Staggered Diff-in-Diff with Binary Dependent Variable

[Maika Schmidt]

Good morning,

we are trying to estimate a staggered treatment diff-in-diff using repeated cross-sectional data. The sample is not huge and time-periods are unequally spaced, so I do not want to use Callaway Sant'Anna. BJS seemed ideal for this setting, but we have several binary dependent variables of interest.
In particular, we want to estimate the effect of our treatment on vaccination rates (data is at the child level, so a binary variable), which are already relatively high. The linear probability model in BJS predicts counterfactual outcomes above 1 and thus creates an artificial negative treatment effect on vaccination rates.

Is there a way to adjust the Wooldridge/Mundlak estimator for a binary dependent variable? I can see from the paper that Wooldridge does not recommend a Logit FE model, but I don't really understand why or what the alternatives are.

TLDR: Would be very grateful for any ideas on how to estimate a staggered DiD with repeated cross-sections and a binary dependent variable with mean close to 1 (or close to zero).

Thank you!
Maika

[FernandoRios]

Hi Maika
I would normally say try jwdid. But if your data has odd timing, may be easier to do this manually
Yes you can use Wooldrige approach with logit as your estimator. just couple of points. Since you have repeated crossection, probably you have enough data to add group fixed effects. So no worries about fixed effects
F

[Maika Schmidt]

Hi Fernando,

thank you so much (once again)! jwdid looks like the perfect option.

We are struggling very much with our estimation strategy because we estimate different outcomes based on two samples with very different structures.
Some of our analysis uses a balanced panel of towns/villages with regular yearly data because it is based on administrative/satellite data. We have 502 towns/villages that were treated over a ten year period and are observed yearly during this period and for a long period before. For this part we can use almost any of the methods, although I am unsure of whether we can do a logit FE (this is a minor problem here because the variables of interest are mostly not binary).

However, the problematic part of the analysis is based on household survey data, where we have many binary outcome variables of interest, and which has a complicated cross-sectional structure:

I observe households from 203 towns/villages which are treated at different points in time over a ten year period:

year_of_treatment | Freq. Percent Cum.
------------+-----------------------------------
2008 | 14 6.90 6.90
2009 | 13 6.40 13.30
2010 | 13 6.40 19.70
2011 | 18 8.87 28.57
2012 | 17 8.37 36.95
2013 | 20 9.85 46.80
2014 | 32 15.76 62.56
2015 | 43 21.18 83.74
2016 | 18 8.87 92.61
2017 | 15 7.39 100.00
------------+-----------------------------------
Total | 203 100.00

The household data comes from household surveys collected in 1993, 2003, 2010, and 2014 (so some of the villages are effectively never treated). The households are different every time (repeated cross-sections), but more than that: most of the villages/towns only show up once in my sample over the years (only one village is observed across all four survey rounds):

copies | observations surplus
----------+---------------------------
1 | 153 0
2 | 74 37
3 | 36 24
4 | 4 3

I am very unsure what fixed effects to add here (especially when using a logit model).
Obviously, I do not want to include location fixed effects, because that would eliminate most of my sample.
I think cohort fixed effects (so for each year of treatment) would make the most sense, but I don't know if the sample is large enough (within each village I observe approximately 12 households). Alternatively, I could either do coarser treatment cohorts (combining several treatment years into one group), or do away with the cohort fixed effects altogether and just include pre-treatment window (pre 2008) controls that have predictive power for treatment timing. (And possibly add fixed effects for regions, which are not (or at least barely) correlated with treatment timing.)

Apologies for the long message and thank you very much for your help!
Maika