Difference-in-Differences with a Continuous Treatment
March 29, 2025
Introduction
\(\newcommand{\E}{\mathbb{E}} \newcommand{\E}{\mathbb{E}} \newcommand{\var}{\mathrm{var}} \newcommand{\cov}{\mathrm{cov}} \newcommand{\Var}{\mathrm{var}} \newcommand{\Cov}{\mathrm{cov}} \newcommand{\Corr}{\mathrm{corr}} \newcommand{\corr}{\mathrm{corr}} \newcommand{\L}{\mathrm{L}} \renewcommand{\P}{\mathrm{P}} \newcommand{\independent}{{\perp\!\!\!\perp}} \newcommand{\indicator}[1]{ \mathbf{1}\{#1\} }\)The DiD sessions yesterday (largely) focused on the case with a binary, staggered treatment
This session: Move from a setting with a binary treatment case to one with a continuous treatment (“dose”)
Some of the arguments involve extending ideas from the binary, staggered treatment case to a setting with continuous treatment
- But we will also face new conceptual issues in this case that do not show up in a setting with a binary treatment
Today’s talk mostly follows Callaway, Goodman-Bacon, and Sant’Anna (2024)
Example:
Effect of \(\underbrace{\textrm{length of school closures}}_{\textrm{continuous treatment}}\) (during Covid) on \(\underbrace{\textrm{students' test scores}}_{\textrm{outcome}}\)
- e.g., (Ager et al. 2024; Gillitzer and Prasad 2023, among others)
Clarifications about Continuous Treatment - What’s Included
For today, mostly emphasize a continuous treatment, but our results also apply to other settings (with trivial modifications):
multi-valued treatments (e.g., effect of state-level minimum wage policies on employment)
binary policy variable with differential “exposure” to it (application in the paper: a binary Medicare policy where different hospitals had more exposure to the policy)
Clarifications about Continuous Treatment - What’s Not Included
We will focus on a setting with a panel data natural experiment
- No units are treated in the first period, some units become treated, with possibly different doses, in the second period
This narrows our focus for today and rules out:
Applications where units are already treated in different amounts in the first period (see: Chaisemartin, d’Haultfœuille, and Vazquez-Bare (2024), Chaisemartin et al. (2025))
Applications where interest in understanding the effect of a binary treatment but the researcher observes aggregate data (sometimes called: “fuzzy DiD”), often using policy changes as an instrument
Today’s Talk
Identification: What’s the same as in the binary treatment case?
Identification: What’s different from the binary treatment case?
Interpreting TWFE Regressions
Empirical Application
1. Identification: What’s the same as in the binary treatment case?
Continuous Treatment Notation
Two time periods: \(t=2\) and \(t=1\)
- No one treated until period \(t=2\)
- Some units remain untreated in period \(t=2\)
Treatment: \(D_i\)
Potential outcomes: \(Y_{it=2}(d)\)
Observed outcomes: \(Y_{it=2}\) and \(Y_{it=1}\)
\[Y_{it=2}=Y_{it=2}(D_i) \quad \textrm{and} \quad Y_{it=1}=Y_{it=1}(0)\]
Parameters of Interest (ATT-type)
Level Effects (Average Treatment Effect on the Treated)
\[ATT(d|d) := \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d]\]
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
Parameters of Interest (ACR-type)
Slope Effects (Average Causal Response on the Treated)
\[ACRT(d|d) := \frac{\partial ATT(l|d)}{\partial l} \Big|_{l=d}\]
- Interpretation: \(ACRT(d|d)\) is the causal effect of a marginal increase in dose local to units that actually experienced dose \(d\)
Aggregated Parameters
Notice that \(ATT(d|d)\) and \(ACRT(d|d)\) are functional parameters
- This is different from \(Y_{it} = \theta_t + \eta_i + \beta^{twfe} D_{it} + e_{it}\)
We can view \(ATT(d|d)\) and \(ACRT(d|d)\) as the “building blocks” for a more aggregated parameter.
