# Diff-in-Diff with different treatment timings for different countries

I am trying to estimate the effect of Universal Primary Education programs on domestic violence in a sample of Sub-Saharan countries. This program has been implemented in a different year in each country (e.g. 1994 in Malawi, 2001 in Tanzania and so on). Thus I am wondering which experimental design is better for this kind of data.

I was thinking about using a Diff-in-Diff but I need a different pre/post dummy for each of the countries of my analysis because the treatment is unique but it has been implemented in each country in a different year. Thus, my idea was to do something like

\begin{gather} Y=\alpha T+\beta\sum_{i=1}^ND_i+\delta\sum_{i=1}^ND_i*T+\epsilon \end{gather} where $T$ is the treatment dummy that is always the same and $D_i$ are the different year dummies. Moreover, I would add country fixed effects to keep constant the cross-country differences.

Is this specification meaningful? Or should I think of something different?

What you are attempting to do is sometimes referred to as staggered difference-in-differences. So you have units which receive a treatment at different points in time. Your specification, $$y_{it} = \alpha_i + \delta_t + \sum_{k} \gamma_k D_{ik} + \epsilon_{it}$$

would be one way of dealing with this, where $y_{it}$ is the outcome for country $i$ in year $t$, $\alpha_i$ and $\delta_t$ are country and year dummies, and $D_{ik}$ is an indicator which is equal to one for country $i$ adopting the policy in all $k\geq t$ periods. This is similar to the approach taken in Stevenson and Wolfers (2006) in the Quarterly Journal of Economics.

If you are interested in plotting the pre-treatment trends, you can standardize the time dimension as $m$ periods before and $g$ periods after the treatment. In this case you have a certain time window around the adoption of the policy, $(m,...,-3,-2,-1,0,1,2,3,...,g)$ where $0$ is the last pre-treatment period. So your data would look something like, for example,

i     t      time    D
1     2001   -2      0
1     2002   -1      0
1     2003    0      0
1     2004    1      1
1     2005    2      1
2     1997   -2      0
2     1998   -1      0
2     1999    0      0
2     2000    1      1
...


and so on. Then you can regress $$y_{ij} = \alpha_i + \lambda_j + \sum_{j = m}^{-1} \pi_j T_{ij} + \sum_{j=1}^{g} \phi_{j}K_{ij} + \epsilon_{ij}$$

where $j = (m,...,3,2,1,0,1,2,3,...,g)$, and $T_{ij}$ are interactions of the treatment indicator (which equals one if country $i$ has ever adopted the policy) and time dummies for all periods before time $0$. Likewise $K_{ij}$ is the treatment indicator interacted with time dummies for all periods after time $0$. When you plot the $\pi_j$ and $\phi_j$, you will then get a graph like this:

Note though that this is a stylized graph. It would be a bit suspicious if ALL your $\pi_j$ are exactly zero, but ideally you should not see any significant coefficients for the treatment before time $0$ (taking into account the confidence intervals). The $\phi_j$ coefficients will then tell you how long it takes for your treatment to reach its full effect, or whether it persists/fades out after implementation.

Another recent (working) paper which uses staggered diff-in-diff and that you can consult is Gipper et al (2016). Hope this helps!

• where is Control group in this setting? Nov 21 '18 at 21:17
• Ran across this when looking for answers to a similar question. FYI, this first approach has been shown to be biased if effects change over time (see Goodman-Bacon 2018). The second approach is basically an event study which is better but requires good data. Mar 15 '19 at 21:56
• Very helpful @Danielle. More generally, can this type of set up be used when there are multiple treatments? Say, some countries got an educational program and others got a monetary aid package? Mar 18 '19 at 19:39
• @andy I'm wondering if perhaps you got the interpretation of 𝜋𝑗 and 𝜙𝑗 reversed? Based on my reading of Angrist (2009) Mostly harmless econometrics eq 5.2.6), 𝜋𝑗 is actually giving you the postreatment effect and 𝜙𝑗 gives you the anticipatory effect. e.g. think about it this way: if you regress Y_0 ~ 𝜋_(-1) and 𝜙_2, this is telling you that the 𝜋 at time -1 has a later effect on Y at time 0, it's is measuing the lag. Similarly, for 𝜙𝑗
– dval
Oct 9 '20 at 19:27
• @dvaaaaaaaalllllll111111111llll You are correct. It doesn’t comport with what is in their book. It doesn’t matter though, as long as you are clear about what is your lead and what is your lag. See a similar question I asked a while ago. Nov 26 '20 at 2:26