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

A standard error for a model that is wrong

Every standard error a regression prints is a statement about a model that is true, and the repair for a model that is not is one of the most-quoted lines in applied work. It is priced here rather than recommended: the robust error reads a spread the model-based one does not, and its 95% interval covers 88.73% at twenty rows, is the worse of the two under mild heteroskedasticity until fifty, and is the smaller of the two whenever the error variance sits in the middle of the design rather than at its edges. The four corrections differ in one number — what each does with a point's leverage — and on a design with a single far-out point that number takes one of them to 0.3191 of the truth and another to 5.1127 times it.
One wrong model, four designs, four slopes. The slope a straight line converges to when the truth is a quadratic, under four covariate distributions, by two routes: the population projection in closed form, and the mean of 2500 fitted slopes at 200 rows apiece. The even spread over [0, 2] gives 1.6000 and the same spread moved to [1, 3] gives 2.6000, while widening it to [0, 4] gives 2.6000 — the same number as the shifted one, because a symmetric design's target is the truth's tangent slope at the design's own mean and does not read the spread at all. An exponential spread with the SAME mean as the first gives 2.6000. So two studies of one world, each fitting the same wrong model, honestly report slopes 1.0000 apart, and neither is making an error.

What a wrong model estimates

A straight line fitted to a curved truth converges on the tangent at its own design's mean. Two honest studies of one world, fitting the same wrong model, report 2.600000 and 1.600000, and neither is in error.

 What each error is a claim about, and what the claim comes out as. Each variance estimate's average over 20000 draws, divided by the variance the slope actually has across those same draws, at 80 rows with the error variance leaning towards the edges of the design (γ = 0.8). One is a standard error that is right. The model-based estimate reads 0.6081 of the spread, so its standard error is 77.98% of the one it should report; the four robust corrections read 0.9576, 0.9821, 0.9961, 1.0362. Two further routes agree with the count and share none of its arithmetic: n times the counted variance is 4.8905 against a population sandwich of 4.9200, and the counted ratio of the two standard errors is 1.2799 against a closed form of 1.2806.

The bread and the filling

The robust standard error is not a safety margin. At one setting of the error variance it is 1.2806 times the model-based one and at another it is 0.8246 times it, and the sign of a single dial decides which.

 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.

 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.

 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.

 The rows are held fixed; only the clusters move. Counted coverage of four 95% intervals for a slope, at five cluster counts with the row count held at 300 throughout and a within-cluster correlation of 0.1, over 6000 draws apiece, with the sizes equal. The interval that counts rows covers 53.42% at 5 clusters — a second closed form says 2Φ(z/√D) − 1 = 54.44% for a design effect of 6.900, and reads nothing about clusters at all. The cluster-robust interval read against a normal covers 74.43% there and 94.20% at 100 clusters; read against a t on G − 1 it covers 85.08% and 94.47%. The number of independent things is the cluster count, and every quantity here is blind to how many rows were typed.

The count that is not the rows

Three hundred rows in five clusters of sixty carry 6.9000 times the variance an independent-rows calculation reports, and the interval that counts rows covers 53.42%. The same five unequal sizes laid out two ways give design effects of 9.3158 and 5.4652.

 One estimator, three answers, and only the reference changes. Coverage of the cluster-robust 95% interval for the slope against the number of clusters, at 30 rows in each. The estimator is identical in all three curves; what differs is the number it is compared against. At 5 clusters it covers 75.05% against a normal, 85.30% against a t on 4 degrees of freedom and 87.95% against a t on 3. At 80 clusters the three agree to within a point. The correction costs nothing: the same standard error, a different table.

The reference the sandwich is read against

The cluster-robust interval covers 75.05% at five clusters and 93.58% at eighty. The same estimate read against a t on G − 2 covers 87.95% at five, and the estimator is unchanged — three hundred rows grouped into five clusters cover 74.28% where the same three hundred grouped into seventy-five cover 94.63%.

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