Aggregated versions of these (into a single number) are \[\begin{align*} ATT^o := \E\Big[ATT(D|D)\Big|D>0\Big] \qquad \qquad ACRT^o := \E\Big[ACRT(D|D)\Big|D>0\Big] \end{align*}\]
\(ATT^o\) averages \(ATT(d|d)\) over the population distribution of the dose
\(ACRT^o\) averages \(ACRT(d|d)\) over the population distribution of the dose
- \(ACRT^o\) is the natural target parameter for the TWFE regression in this case
Identification
“Standard” Parallel Trends Assumption
For all \(d\),
\[\E[\Delta Y(0) | D=d] = \E[\Delta Y(0) | D=0]\]
Identification
“Standard” Parallel Trends Assumption
For all \(d\),
\[\E[\Delta Y(0) | D=d] = \E[\Delta Y(0) | D=0]\]
Then,
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{150pt} \end{aligned} \]
Identification
“Standard” Parallel Trends Assumption
For all \(d\),
\[\E[\Delta Y(0) | D=d] = \E[\Delta Y(0) | D=0]\]
Then,
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{150pt}\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=d] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=d] \end{aligned} \]
Identification
“Standard” Parallel Trends Assumption
For all \(d\),
\[\E[\Delta Y(0) | D=d] = \E[\Delta Y(0) | D=0]\]
Then,
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{150pt}\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=d] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=d]\\ &= \E[\Delta Y | D=d] - \E[\Delta Y | D=0] \end{aligned} \]
This is exactly what you would expect
Using similar arguments, you can show \[ATT^o = \E[\Delta Y | D>0] - \E[\Delta Y | D=0]\]
2. Identification: What’s different from the binary treatment case?
Are we done?
Unfortunately, no
Most empirical work with a continuous treatment wants to think about how causal responses vary across dose
- Plot treatment effects as a function of dose and ask: does more dose tends to increase/decrease/not affect outcomes?
- Average causal response parameters inherently involve comparisons across slightly different doses
There are new issues related to comparing \(ATT(d|d)\) at different doses and interpreting these differences as causal effects
- Unlike the staggered, binary treatment case: No easy fixes here!
Interpretation Issues
Consider comparing \(ATT(d|d)\) for two different doses
\[ \begin{aligned} & ATT(d_h|d_h) - ATT(d_l|d_l) \hspace{350pt} \end{aligned} \]
Interpretation Issues
Consider comparing \(ATT(d|d)\) for two different doses
\[ \begin{aligned} & ATT(d_h|d_h) - ATT(d_l|d_l) \hspace{350pt}\\ & \hspace{25pt} = \E[Y_{t=2}(d_h)-Y_{t=2}(d_l) | D=d_h] + \E[Y_{t=2}(d_l) - Y_{t=2}(0) | D=d_h] - \E[Y_{t=2}(d_l) - Y_{t=2}(0) | D=d_l] \end{aligned} \]
Interpretation Issues
Consider comparing \(ATT(d|d)\) for two different doses
\[ \begin{aligned} & ATT(d_h|d_h) - ATT(d_l|d_l) \hspace{350pt}\\ & \hspace{25pt} = \E[Y_{t=2}(d_h)-Y_{t=2}(d_l) | D=d_h] + \E[Y_{t=2}(d_l) - Y_{t=2}(0) | D=d_h] - \E[Y_{t=2}(d_l) - Y_{t=2}(0) | D=d_l]\\ & \hspace{25pt} = \underbrace{\E[Y_{t=2}(d_h) - Y_{t=2}(d_l) | D=d_h]}_{\textrm{Causal Response}} + \underbrace{ATT(d_l|d_h) - ATT(d_l|d_l)}_{\textrm{Selection Bias}} \end{aligned} \]
“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 \(ACRT(d|d)\)
Interpretation Issues
Intuition:
Difference-in-differences identification strategies result in \(ATT(d|d)\) parameters. These are local parameters and difficult to compare to each
This explanation is similar to thinking about LATEs with two different instruments
Thus, comparing \(ATT(d|d)\) across different values is tricky and not for free
What can you do?
