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

Wald ratio — where it appears

The ratio of the instrument's effect on the outcome to its effect on the treatment. It is the just-identified instrumental estimate written directly, and with a binary instrument and binary treatment it converges on the average effect among the units the instrument moves.

Named by 5 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 A weak instrument gives back the problem it was hired for. The counted mean bias of two-stage least squares at 4 instruments and 200 rows, over 2000 draws a setting, against the standard approximation and against the least-squares inconsistency the instrument was brought in to remove. At π = 0.02 the counted bias is 0.3220 ± 0.0142 where least squares is out by 0.3594 — 89.6% of the way back. At π = 0.3 it is 0.0118 against 0.2647. The approximation, the inconsistency over the population first-stage F, tracks the count at the weak end and sits above it in the middle: 0.968, 0.971, 0.918, 0.810, 0.740, 0.722, 0.846 as the ratio of counted to approximated bias.

Weak, and back where it started

A consistent instrumental estimate at two hundred rows and a concentration parameter of 0.32 is biased by 0.3220 ± 0.0142 against a least-squares inconsistency of 0.3594 — 89.6% of the way back to the problem it was hired to solve. Just identified, it has no mean at all, and that is measured as a rate rather than assumed.

instrument · Exclusion Four numbers, and only one of them is the question. Four quantities in a population where the effect is not the same for everybody: 40.0% compliers, 25.0% always-takers and 35.0% never-takers, with the compliers carrying an effect 1.0000 larger than everybody else's. The population average effect is 0.5000. The compliers' average effect is 1.1000. A valid instrument converges on 1.1000 — the second of those, not the first, and the two differ by 0.6000. Comparing the treated with the untreated as they stand gives 1.4182, wrong by 0.9182 in the same direction, because always-takers start 1.5000 above never-takers before any treatment happens. The instrument removes the selection and changes the question at the same time, and only one of those is reported.

Whose effect it is

With a perfectly valid instrument and no violation of anything, the estimate converges on 1.1000 where the population average effect is 0.5000. The gap is exactly θ(1 − p_c), the always-takers and never-takers cancel out of both halves of the ratio, and five per cent defiers move the answer to 1.2667.

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 The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

A ratio whose interval has to be the whole line

The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.

normal · Clt

Named alongside it

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

First stageInstrumental-variableAsymptotic biasCausal effectConsistencyEndogeneityExclusion restrictionLeast squaresObservational equivalenceTwo-stage least squaresUnmeasured confoundingWeak instrument

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