The triangle that was not the multiplier's

The essay that found the ceiling ends on a bound and a list. The bound is one line: a multiplier resample sets e*ₜ = eₜ·wₜ, so its autocovariance is the residuals’ times the multiplier’s, and for a blocked multiplier the multiplier’s own autocovariance is exactly the triangle (1 − k/ℓ)⁺. Since the residuals are already less persistent than the errors, no block length reaches the truth.

The list is what the bound does not cover. It is stated for resamplings that leave each residual on its own row, and two constructions do not. A block bootstrap takes runs of consecutive residuals and puts them somewhere else. A sieve generates fresh errors from a fitted model. Neither is a multiplier, so neither is bounded by the argument, and both were named as escapes and left unpriced.

Pricing the first one turned up two things, and the second is the sort of finding that only arrives by reading code rather than names.

The attenuation is the join #

Write out what each construction does to a lag-k pair.

A blocked multiplier gives every row in a run of ℓ the same sign. A pair (t, t + k) inside one run carries the same sign twice, so the signs cancel and the pair contributes eₜeₜ₊ₖ. A pair straddling a boundary carries two independent signs and contributes zero in expectation. The fraction of lag-k pairs inside a run is 1 − k/ℓ.

A fixed-length moving block lays down blocks of exactly ℓ consecutive residuals, each drawn from a uniform starting position. A pair inside one block is a pair of residuals at distance k in the original series, and contributes an average over starting positions. A pair straddling a join comes from two independently drawn blocks and contributes the square of the residual mean — zero, because residuals from a fit with an intercept sum to zero. The fraction of lag-k pairs inside a block is 1 − k/ℓ.

The two expressions are different and their values are not. One construction never moves a residual and the other moves every one of them; both have one boundary every ℓ rows; both lose exactly the pairs that straddle one.

At ℓ = 5 the multiplier keeps 0.5033 of the first lag and the fixed block keeps 0.4956, against a triangle prediction of 0.5033. At the fourth: 0.0292 and 0.0280 against 0.0286. Past the block length both are dead — −0.0003 and −0.0005 at the sixth lag, against standard errors of 0.0030. The worst gap between the two constructions over six lags is 1.4 standard errors.

Neither prediction is fitted. The multiplier’s is computed by summing the residuals’ own products over the pairs that share a block, exactly, on each sample; the fixed block’s by averaging over every starting position, exactly, on the same sample.

Dividing the triangle out recovers the residuals #

The four readings carry more than the agreement they were computed to check. Because the prediction is the triangle times the residuals’ own autocorrelation, dividing it back out reads that autocorrelation off directly.

At ℓ = 5 the triangle weights are 0.8, 0.6, 0.4 and 0.2, so the first lag’s 0.5033 is a residual autocorrelation of 0.629 and the fourth’s 0.0286 is 0.143.

Those two are worth putting beside each other. A geometric decay from the first would put the fourth at 0.629⁴ = 0.157, and the measured value is nine per cent below it. So the residuals decay slightly faster than a first-order process, which is the direction the shortfall against the errors predicts: fitting a mean and a design takes proportionally more out of the longer lags than of the shorter ones.

That is a second reading of the same four numbers and it costs nothing. The figure was drawn to establish that two constructions agree; the same arithmetic says what they agree about.

What a block of five actually keeps #

The triangle’s cost is usually quoted at one lag, and the quantity that matters to a long-run variance is the whole sum. Filling lags two and three geometrically at 0.629 — an interpolation, and flagged as one — the residuals’ own dependence sums to 1.70 over all lags, and what a block of five delivers is

$$\sum_{k\ge1}\left(1 - \tfrac{k}{5}\right)^{+}\hat\rho(k) = 0.503 + 0.238 + 0.100 + 0.029 = 0.869.$$

Fifty-one per cent. A block length of five discards half of the dependence the residuals still have, before any of the separate shortfall between residuals and errors is counted.

Note that fifty-one is higher than the triangle’s own average over the four live lags, which is 0.5 flat. The weights that survive are the ones multiplying the largest autocorrelations, so the retained share always beats the unweighted taper — and the gap between the two is small here because 0.629 is not a very persistent process.

The block length ninety per cent would take #

Running the same sum backwards gives the length rather than the loss, and the answer is a long way from five.

For a geometric residual sequence the taper costs $\frac{1}{\ell}\sum_k k\hat\rho(k) = \frac{1}{\ell} \cdot \frac{\hat\rho}{(1-\hat\rho)^2}$, which at 0.629 is 4.57 / ℓ against a total of 1.70. Setting the retained share to nine tenths gives

$$\ell \approx \frac{4.57}{0.1 \times 1.70} \approx 27.$$

Twenty-seven rows, five and a half times the block length in the figure, to keep nine tenths of what the residuals have — and the residuals are already short of the errors, so twenty-seven rows buys nine tenths of a quantity that is itself below the truth.

