yearly from 1980-1986

Outcome is capital/labor ratio

* proxy using the depreciation share of total operating expenses (avg. 4.5%)

* our setup: collapse to two periods by taking average in pre-treatment periods and average in post-treatment periods

Dose is "exposure" to the policy

* the number of Medicare patients in the period before the policy was implemented

* roughly 15% of hospitals are untreated (have essentially no Medicare patients)

* AF provide results both using and not using these hospitals as (good) it is useful to have untreated hospitals (bad) they are fairly different (includes federal, long-term, psychiatric, children's, and rehabilitation hospitals)

---

# Bin Scatter

---

# Results

`$TWFE$`: 1.14 (s.e.: 0.10)

* "Given that the average hospital has a 38 percent Medicare share prior to PPS, this estimate suggests that in its first three years, the introduction of PPS was associated with an increase in the depreciation share of about 0.42 (=1.13*0.38)...corresponds to a sizable 10 percent increase in the capital-labor ratio of the average Medicare share hospital."

* Probably most natural comparison for this parameter is `$\widehat{ATT}(d=0.38|d=0.38) = 0.87$` (s.e.: 0.09).

Can also use the TWFE estimate to think about causal responses

* `$\widehat{ACRT}^O:$` -0.078 (s.e.: ??)

Finally, if only want to exploit variation in the dose

* `$TWFE$` (no untreated): 0.25 (s.e. 0.18)

* Our approach: `$ATT(d|d)$` not available in this case, `$ACRT^O$` does not change

---

# ATT Plot

---

# ACR(T) Plot

---

# Density weights vs. TWFE weights

---

# TWFE Weights with and without Untreated Group

---

# Conclusion

* There are a number of challenges to implementing/interpreting DID with a multi-valued or continuous treatment

* Issues related to TWFE are (mostly) anticipated at this point

* But (in my view) the main new issue here is that justifying interpreting comparisons across different doses as causal effects requires stronger assumptions than most researchers probably think that they are making

* Link to paper: [https://arxiv.org/abs/2107.02637](https://arxiv.org/abs/2107.02637)

* Other Summaries: &nbsp; (i) [Five minute summary](https://bcallaway11.github.io/posts/five-minute-did-continuous-treatment) &nbsp; &nbsp; &nbsp; (ii) [Pedro's Twitter](https://twitter.com/pedrohcgs/status/1415915759960690696)

* Comments welcome: [brantly.callaway@uga.edu](mailto:brantly.callaway@uga.edu)

* Code: in progress

---

# Ex. Mixture of Normals Dose

![](data:image/png;base64,#did_continuous_treatment_files/figure-html/unnamed-chunk-5-1.png)

[Back](#issues)

---

# Ex. Exponential Dose
![](data:image/png;base64,#did_continuous_treatment_files/figure-html/unnamed-chunk-6-1.png)

[Back](#issues)

---

# More General Case Multiple periods, variation in treatment timing

---

# Setup

* Staggered treatment adoption

* If you are treated today, you will continue to be treated tomorrow
  
  * Note relatively straightforward to relax, just makes notation more complex
  
  * Can allow for treatment anticipation too, but ignoring for simplicity now
  
  * Once become treated, dose remains constant (could probably relax this too)
  
---

# Setup

* Additional Notation:

* `$G_i$` &mdash; a unit's "group" (the time period when unit becomes treated)
  
  * Potential outcomes `$Y_{it}(g,d)$` &mdash; the outcome unit `$i$` would experience in time period `$t$` if they became treated in period `$g$` with dose `$d$`
  
  * `$Y_{it}(0)$` is the potential outcome corresponding to not being treated in any period
  
---

# Parameters of Interest

Level Effects:

$$ ATT(g,t,d|g,d) := \E[Y_t(g,d) - Y_t(0) | G=g, D=d] \ \ \ \textrm{and} \ \ \ ATE(g,t,d) := \E[Y_t(g,d) - Y_t(0) ]$$
--

Slope Effects:

`$$ACRT(g,t,d|g,d) := \frac{\partial ATT(g,t,l|g,d)}{\partial l} \Big|_{l=d} \ \ \ \textrm{and} \ \ \ ACR(g,t,d) := \frac{\partial ATE(g,t,d)}{\partial d}$$`

---

# Parameters of Interest

These essentially inherit all the same issues as in the two period case

* Under a multi-period version of "standard" parallel trends, comparisons of `$ATT$` across different values of dose are hard to interpret

* They contain selection bias terms

--
  
* Under a multi-period version of "strong" parallel trends, comparisons of `$ATE$` across different values of dose straightforward to interpret

* But this involves a much stronger assumption

Expressions in remainder of talk are under "strong" parallel trends

* Under "standard" parallel trends, add selection bias terms everywhere
---

# Parameters of Interest

Often, these are high-dimensional and it may be desirable to "aggregate" them

* Average by group (across post-treatment time periods) and then across groups

`$\rightarrow$` `$ACR^{overall}(d)$` (overall average causal response for particular dose)

* Average `$ACR^{overall}(d)$` across dose

`$\rightarrow$` `$ACR^O$` (this is just one number) and is likely to be the parameter that one would be targeting in a TWFE regression

--
  
* Event study: average across groups who have been exposed to treatment for `$e$` periods

`$\rightarrow$` For fixed `$d$`
  
  `$\rightarrow$` Average across different values of `$d$` `$\implies$` typical looking ES plot
  
