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

Leave-one-out — where it appears

Weighting a point's squared residual by 1/(1 − h)², which approximates what the residual would have been had the fit not seen that point. It is the dearest of the four leverage corrections and the only one that over-states the variance, which is what buys its coverage at small samples.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Robust, at the sample sizes it is reached for. Counted coverage of five 95% intervals for a slope, at six sample sizes, under an error variance leaning towards the edges of the design (γ = 0.8), over 20000 draws at the small end. The model-based interval sits at about 87.06% everywhere and does not improve with the sample, because it is a claim about a variance it is not estimating. The robust ones do improve: HC0 covers 88.73% at 20 rows, 92.70% at 50 and 94.93% at 1,000. Its promise is asymptotic and its use is not, and the gap between those two facts is this picture. The leave-one-out correction read against a t on n − 2 is the only line that is near its promise at the small end: 94.55% at 20 rows.

Robust is not free

A robust standard error's promise is asymptotic and its use is not. Its 95% interval covers 88.73% at twenty rows, and under mild heteroskedasticity it is the worse of the two intervals until a hundred.

sandwich · Misspecification What each correction is worth, exactly. Each variance estimate's expectation under a constant error variance, divided by the variance the slope actually has, at 20 rows on an even design, by two routes: the closed form E[eᵢ²] = σ²(1 − hᵢᵢ) carried through each correction's own weight, and the mean of 20000 counted estimates. The maximum leverage here is 0.1857 and the design's fourth-moment share Σu⁴/(Σu²)² is 0.0897, which is the only thing the closed form reads. HC0 comes out at 0.8603 — short by construction, since its factor is exactly 1 − 1/n − Σu⁴/(Σu²)². HC1 reaches 0.9559, HC2 is exactly 1.0000 at every design and every sample size, and HC3 overshoots to 1.1647. On an even design the four are within a fifth of each other and the choice barely matters.

Three corrections and a leverage

On an even design of twenty rows the four robust corrections read 0.8603, 0.9559, 1.0000 and 1.1647 of the truth and the choice barely matters. Add one point at x = 8 and they read 0.3191, 0.3419, 1.0000 and 5.1127.

sandwich · Misspecification The price of insurance is noise, not coverage. What a robust standard error costs under a constant error variance — the case where the model-based one is exactly right — at five sample sizes over 20000 draws at the small end. It is not coverage: the leave-one-out interval read against a t on n − 2 covers 95.52% at 20 rows against the model-based 95.06%. It is a wider interval, by a factor of 1.0689 at 20 rows falling to 1.0049 at 250, and it is a variance estimate 2.57 times as variable at 20 rows and 1.85 times at 250 — against a denominator that is exactly V²·2/(n − 2), so only the numerator is counted. The uncorrected estimate is the cheaper of the two at small samples and the dearer at large: 1.31 against 1.76.

Right for the wrong reason

A robust standard error costs no coverage where the risk is absent — 95.52% against 95.06% at twenty rows. It costs a 6.89% wider interval and a variance estimate 2.572 times as variable, and the pre-test that would avoid paying recovers 15.9% of what the insurance is worth.

sandwich · Misspecification Two far rows, and the line with one of them deleted. Twenty clean points and two rows near x = 9. The slope is −0.511 with every row, −0.376 with one far row deleted, and 0.495 with both deleted. Deleting one of them barely moves the line, because the other is still there.

Two points that hide each other

One far observation among twenty-one has a Cook's distance of 24.1. Put a second beside it and the two read 0.966 and 0.772, neither crossing 1, while together they reverse the slope and deleting both moves the fit by 53.3.

regression · Leverage Three intervals as one strength is spread thinner, at a concentration of 8. Coverage of three nominal 95.0% intervals on the same 1000 draws of 200 rows at each count, when a total concentration parameter of 8 is spread over 1 to 32 instruments. Two-stage least squares covers 97.2%, 96.4%, 94.0%, 86.7%, 73.2%, 51.5%. Building each row's fitted treatment from a first stage that never saw that row covers 97.1%, 97.2%, 98.3%, 97.8%, 97.9%, 98.7%. Limited-information maximum likelihood with its conventional standard error covers 97.2%, 96.7%, 95.7%, 90.9%, 85.2%, 79.0%. At one instrument the likelihood estimator is two-stage least squares exactly, which is why the first readings of those two agree to the last draw.

Leaving each row out of its own first stage

Spread a fixed first-stage strength over thirty-two instruments and two-stage least squares covers 51.5%. Build each row's fitted treatment from a first stage that never saw that row and the same draws cover 98.7% — through an interval 5.99 times as wide, around an estimate that misses by more than the whole effect on 34.7% of draws. At eight times the strength the same repair covers 95.3% and costs a width factor of 1.66.

instrument · Exclusion

Named alongside it

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

Closed formInterval widthLeverageMonte CarloRobust standard errorConfidence intervalCoverageEstimated varianceFinite-sample correctionHat matrixHeteroskedasticityHeteroskedasticity-consistent

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