What a sample shows, and what the algebra does

The difference between a rectangular block's implied long-run variance and a trapezoidal one's, as a share of the truth. Above the axis the rectangle is less biased and below it the trapezoid is. The heavy line is exact — computed from the law's own autocovariances — and it crosses at 19.2. The others are what samples of 120, 240, 480, 960 rows report, and every one of them exaggerates whichever window is ahead: at ℓ = 20, where the exact difference is 0.28 points, a sample of 120 rows shows 4.31 points — 15 times larger. That is the number the earlier reading of this comparison was missing: three tenths of a point is what the algebra says and not what a hundred and twenty rows report.

A trapezoidal block at 120 rows, with the error split into the two things it is made of. The bias falls with the block length, because a longer block attenuates less, and it flattens at 23.2% because the sample's own autocovariances are short whatever window is applied to them. The spread rises with it, because a longer block means fewer of them. Their sum in quadrature has a minimum at ℓ = 16, which is not where either of the two has one. The faint line is the rectangle's total error, for scale: it is above the trapezoid's from ℓ = 12 onwards.

The number of draws each instrument needs to separate a rectangular block from a trapezoidal one at two standard errors, at a block length of 20 and 120 rows — measured from each instrument's own spread on the same draws. The implied variance needs 7.0 and the 95% point needs 20.2, a factor of 2.90 at this block length. There is a closed form beside it and it does not depend on either the scale or the size of the gap: the standard error of a p-quantile is √(p(1−p))/f(q) over √B where a standard deviation's is σ/√(2B), which at the 95% point of a nearly normal reference distribution is 3.30 times as many draws for the same statement. And the quantile route needs every one of those draws resampled, where the variance route needs none.