30.14 Mediation Under DiD

Mediation analysis helps determine whether a treatment affects the outcome directly or through an intermediate variable (mediator). In a DiD framework, this allows us to separate:

1. Direct effects: The effect of the treatment on the outcome independent of the mediator.
2. Indirect (mediated) effects: The effect of the treatment that operates through the mediator.

This is useful when a treatment consists of multiple components or when we want to understand mechanisms behind an observed effect.

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30.14.1 Mediation Model in DiD

To incorporate mediation, we estimate two equations:

Step 1: Effect of Treatment on the Mediator

$$M_{it} = \alpha + \gamma Treat_i + \lambda Post_t + \delta (Treat_i \times Post_t) + \epsilon_{it}$$ where:

• $M_{it}$ = Mediator variable (e.g., job search intensity, firm investment, police presence).
• $\delta$ = Effect of the treatment on the mediator (capturing how the treatment changes $M$).

Step 2: Effect of Treatment and Mediator on the Outcome

$$Y_{it} = \alpha' + \gamma' Treat_i + \lambda' Post_t + \delta' (Treat_i \times Post_t) + \theta M_{it} + \epsilon'_{it}$$ where:

• $Y_{it}$ = Outcome variable (e.g., employment, crime rate, firm performance).
• $\theta$ = Effect of the mediator on the outcome.
• $\delta'$ = Direct effect of the treatment (controlling for the mediator).

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30.14.2 Interpreting the Results

• If $\theta$ is statistically significant, it suggests that mediation is occurring—that is, the treatment affects the outcome partly through the mediator.
• If $\delta'$ is smaller than $\delta$, this indicates that part of the treatment effect is explained by the mediator. The remaining portion of $\delta'$ represents the direct effect.

Thus, we can decompose the total treatment effect as:

$$\text{Total Effect} = \delta' + (\theta \times \delta)$$

where:

• $\delta'$ = Direct effect (holding the mediator constant).

• $\theta \times \delta$ = Indirect (mediated) effect.

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30.14.3 Challenges in Mediation Analysis for DiD

Mediation in a DiD setting introduces several challenges that require careful consideration:

1. Potential Confounding of the Mediator

• A key assumption is that no unmeasured confounders affect both the mediator and the outcome.
• If such confounders exist, estimates of $\theta$ may be biased.

2. Mediator-Outcome Endogeneity

• If the mediator is itself influenced by unobserved factors correlated with the outcome, it introduces endogeneity, making direct OLS estimates of $\theta$ problematic.
• For example, in a crime policy evaluation:
  • The number of police officers (mediator) may be influenced by crime rates (outcome), leading to reverse causality.

3. Interaction Between Multiple Mediators

• If there are multiple mediators (e.g., a policy that increases both police presence and surveillance cameras), they may interact with each other.
• A useful test is to regress each mediator on treatment and other mediators. If a mediator predicts another, their effects are not independent, complicating interpretation.

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30.14.4 Alternative Approach: Instrumental Variables for Mediation

One way to address mediator endogeneity is to use instrumental variables, where treatment serves as an instrument for the mediator:

Two-Stage Estimation:

1. First Stage: Predict the Mediator Using the Treatment $$M_{it} = \alpha + \pi Treat_i + \lambda Post_t + \delta (Treat_i \times Post_t) + \nu_{it}$$
2. Second Stage: Predict the Outcome Using the Instrumented Mediator $$Y_{it} = \alpha' + \gamma' Treat_i + \lambda' Post_t + \phi \hat{M}_{it} + \epsilon'_{it}$$

• Here, $\hat{M}_{it}$ (predicted values from the first stage) replaces $M_{it}$, eliminating endogeneity concerns if the exclusion restriction holds (i.e., treatment only affects $Y$ through $M$).

Key Limitation of IV Approach

• The IV strategy assumes that treatment affects the outcome only through the mediator, which may be too strong of an assumption in complex policy settings.

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