The gap a sample shows

A block weighted inside itself establishes what a tapered block attenuates, exactly: the window’s normalised self-convolution, whose first-order term is the sum of the squared values at the two ends and nothing else about the shape. It then compares a trapezoidal block against a rectangular one on the implied bias, and reports that the taper is worse at every block length a hundred and twenty rows can afford, with the two crossing at ℓ = 20 and a difference there of three tenths of a point — too small for its critical-value table to resolve.

Two of those three statements are about the wrong quantity.

The exact gap and the sampled one #

The exact bias is Σ_k κ(k)γ(k) against Σ_k γ(k), with γ the law’s own autocovariances. There is no sample in it. Under a first-order autoregression at 0.7 the two windows’ shares differ by 5.59, 3.02, 1.06, −0.28, −1.20 and −2.15 points across block lengths of 8 to 32, crossing at 19.15.

Now the same quantity computed the way a practitioner computes it, from a sample’s own autocovariances. At a hundred and twenty rows the differences are 4.30, 0.94, −2.00, −4.31, −6.05 and −7.94, and the crossing is at 13.28.

At ℓ = 20 the exact difference is 0.28 points and the sampled one is 4.31 — fifteen times larger, and in the same direction.

Why the sample exaggerates #

κ does not depend on the sample. What does is γ̂.

A sample autocovariance at lag k averages n − k products and subtracts a sample mean, so it is attenuated, and worst at the longest lags. The window is applied to that attenuated sequence rather than to the law’s, and a window that discards the long lags loses less of what has already been lost.

The trapezoid keeps more at short lags and less at long ones than the rectangle does. So on a sequence whose long lags are already too small, the trapezoid is weighting the reliable part more heavily and the unreliable part less — which is an advantage the exact calculation cannot see, because in the exact calculation the long lags are not unreliable, they are correct.

That is the whole mechanism, and it has a prediction attached: the exaggeration should shrink as the sample grows, because γ̂’s attenuation does.

It does. At ℓ = 20 the sampled gap runs −4.31, −2.50, −1.42 and −0.77 at 120, 240, 480 and 960 rows, against an exact −0.28. And the crossing walks out towards the exact one: 13.28, 14.97, 16.39 and 17.99, against 19.15.

The crossing a finite sample shows is at a shorter block length than the asymptotic one, and it walks out as the sample grows. Which is a statement about the instrument rather than about the windows, and it reverses the earlier conclusion at the sample size the earlier conclusion was made about.

The exaggeration is linear in the block length #

Subtracting the two rows gives the sample’s contribution on its own, and it has a simple shape.

At block lengths of 8, 12, 16, 20, 24 and 32 — which the two crossings identify, since 16 + 4 × (1.06/1.34) = 19.16 and 12 + 4 × (0.94/2.94) = 13.28 — the sampled difference is below the exact one by

1.29, 2.08, 3.06, 4.03, 4.85 and 5.79 points.

Between eight lags and twenty-four that is almost exactly 0.21 points per lag, straight. It flattens only over the last stretch, where the window is a quarter of the sample and the attenuation has run out of lags to accumulate over.

Linear is what the $(1 - k/n)$ attenuation has to produce: each extra lag the window admits adds one more term that the sample reports short, and the shortfall per term is roughly constant across the range where $\gamma(k)$ is still substantial. It also says how the effect should scale — halving with the sample length — so the same slope at two hundred and forty rows should be about 0.10 points per lag.

Why the crossing moves as far as it does #

The crossing shifts from 19.15 to 13.28, nearly six lags, and the reason is not that the curve was displaced. It is that the sampled curve is steeper.

Near the crossing the exact difference falls at about 0.23 points a lag. The sampled one falls at about 0.435 — nearly twice as fast, because the exact curve’s own decline and the attenuation’s 0.21 a lag add together.

A curve twice as steep crosses zero at a very different place for the same displacement, which is why a sub-point discrepancy at one length becomes a six-lag error in the answer.

And why the ratio at ℓ = 20 is the wrong summary #

The fifteen-fold figure deserves a caveat, because it is a ratio taken near a zero.

At the exact crossing of 19.15 the exact difference is zero by construction and the sampled difference is about −4.1, so the ratio there is infinite. Read at 24 instead it is 6.05/1.20 = 5.0; at 32, 7.94/2.15 = 3.7; at 8, 4.30/5.59 = 0.77.

