“Tienes IMSS” — you already have it, you just didn't know.
Public high-school and college students had been technically entitled to national health coverage (IMSS) since 1998, registered in bulk by their schools under shared numbers nobody used. In 2016, a joint government campaign gave each of 6.6 million students their own permanent social-security number — the same one formal employers use to hire — and told them, loudly, that they had it.
A nationwide rollout has no control group. It has a control gradient.
Everyone was eligible at once, so there's no state that got the policy and no state that didn't. But some states started far more informal than others. I compare birth cohorts still in school in 2016 against slightly older cohorts already past school — and ask whether that cohort gap widens more in the states where informality, and so the stakes, were highest.
How the comparison is built, formally
Long difference, by state s: Ys,post − Ys,pre = β·Is,pre + Xs,preΓ + α + εs, where Is,pre is the state's 2013–15 informality rate and Y is a labor-market outcome for the exposed-minus-unexposed cohort gap. This clean two-group comparison is why the long difference excludes the two partially exposed cohorts (1999–2000, one and two years of high-school exposure respectively). A pooled panel version keeps them: it adds cohort, state, and quarter fixed effects and interacts pre-policy informality with a continuous cohort-level exposure function, 2016–2019, so partial exposure enters with partial weight rather than being dropped. Weights use the square root of the surveyed population in each cell; standard errors are wild-cluster-bootstrapped by state (32 clusters).
Maybe poor states were just going to catch up anyway.
The main threat isn't a confound in the usual sense — it's mean reversion: high-informality (poorer) states converging toward richer ones regardless of any policy. If that convergence already existed among cohorts too old to be exposed, it would show up as drift in this figure before the eligibility ramp.
- Gradients sit in a narrow, flat band among cohorts too old to be exposed, and break exactly at the eligibility ramp — not gradually before it.
- A placebo design run entirely on untreated cohorts (1995–98 vs. 1991–94) returns null or opposite-signed estimates.
- Pre-2016 gaps between high- and low-informality states move in parallel, quarter by quarter, before rollout.
That's reassuring — but a pass/fail glance at a chart is a weak test with only 32 states, and treating it as a screen would itself distort the inference that follows. So the next step doesn't ask whether the assumption holds. It asks how much it could fail and the result would still stand.
Stop asking “did trends stay parallel?” Ask “how far off could they be?”
Every cohort-comparison design like this one rests on one assumption: absent the policy, the untreated cohorts' trend is a good stand-in for what the treated cohorts would have done. Nobody can observe that counterfactual directly — and with 32 states, a chart like the one on the last slide can't rule out a violation large enough to matter.
Rambachan & Roth's fix: instead of assuming the violation is exactly zero, let it be anything up to a size M̄, scaled relative to the largest wobble actually observed among cohorts that couldn't have been treated. Then ask — as M̄ grows from zero — at what point does the confidence set for the effect start to include zero. That crossing point is the breakdown value. A big one means the result would survive a violation much larger than anything already visible in the data; a small one means it's fragile.
Schematic — illustrates the logic, not the paper's actual estimates.
Applied to this design — birth cohort as event time, normalized on the last fully unexposed cohort — the formality result survives a post-eligibility violation of parallel cohort trends up to 15% larger than the worst violation already visible among untreated cohorts. The weaker participation margin breaks under a much smaller one. Under a linear-drift restriction that extrapolates the untreated trend forward instead of assuming it away, the formality estimate rises, since that drift already runs against the result.
Implementation notes
The procedure needs a full covariance matrix, which the paper's usual wild-cluster bootstrap doesn't supply — cluster-jackknife (CV3) standard errors are used instead, the natural choice with 32 clusters. Using CRV1 instead gives M̄ = 1.38 for formality; normalizing on the following cohort instead of the last unexposed one gives 0.78. Both alternatives point the same direction.
