#StackBounty: Difference-in-difference model with mediators: Estimating the effect of different elements of a policy

Bounty: 100

How do I conduct a mediation analysis in a difference-in-difference setting? For example, a city selects some neighborhoods for a new crime fighting strategy (the treatment $D$) that involves an increase in the number of police officers on the street (mechanism $M_1$), additional surveillance cameras (mechanism $M_2$), an increase in misdemeanor arrests ($M_3$) and potentially other unmeasured components (e.g. change in leadership etc).

To estimate the effect of $D$ on crime, I can use a simple difference in difference model.

$$ y_{jt} = delta_j + gamma_t + phi D_{jt} + epsilon_{jt} $$

$y$ is the crime rate by neighborhood $j$ and year $t$. $D_{jt}$ signifies the neighborhoods and years in which the new crime fighting strategy was in place. The relevant coefficient for the overall effect of the treatment is $phi$.

Now my question: How do I determine which element of the policy was effective (increase in officers, additional cameras, increase in misdemeanor arrests)? I see two relevant questions here:

  • Did $M_x$ mediate the relation between $D$ and $Y$?
  • What is the effect of $M_x$ on $y$ and can I use the variation in $M_x$ created by the policy to estimate the effect of $M_x$?

Get this bounty!!!

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