Concept

Partial identification — where it appears

Reporting the set of parameter values the data cannot rule out rather than a single estimate. It is the honest answer when the assumptions on hand pin a parameter only to a range, and the range can be wide enough to contain values with opposite signs.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The first stage an instrument needs is set by the violation nobody can see. The error each estimator converges on when the instrument has a direct effect of 0.05 on the outcome — a path the exclusion restriction asserts is zero and no sample can check. The instrument's error is δ/π exactly, so it is the reciprocal of the very quantity that made the method work: 1.0000 at a first stage of 0.05 and 0.0833 at 0.60. Least squares carries the confounding instead, at 0.3440 at a first stage of 0.30. The two cross at π = 0.1389, and the crossing is exactly δ times 2.7778 — the first stage an instrument needs is proportional to the violation it is assumed not to have, and below that line the method being corrected is the better estimator.

The assumption nothing tests

An instrument buys a causal effect with an assumption no sample can check, and the price is set by the same quantity that made the method work. The first stage it needs is 2.7778 times the violation it is assumed not to have, so a direct effect of 0.05 demands a first stage of 0.1389 and least squares wins below it.

instrument · Exclusion

The interval that over-covers when the instrument fails. Counted coverage of two nominal 95.0% intervals for the same causal effect, read off the same 2000 draws of 200 rows at each first stage. The exact Anderson–Rubin set covers 95.3% at every setting — flat, because the statistic it inverts is built from y − tβ, which contains no π at all, and is therefore the same number on the same draw whatever the instrument is worth. The conventional interval covers 99.1% at π = 0.02 and 95.6% at π = 0.6: it goes wrong at the weak end by covering too MUCH, at a median width of 7.320, because its standard error is computed from residuals taken at an estimate that has itself gone wrong. A weak instrument does not make this interval lie about its coverage; it makes it useless while telling the truth.

What the first stage does not know

A single weak instrument does not make the conventional interval undercover — it makes it cover 99.1% at a width of 7.320. Where the promise actually breaks is many instruments — coverage falls from 97.2% to 51.5% while the median width falls from 1.454 to 0.583.

instrument · Exclusion

Two worlds, one Kaplan–Meier curve, two truths. World A gives each subject a frailty with mean one and variance 1, and multiplies both its event hazard (0.35) and its dropout hazard (0.5) by it, so the subjects likeliest to leave are the ones likeliest to fail. World B has independent event and dropout times whose hazards are world A's crude hazards. Kaplan–Meier over 1000 studies of 400 gives the same curve from both — 0.7750 and 0.7763 at t = 1; 0.6627 and 0.6641 at t = 2; 0.5924 and 0.5932 at t = 3; 0.5047 and 0.5042 at t = 5 — and that curve is world B's truth, 0.5052 at t = 5. World A's truth is 0.3636 there. The dashed lines are the two bounds that assume nothing, from every dropout failing on leaving (0.1905 at t = 5) to none ever failing (0.6667).

A dropout the data cannot see

Two worlds leave the same record to the last detail a study can write down — the same times, the same share ending in the event, the same share leaving first — and a log-rank test between them rejects at its own 5% level at every sample size from a hundred to sixteen hundred. Kaplan–Meier converges on 0.5052 at t = 5 from both. The truth is 0.5052 in one and 0.3636 in the other, and what is left to argue about is where between two bounds to stand.

survival · Censoring

Named alongside it

The objects these essays reach for when they reach for this one.

First stage Non-identifiability Observational equivalence Anderson rubin Asymptotic bias Causal effect Cause specific hazard Censoring Closed form Competing risks Concentration parameter Conditional coverage

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