One idea, just recover \(ATT(d|d)\) and interpret it cautiously (interpret it by itself not relative to different values of \(d\))
If you want to compare them to each other, it will come with the cost of additional (structural) assumptions
Introduce Stronger Assumptions
“Strong” Parallel Trends Assumption
For all doses d and l,
\[\mathbb{E}[Y_{t=2}(d) - Y_{t=1}(0) | D=l] = \mathbb{E}[Y_{t=2}(d) - Y_{t=1}(0) | D=d]\]
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
Introduce Stronger Assumptions
Strong parallel trends is equivalent to a certain restriction on treatment effect heterogeneity. Notice:
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{200pt} \ \end{aligned} \]
Introduce Stronger Assumptions
Strong parallel trends is equivalent to a certain restriction on treatment effect heterogeneity. Notice:
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{200pt} \\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=d] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=d] \ \end{aligned} \]
Introduce Stronger Assumptions
Strong parallel trends is equivalent to a certain restriction on treatment effect heterogeneity. Notice:
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{200pt} \\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=d] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=d] \\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=l] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=l] \ \end{aligned} \]
Introduce Stronger Assumptions
Strong parallel trends is equivalent to a certain restriction on treatment effect heterogeneity. Notice:
\[ \begin{aligned} ATT(d|d) &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=d] \hspace{200pt} \\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=d] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=d] \\\ &= \E[Y_{t=2}(d) - Y_{t=1}(0) | D=l] - \E[Y_{t=2}(0) - Y_{t=1}(0) | D=l] \\\ &= \E[Y_{t=2}(d) - Y_{t=2}(0) | D=l] = ATT(d|l) \end{aligned} \]
Since this holds for all \(d\) and \(l\), it also implies that \(ATT(d|d) = ATE(d) = \E[Y_{t=2}(d) - Y_{t=2}(0)]\). Thus, under strong parallel trends, we have that
\[ATE(d) = \E[\Delta Y|D=d] - \E[\Delta Y|D=0]\]
RHS is exactly the same expression as for \(ATT(d|d)\) under “standard” parallel trends, but here
assumptions are different
parameter interpretation is different
Comparisons across dose
ATE-type parameters do not suffer from the same issues as ATT-type parameters when making comparisons across dose
\[ \begin{aligned} ATE(d_h) - ATE(d_l) &= \E[Y_{t=2}(d_h) - Y_{t=2}(0)] - \E[Y_{t=2}(d_l) - Y_{t=2}(0)] \end{aligned} \]
Comparisons across dose
ATE-type parameters do not suffer from the same issues as ATT-type parameters when making comparisons across dose
\[ \begin{aligned} ATE(d_h) - ATE(d_l) &= \E[Y_{t=2}(d_h) - Y_{t=2}(0)] - \E[Y_{t=2}(d_l) - Y_{t=2}(0)]\\ &= \underbrace{\E[Y_{t=2}(d_h) - Y_{t=2}(d_l)]}_{\textrm{Causal Response}} \end{aligned} \]
Thus, recovering \(ATE(d)\) side-steps the issues about comparing treatment effects across doses, but it comes at the cost of needing a (potentially very strong) extra assumption
Given that we can compare \(ATE(d)\)’s across dose, we can recover slope effects in this setting
\[ \begin{aligned} ACR(d) := \frac{\partial ATE(d)}{\partial d} \qquad &\textrm{or} \qquad ACR^o := \E\Big[ACR(D) \Big| D>0\Big] \end{aligned} \]
Additional Comments
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
- These issues extend to TWFE regressions
This suggests targeting ATE-type parameters
- Comparisons across doses do not contain selection bias terms
- But identifying ATE-type parameters requires stronger assumptions
3. Interpreting TWFE Regressions
TWFE Regressions in this Context
Consider the same TWFE regression as before: \[\begin{align*} Y_{it} = \theta_t + \eta_i + \beta^{twfe} D_{it} + e_{it} \end{align*}\] We show that \[\begin{align*} \beta^{twfe} = \int_{\mathcal{D}_+} w(l) m'_\Delta(l) \, dl \end{align*}\] where \(m_\Delta(l) := \E[\Delta Y|D=l] - \E[\Delta Y|D=0]\) and \(w(l)\) are weights
Under standard parallel trends, \(m'_{\Delta}(l) = ACRT(l|l) + \textrm{local selection bias}\)
Under strong parallel trends, \(m'_{\Delta}(l) = ACR(l)\).
About the weights: they are all positive, but have some strange properties (e.g., always maximized at \(l = \E[D]\) (even if this is not a common value for the dose))
- \(\implies\) even under strong parallel trends, \(\beta^{twfe} \neq ACR^o\).
TWFE Regressions in this Context
Other issues can arise in more complicated cases
For example, suppose you have a staggered continuous treatment, then you will additionally get issues that are analogous to the ones we discussed earlier for a binary staggered treatment
In general, things get worse for TWFE regressions with more complications
Estimation - What should you do?
Level Effects - no issues related to selection bias
For \(ATT^o\): Binarize treatment, \(ATT^o = \E[\Delta Y | D > 0] - \E[\Delta Y | D=0]\).