That is the ceiling in its most concrete form. The triangle is not a small correction that a moderately long block removes; it is a loss that closes like 1/ℓ against a sample of a hundred and twenty rows, and a block of twenty-seven is already more than a fifth of the series.

Which is not a fact about the moving #

That settles the first item on the list against the expectation. The reasoning that made a block bootstrap look like an escape was: the bound is proved for a construction that multiplies a residual in place, so a construction that relocates residuals is outside the proof. True, and irrelevant — being outside a proof is not being outside its conclusion, and the conclusion here follows from the block structure rather than from the multiplication.

Nothing is gained by moving the rows. What is lost is the design: a multiplier keeps eₜ on row t, so whatever that row’s variance was travels with it, and a block bootstrap puts a residual generated at one design point onto another.

And the construction in this table is a different one #

The block bootstrap this collection has been running since its resampling table was written is not the fixed-length construction above. Its index restarts at a uniform position with probability 1/ℓ at each step and otherwise advances, so its runs are geometric with mean ℓ rather than exactly ℓ long. That is the stationary bootstrap, it is a real construction with a real name, and it is the better default — a fixed grid of blocks makes the resampled series non-stationary.

It has a different taper. A lag-k pair survives when none of the k intervening steps restarted, so

$$\operatorname{E}[\gamma^*(k)] = \hat\gamma(k)\,\left(1 - \tfrac{1}{\ell}\right)^{k}$$

— geometric where a fixed block is triangular. The two agree at the first lag, where both are 1 − 1/ℓ, and nowhere else. At ℓ = 5 and the fourth lag the geometric keeps 0.0528 where the triangle keeps 0.0280, which is nearly double; and past ℓ the separation is total, because the triangle is exactly zero there and the geometric is not. At the sixth lag the stationary bootstrap still carries 0.0081 against a standard error of 0.0030.

Every number in the two sections above was first computed on the assumption that this construction was the fixed-length one, and the build refused the figure. The name was not the mechanism, and the docstring that described it — “takes runs of consecutive residuals” — was true of both and distinguished neither. The fixed-length construction now exists beside it, because the difference between the two tapers is a measurement rather than a footnote.

A table of critical values that does not order by taper #

Here is where the tidy story stops. If what a reference distribution is worth were a function of how much dependence its resample carries, five constructions with five tapers would order themselves by taper. They do not.

Where only the rows repeat: the plain wild bootstrap keeps nothing beyond lag zero and is short by 44.0%. The fixed block and the stationary bootstrap have visibly different tapers and land at 10.5% and 10.3% short — indistinguishable, against standard errors of about 2.5% on those shares. And the blocked multiplier, which shares the fixed block’s taper exactly, is short by 18.1% — nearly twice as much as the construction it agrees with about every autocovariance.

An expected autocovariance is not a reference distribution. The critical value is the 95th percentile of a maximum over a table of comparisons, and that depends on far more than the second moments of the resampled errors. Two of the differences here are visible in the mechanisms: the multiplier’s blocks sit on a fixed grid of rows, so the pairs it keeps are a particular subset of offsets rather than a random sample of them, and its resampled series is therefore more variable from draw to draw than its expectation suggests; and the two moving constructions average over starting positions, which smooths that away.

The identity in the first section is an identity in expectation over samples. On any one sample the two constructions differ, and a critical value is an average of a nonlinear functional over samples. Both statements are true and only the first is about the tapers.

What does order the table #

One thing does run cleanly through all five rows, and it is not the dependence at all.

Add a variance that depends on the design, and every construction that moves a residual off its row gets worse: the fixed block goes from 10.5% short to 17.1%, the stationary bootstrap from 10.3% to 17.5%, and the sieve from 2.9% over to 7.8% short. Every construction that leaves the residual where it was gets better: the multiplier from 18.1% to 11.7% short, and even the plain wild bootstrap from 44.0% to 37.6%.

That is a five-row statement with one mechanism behind it, and it is the reason the multiplier exists. The resampled error on row t has the size row t’s own variance produced, so the reference distribution is built from series whose heteroskedasticity matches the design. Move the residual and every row carries the sample’s average variability instead.

So the inventory the field arrives at is not which construction keeps the most dependence. It is:

• a blocked multiplier truncates the dependence at a triangle and keeps the row;
• a fixed-length block has the identical triangle and loses the row;
• a stationary bootstrap has a geometric taper, keeps more at every lag past the first, and loses the row;
• and a sieve is bounded by nothing in the sample, and loses the row.