So the ratio runs from under one to unbounded across the range, and quoting it at any particular length says as much about the length as about the effect. The difference is the quantity that behaves — 0.21 points a lag, everywhere in the range — and it is the one a reader should carry.

The instrument that makes it visible #

None of this would be measurable through the instrument the earlier comparison used, and the reason is worth one paragraph because it is what the field’s third essay is about.

A resample’s implied long-run variance is Σ_k κ(k)γ̂(k) — the attenuation times the sample’s own autocovariances — and it can be computed from the sample without resampling at all. The identity that makes it available is the one the taper field establishes: a block resample’s autocovariance at lag k is κ(k) times the residuals’. So the whole second moment of the reference distribution is a sum over lags rather than a variance of a few hundred resampled means.

That is not merely cheaper. A quantity computed rather than sampled has no resampling noise in it, so what is left in it is the noise in γ̂ — which is exactly the thing being measured. An instrument that added its own noise to that would be measuring a sum of two things and calling it one.

The check that the identity is right is the resampler itself: four hundred realised resamples of a two-hundred-row series agree with the computed value to within a tenth on three windows at two block lengths, and the two share nothing but the series.

What this does to the earlier reading #

The earlier comparison said three things and they are worth taking one at a time.

The taper’s attenuation is its window’s self-convolution, and κ′(0) is the squared end values. That is exact and unaffected. Nothing here touches the algebra.

The taper is more biased than the plain block at every block length a hundred and twenty rows afford. At ℓ = 8 that is true of the sampled quantity as well — 4.30 points in the rectangle’s favour — and at ℓ = 16 it is already false, where the exact calculation says it is still true. The statement holds over a shorter range than it was made over.

The difference at the crossing is three tenths of a point, which twenty thousand draws could not resolve. That is the exact difference. What a hundred and twenty rows report at that block length is 4.31 points, and an instrument that reads the implied variance rather than a critical value resolves it in seven draws.

The number that made the comparison look unresolvable was the wrong number, and it was the wrong number because it was the right answer to the asymptotic question.

What the crossing is worth #

A crossing point is a place where two rules change places and it says nothing about how much either is worth. That question has an answer here and it is not the flattering one for either window.

At a hundred and twenty rows the rectangle’s implied long-run variance is 28.2% low at ℓ = 20 and the trapezoid’s 23.9%. Both are wrong by a quarter of the quantity they estimate. The best either can manage, at its own best block length on the bias, is 28.2% and 23.2% — so the crossing is a crossing between two badly biased estimates, and moving from one to the other buys four points of a shortfall that started at twenty-eight.

That is the honest scale of the whole comparison. Choosing the window is worth about a sixth of what the estimate is wrong by, and the rest is the sample’s own attenuation — which no window repairs, because every window is a weighting of the same short sequence.

A dial that moves four points of a twenty-eight point error is a dial worth setting and not worth much. Which is a sentence the exact calculation could not produce either, because in the exact calculation the shortfall at ℓ = 20 is 13.7% rather than 28.2% and the dial looks twice as important as it is.

Both routes, and where they meet #

The two calculations are not in competition; they answer different questions and the sample size is what moves between them.

The exact bias is what a resample would imply if the autocovariances it is built from were the law’s. It is the quantity that decides the order of the bias — first order for a window with non-zero ends, second order for one that reaches zero — and orders are asymptotic statements, so the exact route is the right route for them.

The sampled bias is what a resample implies from the series in front of somebody. It contains the exact bias plus the interaction between the window and γ̂’s own attenuation, and that second term is O(1/n) and dominates at n = 120.

They meet: at 960 rows the sampled gap at ℓ = 20 is −0.77 against an exact −0.28, and at 1,920 rows it is −0.51. The convergence is slow, which is what O(1/n) against a difference of 0.28 points looks like.

Why not simply use a longer sample #

The taper field names a longer sample as one of the two honest routes out, and the sweep here runs to nine hundred and sixty rows, so it is worth saying why that is a diagnostic rather than a recommendation.

The comparison is being made because a practitioner has a series and wants a reference distribution from it. The sample length is not a dial: it is the data. Running the same measurement at 240, 480 and 960 rows says what the finite-sample term is doing and confirms it decays, and it does not help anybody whose series is a hundred and twenty rows long.

