30.13 Multiple Treatments

In some settings, researchers encounter two (or more) treatments rather than a single treatment and control group. This complicates standard DiD estimation, but a properly structured model ensures accurate identification.

Additional References

• (Fricke 2017): Discusses identification challenges in multiple treatment settings.
• (Clement De Chaisemartin and D’haultfœuille 2023): Provides a video tutorial (YouTube) and code (Google Drive) for implementing multiple-treatment DiD models.

Key Principles When Dealing with Multiple Treatments

1. Always include all treatment groups in a single regression model.

   • This ensures proper identification of treatment-specific effects while maintaining a clear comparison against the control group.

2. Never use one treated group as a control for the other.

   • Running separate regressions for each treatment group can lead to biased estimates because each treatment group may differ systematically from the control group in ways that a separate model cannot fully capture.

3. Compare the significance of treatment effects ($\delta_1$ vs. $\delta_2$).

   • Instead of assuming equal effects, we should formally test whether the effects of the two treatments are statistically different using an F-test or Wald test:

   $$H_0: \delta_1 = \delta_2$$

   • If we reject $H_0$, we conclude that the two treatments have significantly different effects.

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30.13.1 Multiple Treatment Groups: Model Specification

A properly specified DiD regression model with two treatments takes the following form:

$$\begin{aligned} Y_{it} &= \alpha + \gamma_1 Treat1_{i} + \gamma_2 Treat2_{i} + \lambda Post_t \\ &+ \delta_1(Treat1_i \times Post_t) + \delta_2(Treat2_i \times Post_t) + \epsilon_{it} \end{aligned}$$

where:

• $Y_{it}$ = Outcome variable for individual $i$ at time $t$.
• $Treat1_i$ = 1 if individual $i$ is in Treatment Group 1, 0 otherwise.
• $Treat2_i$ = 1 if individual $i$ is in Treatment Group 2, 0 otherwise.
• $Post_t$ = 1 for post-treatment period, 0 otherwise.
• DiD coefficients:
  • $\delta_1$ = Effect of Treatment 1.
  • $\delta_2$ = Effect of Treatment 2.
• $\epsilon_{it}$ = Error term.

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30.13.2 Understanding the Control Group in Multiple Treatment DiD

One common concern in multiple-treatment DiD models is how to properly define the control group. A well-specified model ensures that:

• The control group consists only of untreated individuals, not individuals from another treatment group.
• The reference category in the regression represents the control group (i.e., individuals with $Treat1_i = 0$ and $Treat2_i = 0$).
• If $Treat1_i = 1$, then $Treat2_i = 0$ and vice versa.

Failing to correctly specify the control group could lead to incorrect estimates of treatment effects. For example, omitting one of the treatment indicators could unintentionally redefine the control group as a mix of treated and untreated individuals.

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30.13.3 Alternative Approaches: Separate Regressions vs. One Model

A common question is whether to run one large regression including all treatment groups or to run separate DiD models on subsets of the data. Each approach has implications:

1. One Model Approach (Preferred)

• Running one comprehensive regression allows for direct comparison between treatment effects in a statistically valid way.
• The interaction terms ($\delta_1, \delta_2$) ensure that each treatment effect is estimated relative to a common control group.
• The F-test (or Wald test) enables a formal test of whether the two treatments have significantly different effects.

2. Separate Regressions Approach

• Running separate DiD models for each treatment group can still be valid, but:
  • The estimated treatment effects are less efficient because they come from separate samples.
  • Comparisons become less straightforward, as they rely on confidence interval overlap rather than direct hypothesis testing.
  • If homoscedasticity holds (i.e., equal error variances across groups), the separate regressions approach is unnecessary. The combined model is more efficient.

Thus, unless there is strong justification for separate regressions (e.g., significant heterogeneity in error variance), the one-model approach is preferred.

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30.13.4 Handling Treatment Intensity

In some cases, treatments differ not just in type, but also in intensity (e.g., low vs. high treatment exposure). If we observe different levels of treatment intensity, we can model it using a single categorical variable rather than multiple treatment dummies:

Rather than coding separate dummies for each treatment group, we define a multi-valued treatment variable:

$$Y_{it} = \alpha + \sum_{j=1}^{J} \beta_j (Treatment_j \times Post_t) + \lambda Post_t + \epsilon_{it}$$

where:

• $Treatment_j$ is a categorical variable indicating whether an individual belongs to the control group, low-intensity treatment, or high-intensity treatment.
• This approach allows for cleaner implementation and avoids excessive interaction terms.

This approach has the advantage of:

• Automatically setting the control group as the reference category.

• Ensuring correct interpretation of coefficients for different treatment levels.

30.13.5 Considerations When Individuals Can Move Between Treatment Groups

One potential complication in multiple-treatment DiD settings is when individuals can switch treatment groups over time (e.g., moving from low-intensity to high-intensity treatment after policy implementation).

• If movement is rare, it may not significantly affect estimates.

• If movement is frequent, it creates a challenge in causal identification because treatment effects might be confounded by self-selection.

A possible solution is to use an intention-to-treat (ITT) approach, where treatment assignment is based on the initially assigned group, regardless of whether individuals later switch.

30.13.6 Parallel Trends Assumption in Multiple-Treatment DiD

• Just as in standard DiD, a key assumption in multiple-treatment DiD models is that the treatment and control groups would have followed parallel trends in the absence of treatment.

• With multiple treatments, we must check pre-trends separately for each treated group against the control group.

• If pre-treatment trends are not parallel, we may need to adopt alternative methods such as synthetic control models or event study analyses.

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References

De Chaisemartin, Clement, and Xavier D’haultfœuille. 2023. “Two-Way Fixed Effects and Differences-in-Differences Estimators with Several Treatments.” Journal of Econometrics 236 (2): 105480.

Fricke, Hans. 2017. “Identification Based on Difference-in-Differences Approaches with Multiple Treatments.” Oxford Bulletin of Economics and Statistics 79 (3): 426–33.