Nothing here keeps both. That is close to a statement about what a residual is: to keep the row is to reuse the residual where it was, and to reuse it where it was is to be limited by what it contains.

Why the argument is about expectations, and what that leaves out #

The identity in the first section is an identity between expected autocovariances, conditional on the residuals, and that is narrower than it sounds in one direction and wider in another.

Narrower: two constructions with the same expectation can have different variances, and these do — which is what the critical values are showing. A blocked multiplier reuses the same residual in the same place on every resample and randomises only ⌈n/ℓ⌉ signs; a moving block redraws the contents as well as the arrangement.

Wider: an expectation conditional on the residuals is a statement about this sample rather than about the process, so the identity is not asymptotic and does not need the sample to be large. It is arithmetic on the hundred and one residuals in hand, recomputed on every draw from that draw’s own residuals, so a systematic departure could not average away.

What the identity licenses, and what it does not #

Two constructions agreeing on their expected autocovariance to 1.4 standard errors over six lags is a strong statement about a weak object, and separating those two halves is the whole lesson of the table underneath it.

What has been established is an identity between first and second moments of the resampled series. That is genuinely all an expected autocovariance is. It says that if the only thing a downstream statistic reads is a variance — a long-run variance, a sandwich, a plug-in standard error — then the fixed-length block and the blocked multiplier are interchangeable and the choice between them is arbitrary. For a great many uses that is the end of the matter, and the identity is worth having for exactly that reason.

What has not been established is anything about a critical value, because a critical value is a functional of the whole resampled path — a quantile of a distribution, not a moment of it. Two constructions can match on every second moment and differ in every higher one, and the fixed block and the blocked multiplier do: one draws whole contiguous stretches of residuals and keeps their joint shape, the other keeps each residual on its own row and multiplies it by an independent sign shared across a window. Those produce different-looking series with the same covariance.

The table says so plainly and it is worth reading in that order rather than as a surprise. The two constructions sharing a taper exactly land 18.1% and 10.5% short, nearly a factor of two apart. The two whose tapers visibly differ — the fixed block’s triangle and the stationary bootstrap’s geometric decay, which disagree by a factor of two by the fourth lag — land at 10.5% and 10.3%, indistinguishable against standard errors of about 2.5%. An expected autocovariance is not a reference distribution, and there is no ordering by taper to be found because there was never a reason to expect one.

What does run through the table is the row. Every construction that moves a residual off the row it was computed on pays when the variance depends on the design, and every construction that leaves it there does not — which is a statement about mechanism rather than about moments, and the only one of the two kinds that predicted anything.

What is claimed here, and what is not #

This essay takes whether moving the rows escapes the multiplier’s ceiling. The claims are that a fixed-length moving block and a blocked multiplier have the same expected autocovariance, agreeing to 1.4 standard errors over six lags and both sitting on γ_resid(k)(1 − k/ℓ)⁺ and both exactly zero past ℓ; that the block bootstrap this collection actually runs is a stationary bootstrap whose taper is γ_resid(k)(1 − 1/ℓ)^k, agreeing with the triangle at the first lag and nowhere else; that the critical values do not order by taper, since two constructions with different tapers are indistinguishable at 10.5% and 10.3% short while two with the same taper differ by nearly a factor of two; and that what does run through the table is the row — adding a design defect costs every construction that moves a residual and costs neither that leaves it.

What stays out and is named as a decision: the tapered block bootstrap, which weights within each block so that the resampled series’ spectral estimate is smoothed. It has a third taper again, it is the construction most of the long-run-variance literature actually uses, and nothing here measures it.

The boundary against the essay that proved the ceiling is that it is about what a multiplier can keep and this one is about whether a different construction keeps more.

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

Three claims are gated. The two constructions are required to agree with each other to within three standard errors at every lag, and each to agree with the prediction its own mechanism gives — agreement between the two alone would be consistent with both being wrong in the same way. The fixed block is required to be dead past ℓ, which is the half of the triangle a geometric taper cannot imitate. And the stationary bootstrap is required to sit on the geometric, to meet the other two at the first lag, and to be measurably alive past ℓ.

Two refusals bite. A fixed-length moving block offered as the escape from the multiplier’s ceiling is rejected, because its expected autocovariance is the multiplier’s at every lag. And a block bootstrap’s taper assumed triangular because the word block is in its name is rejected — the construction in this collection’s own table keeps 0.0081 at a lag where a triangle says it keeps nothing, and the two agree at the first lag, which is the one anybody checks.