---

# TWFE in More General Case

---

# TWFE Regression

Consider the same TWFE regression as before

`$$Y_{it} = \theta_t + \eta_i + \beta^{twfe} \cdot D_i \cdot Treat_{it} + v_{it}$$`

---

# Running Example

---

# How should `$\beta^{twfe}$` be interpreted?

We show in the paper that `$\beta^{twfe}$` is a weighted average of the following terms:

`$$\delta^{WITHIN}(g) = \frac{\textrm{cov}(\bar{Y}^{POST}(g) - \bar{Y}^{PRE(g)}(g), D | G=g)}{\textrm{var(D|G=g)}}$$`

* Comes from within-group variation in the amount of dose

* This term is essentially the same as in the baseline case and corresponds to a reasonable treatment effect parameter under strong parallel trends

* Like baseline case, (after some manipulations) this term corresponds to a "derivative"/"ACR"

* Does not show up in the binary treatment case because there is no variation in amount of treatment

---

# How should `$\beta^{twfe}$` be interpreted?

---

# How should `$\beta^{twfe}$` be interpreted?

---

# `$\beta^{twfe}$` weighted average, term 2 of 4

For `$k > g$` (i.e., group `$k$` becomes treated after group `$g$`),

`$$\delta^{MID,PRE}(g,k) =  \frac{\E\left[\big(\bar{Y}^{MID(g,k)} - \bar{Y}^{PRE(g)}\big) | G=g\right] - \E\left[\big(\bar{Y}^{MID(g,k)} - \bar{Y}^{PRE(g)}\big) | G=k \right]}{\E[D|G=g]}$$`

* Comes from comparing path of outcomes for a group that becomes treated (group `$g$`) relative to a not-yet-treated group (group `$k$`)

* Corresponds to a reasonable treatment effect parameter under strong parallel trends

* Denominator (after some derivations) ends up giving this a "derivative"/"ACR" interpretation

* Similar terms show up in the case with a binary treatment

---

# `$\beta^{twfe}$` weighted average, term 2 of 4

---

# `$\beta^{twfe}$` weighted average, term 3 of 4

For `$k > g$` (i.e., group `$k$` becomes treated after group `$g$`),

$$
`\begin{aligned}
\delta^{POST,MID}(g,k) &=  \frac{\E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{MID(g,k)}\big) | G=k\right] - \E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{MID(g,k)}\big) | D=0 \right]}{\E[D|G=k]} \\
&- \left(\frac{\E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{PRE(k)}\big) | G=g\right] - \E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{PRE(g)}\big) | D=0 \right]}{\E[D|G=k]} \right.\\
& \hspace{25pt} - \left.\frac{\E\left[\big(\bar{Y}^{MID(g,k)} - \bar{Y}^{PRE(k)}\big) | G=g\right] - \E\left[\big(\bar{Y}^{MID(g,k)} - \bar{Y}^{PRE(g)}\big) | D=0 \right]}{\E[D|G=k]} \right)
\end{aligned}`
$$
---

# `$\beta^{twfe}$` weighted average, term 3 of 4

For `$k > g$` (i.e., group `$k$` becomes treated after group `$g$`),

$$
`\begin{aligned}
\delta^{POST,MID}(g,k) &= \frac{\E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{MID(g,k)}\big) | G=k\right] - \E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{MID(g,k)}\big) | D=0 \right]}{\E[D|G=k]} \\
&- \textrm{Treatment Effect Dynamics for Group g}
\end{aligned}`
$$
* Comes from comparing path of outcomes for a group that becomes treated (group `$k$`) to paths of outcomes of an already treated group (group `$g$`)

* In the presence of treatment effect dynamics (these are not ruled out by any parallel trends assumption), this term is problematic

* This is similar-in-spirit to the problematic terms for TWFE with a binary treatment

---

# `$\beta^{twfe}$` weighted average, term 3 of 4

---

# `$\beta^{twfe}$` weighted average, term 4 of 4

For `$k > g$` (i.e., group `$k$` becomes treated after group `$g$`),

$$
`\begin{aligned}
\delta^{POST,PRE}(g,k) = \frac{\E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{PRE(g)}\big) | G=g\right] - \E\left[\big(\bar{Y}^{POST(k)} - \bar{Y}^{PRE(g)}\big) | G=k \right]}{\E[D|G=g] - \E[D|G=k]}
\end{aligned}`
$$
* Comes from comparing path of outcomes for groups `$g$` and `$k$` in their common post-treatment periods relative to their common pre-treatment periods

* In the presence of heterogeneous causal responses (causal response in same time period differs across groups), this term ends up being (partially) problematic too

* Only shows up when `$\E[D|G=g] \neq \E[D|G=k]$`

* No analogue in the binary treatment case

---

# `$\beta^{twfe}$` weighted average, term 4 of 4

---

# Summary of TWFE Issues

* Issue \#1: Selection bias terms that show up under standard parallel trends

`$\implies$` to interpret as a weighted average of any kind of causal responses, need to invoke (likely substantially) stronger assumptions
  
--
  
* Issue \#2: Weights

* Negative weights possible due to (i) treatment effect dynamics or (ii) heterogeneous causal responses across groups
  
  * Are (undesirably) driven by estimation method

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Weights issues can be solved by carefully making desirable comparisons and user-chosen appropriate weights

Selection bias terms are more fundamental challenge