DID designs for repeated cross sections, a binary response variable, and common timing.

[Daniel Lipsey]

Hi all,

I am trying to figure out what my options are w.r.t difference in differences designs with common timing, repeated cross sections, and a binary outcome.

I have individual patient-level repeated cross sections from 85 clinics over T = 26 months (and counting). On average, each clinic has about 75 patients per month, but there is wide variation. The two treated clinics are much larger, averaging ~900 patients per month. I am hoping to stratify by specialty (I have 4). On average, within each specialty-clinic-month cell there are about 22 patients (but among treated clinics there are close to 240 on average). The two large clinics I mentioned were treated on the exact same date, so I'm dealing with common timing. I have 6 months of post-treatment observations and 20 months of pre-treatment observations.

I am planning on estimating LPMs with clinic and month dummies (both a static (i.e. T = 2 case) and dynamic TWFE specification), but I was also hoping to estimate a non-linear model.

I am confused about a few things:

(1) can I apply pooled QMLE as described by Wooldridge 2023 (see link) with repeated cross-sections because I don't have staggered timing?

(2) if not, will a GLM with a logit or probit link function yield consistent estimates? I am concerned about/don't fully understand: the incidental parameters problem I might run into with the clinic and time dummies AND interpreting interaction terms (computing cross-partial effects) in a non-linear setting.

Thank you!

Jeffrey M Wooldridge, Simple approaches to nonlinear difference-in-differences with panel data, The Econometrics Journal, Volume 26, Issue 3, September 2023, Pages C31–C66, https://doi.org/10.1093/ectj/utad016

[Maxence Morlet]

I can help you on your second question. I would probably go for LPM in your case. Then perhaps apply the trimmed estimator of Horrace and Oaxaca (2006) if you have a lot of out of sample predicted values.

I would not use probit. Logit on the other hand has a sufficient statistic for the incidental parameter, here clinic fixed effects. This eliminates the IPP, all else equal: this would be xtlogit, fe if my memory serves.

You are correct: interaction terms in nonlinear models are very complicated. Here are two references that should help:

INTERACTION TERMS IN POISSON AND LOG LINEAR REGRESSION MODELS - Shang - 2018 - Bulletin of Economic Research - Wiley Online Library

Interaction terms in logit and probit models - ScienceDirect

[Daniel Lipsey]

Thank you, Maxence! Will read up on all of the above.

If anyone has thoughts about potential for pooled QMLE with repeated cross sections please let me know. Thanks again!

[George Ford]

I think pooled QMLE would be suitable. Do both LPM and QMLE and compare. I doubt much difference between the two.

[Daniel Lipsey]

Thank you, Dr. Ford!