For \(ATT(d|d)\): Nonparametrically estimate \(m_\Delta(d) = \E[\Delta Y|D=d]-\E[\Delta Y|D=0]\)
- This is not actually too hard to estimate. No curse-of-dimensionality, etc.
Slope Effects - must deal with selection bias
Nonparametrically estimate derivative of \(m_\Delta(d)\)
For \(ACR(d)\): Under strong parallel trends, derivative is equal to \(ACR(d)\)
For \(ACR^o\): Average \(ACR(D)\) over \(D>0\); i.e., \(ACR^o = \E[ACR(D)|D>0]\)
Additional Comments:
- Changing the estimation strategy helps with the weights, but it does not fix the issues related to standard vs. strong parallel trends
Estimation - What should you do?
Level Effects - no issues related to selection bias
For \(ATT^o\): Binarize treatment, \(ATT^o = \E[\Delta Y | D > 0] - \E[\Delta Y | D=0]\).
For \(ATT(d|d)\): Nonparametrically estimate \(m_\Delta(d) = \E[\Delta Y|D=d]-\E[\Delta Y|D=0]\)
- This is not actually too hard to estimate. No curse-of-dimensionality, etc.
Slope Effects - must deal with selection bias
Nonparametrically estimate derivative of \(m_\Delta(d)\)
For \(ACR(d)\): Under strong parallel trends, derivative is equal to \(ACR(d)\)
For \(ACR^o\): Average \(ACR(D)\) over \(D>0\); i.e., \(ACR^o = \E[ACR(D)|D>0]\)
Additional Comments:
- It’s relatively straightforward to extend this strategy to settings with multiple periods and variation in treatment timing by extending existing work about a staggered, binary treatment
4. 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.
In the paper, we provide some theoretical arguments concerning properties of production functions that suggests that strong parallel trends holds.
Data
Annual hospital-reported data from the American Hospital Association, 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 fraction 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
ATE(T) Plot
\(\widehat{ATT}^o = 0.80~~(\textrm{s.e}.=0.05)\)
ACR(T) Plot
Results
Results
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 continuous treatment
The extension to multiple periods and variation in treatment timing is relatively straightforward (at least proceeds along the “expected” lines)
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
Thank you
Appendix
References
Can you relax strong parallel trends?
Some ideas:
- It could be reasonable to assume that you know the sign of the selection bias. This can lead to (possibly) informative bounds on differences/derivatives/etc. between \(ATT(d|d)\) parameters
- If interest is mainly in \(ACRT\)’s, then can weaken the assumption to something like “local” strong parallel trends
Strong parallel trends may be more plausible after conditioning on some covariates.
- For length of school closure, strong parallel trends probably more plausible conditional on being a rural county in the Southeast or conditional on being a college town in the Midwest.
Positive Side-Comments: No untreated units
It’s possible to do some versions of DiD with a continuous treatment without having access to a fully untreated group.
In this case, it is not possible to recover level effects like \(ATT(d|d)\).
However, notice that \[\begin{aligned}& \E[\Delta Y | D=d_h] - \E[\Delta Y | D=d_l] \\ &\hspace{50pt}= \Big(\E[\Delta Y | D=d_h] - \E[\Delta Y(0) | D=d_h]\Big) - \Big(\E[\Delta Y | D=d_l]-\E[\Delta Y(0) | D=d_l]\Big) \\ &\hspace{50pt}= ATT(d_h|d_h) - ATT(d_l|d_l)\end{aligned}\]
In words: comparing path of outcomes for those that experienced dose \(d_h\) to path of outcomes among those that experienced dose \(d_l\) (and not relying on having an untreated group) delivers the difference between their \(ATT\)’s.
Still face issues related to selection bias / strong parallel trends though
Positive Side-Comments: Alternative approaches
Strategies like binarizing the treatment can still work (though be careful!)
If you classify units as being treated or untreated, you can recover the \(ATT^o\) by comparing mean outcome for treated relative to untreated.
On the other hand, if you classify units as being “high” treated, “low” treated, or untreated — our arguments imply that selection bias terms can come up when comparing effects for “high” to “low”
Negative Side-Comment: 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. In particular, the relevant information that we have for testing each one is the same
- In effect, the only testable implication of strong parallel trends in pre-treatment periods is standard parallel trends.
Ex. Mixture of Normals Dose
Ex. Exponential Dose