What the sweep licenses is the opposite of a recommendation. The window that should be chosen depends on the sample length, and it depends on it through a term nobody was computing: at a hundred and twenty rows the crossing is at 13 and at nine hundred and sixty it is at 18, so a practitioner told to use blocks of sixteen would want a trapezoid on the short series and a rectangle on the long one.

That is a dependence a rule of thumb cannot carry, and it is why the field’s conclusion is a measurement to be repeated rather than a window to prefer.

What the field this replaces got right #

It is worth being precise about what was wrong, because most of it was not.

The taper field computes the exact bias correctly, identifies the order correctly, produces the refutation case correctly — a trapezoid cut off at half height, which is tapered, smooth and buys nothing — and reports its own instrument’s limitation accurately. Its conclusion is stated as the taper’s advantage has not arrived at any block length this sample size affords, and it names the two honest routes out: a longer sample, or a measurement of the implied variance rather than of a quantile.

Both of those are taken here, and taking them changes the conclusion. That is the intended use of a named deferral, and it is what makes the earlier statement a correct statement of what its instrument could see rather than an error.

What did go wrong is one sentence of reading: the exact difference at the crossing was quoted as though it were what a comparison at that sample size would have to resolve. An exact quantity and a finite-sample one were treated as the same number, which is the same shape of mistake as reading a residual order as an error order one field along.

What is claimed here, and what is not #

This essay takes what a sample reports about two block windows, against what the algebra says. The claims are that the exact difference between a rectangular and a trapezoidal block’s implied long-run variance crosses zero at a block length of 19.15 under a first-order autoregression at 0.7; that a sample of a hundred and twenty rows reports the crossing at 13.28 and a difference of 4.31 points at ℓ = 20 where the exact difference is 0.28; that the exaggeration shrinks as the sample grows, with the sampled crossing at 14.97, 16.39 and 17.99 at 240, 480 and 960 rows; and that the mechanism is the attenuation in the sample autocovariances the window is applied to, which the window that discards the long lags loses less of.

What stays out, and is named as a decision: a closed form for the finite-sample term. The expectation of a sample autocovariance under a known law is computable — the laws field computes it — and combining it with κ would give the sampled bias exactly rather than by drawing. It is not done because the sampled quantity here is a ratio of two estimated things and its expectation is not the ratio of the two expectations, which is the refusal that essay records, and doing it properly is a different calculation from the one that would be quick.

Also out: other laws. Everything is at a first-order autoregression at 0.7, which is the world the taper field measures in. Whether the exaggeration is the same size under a moving average or under long memory is not measured and would be the natural next sweep.

The boundary against the taper field is that it computes the exact bias and this one computes what a sample of a given length reports of it. Neither number is wrong; they are answers to different questions, and only one of them is the question a practitioner with a series is asking.

What comes next from here #

The finite-sample term this essay isolates has two consequences the rest of the field takes up.

A comparison at a shared block length is the wrong comparison. Each window has its own best block length and they are not the same, so comparing two windows at one ℓ is comparing two rules with one of them mis-tuned. What that does to the ordering is the next essay, and the short answer is that it reverses at a hundred and twenty rows.

A bias is half of an error. Everything above is about bias, because that is what the taper field compares. A window that reaches zero at its ends attenuates less at short lags and uses less of each block, so it trades bias for spread — and the block length minimising one is not the one minimising the other.

Both of those are about reading the right quantity, which is the same complaint this essay makes about the exact gap, one level along. The pattern the field ends up with is that the taper’s case was decided three times on three quantities and none of them was the one a practitioner is choosing between.

The checks, and the refusals that make them mean something #

Three claims are gated. The implied variance computed from a sample’s autocovariances is required to agree with the variance of four hundred realised resamples, on three windows at two block lengths — one route is an identity about what a block keeps and the other is a sampler that has never heard of it. The exact bias computed here is required to equal the exact bias computed in the taper field, to nine decimal places, at every block length and window, because the two files must not drift apart. And the sampled gap at ℓ = 20 is required to exceed the exact one by more than a factor of three at a hundred and twenty rows and to shrink towards it as the sample grows.

The refusal is the sentence this essay corrects. An asymptotic gap quoted as what a finite sample shows is refused, with the exact 0.28 points printed beside the 4.31 a hundred and twenty rows report, and with the mechanism: the autocovariances the window is applied to are themselves attenuated, and the window that discards the long lags loses less of them.