Normal Distribution
nothing is called 95% until it has been counted
Concept

Exclusion restriction — where it appears

The assumption that an instrument affects the outcome only through the treatment. It is not testable from the data, because a world in which it holds and a world in which it fails can have identical joint distributions of instrument, treatment and outcome, and the estimator reports the same number in both.

Named by 2 essays across one field — 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 same error, caught or invisible, by how it is arranged. The overidentification test's rejection rate against the error the violation actually puts into the estimate, so the two rows are the same estimate being equally wrong. With the whole violation on one instrument the test keeps its size at 5.0% under the null and reaches 86.4% by an error of 0.800. With both instruments violating in the same ratio as their first stages the two Wald ratios are identical, the test has nothing to compare, and it rejects at 5.8% at that same error — its own size. Over 1000 draws of 300 rows at each setting, at a nominal 5.0%. The test is a comparison between instruments and it was never a check on either.

Two instruments that disagree

The overidentification test keeps its size at 5.0% and reaches 86.4% power against a violation carried by one instrument. Against the same error carried by both in proportion to their first stages it rejects on 4.6% of draws — its own size — while the estimate is wrong by 0.3000, which is 94.2% of the confounding the instruments were brought in to remove.

instrument · Exclusion

Named alongside it

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

Asymptotic biasFirst stageInstrumental-variableObservational equivalenceUnmeasured confoundingWald ratioCausal effectChi squaredClosed formConsistencyCritical valueEndogeneity

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