An interval that carries its scale

The field that read one block bootstrap three ways ends on the reading a reader is actually promised, and it is the one that comes out worst: not one of its eight cells reaches 95%. They run from 80.8% to 91.0%, so the choice between two block windows that the field spends three essays on is a choice inside a shortfall four times larger than the choice.

Its last paragraph names the repair it did not run. The interval measured there is a percentile interval — the 2.5% and 97.5% points of the resampled means, used directly — and a percentile interval inherits the resampled distribution’s skewness and its scale error together. The standard fix for exactly that is to studentise: resample a t-statistic rather than a mean, so that each resample carries its own scale.

It is one extra variance per resample. This essay is the construction; the two after it are what it does.

Three intervals from one resampling #

Everything is built from the same resamples, at the same block lengths, under the same two windows, on the same draws. That is the earlier field’s own discipline carried forward: three intervals from three loops could differ because of three sets of resamples, and three intervals from one cannot.

A normal interval. x̄ ± 1.96 σ̂/√n, with σ̂ the block-means variance below. No resampling in the interval at all. It is here as the thing a bootstrap is supposed to beat, which is the comparison nobody makes.

A percentile interval. The earlier field’s, reproduced rather than cited: the 2.5% and 97.5% points of the resampled means.

A studentised interval. The 2.5% and 97.5% points of √n(x̄* − x̄)/σ̂*, with σ̂* computed on each resample by the same estimator that produced σ̂, and the interval read back as x̄ − q σ̂/√n.

The variance estimator, and why it is the block one #

Studentising needs a scale for each resample, and the choice of estimator is where a bootstrap-t is usually made worse than the interval it replaces.

A moving-block resample is k = ⌈n/ℓ⌉ blocks laid end to end, and the blocks are drawn independently. So the resampled mean is an average of k independent block means, and its variance is estimated by ℓ times the sample variance of those k means.

That is the estimator the resampling scheme itself implies. It is O(n) per resample. And — the part that matters — the same functional is applied to the original series and to every resample: split into ⌊n/ℓ⌋ non-overlapping blocks, take their means, scale by ℓ.

Studentising with two different estimators is the commonest way to make a bootstrap-t worse than a percentile interval, because the t-statistic is then not a ratio of two estimates of one thing and its resampled distribution is not the distribution of the statistic being read.

A tail of fewer than ℓ rows is dropped rather than counted short, because a block mean over three rows and a block mean over twenty have different variances and averaging them would put a bias in the scale that grows with how badly ℓ divides n.

What studentising is for #

The word carries an argument and it is worth spelling out, because the reason to expect it to help is a good one.

A percentile interval takes the resampled statistic’s own quantiles and uses them directly. That works when the resampled distribution is a good approximation to the statistic’s, and it inherits everything that is wrong with the approximation — including, in particular, that the resampled distribution’s scale is estimated from the same sample that produced the point estimate.

A studentised interval divides that away. Each resample’s statistic is standardised by its own scale, so what is resampled is a quantity that is approximately pivotal: its distribution depends less on the unknown parameters than the raw statistic’s does. A quantile of a pivot transfers to the sample; a quantile of a non-pivot carries the sample’s own scale error with it.

That is the textbook argument and it is not wrong. What it assumes is that the scale is estimated well enough that dividing by it removes more variation than it adds, and the next essay is what happens when it is not.

Under dependence the scale is a long-run variance, which is the hardest thing in this whole line of fields to estimate — the earlier fields report errors of forty to seventy per cent on exactly that quantity — so a repair that turns on dividing by it is a repair that inherits a large problem.

The check on the scale #

The estimator has a closed form on a series built to have one, and it is checked against it.

Take a series that is k blocks of a known constant, laid end to end. Its block means are those constants, so ℓ times their sample variance is exactly ℓ times the variance of the constants. The two agree to a part in 10¹².

The same series with a tail of three extra rows gives the identical answer, which is the check that the tail is dropped. And a series with fewer than two whole blocks returns nothing rather than a number, which is the check that a variance of one number is refused rather than reported as zero.

An off-by-one in the block loop would produce a scale that is merely wrong: every interval would still be an interval and every coverage would still be a number between nought and one.

The world every number is in #

One setting is fixed throughout and inherited rather than chosen: a first-order autoregression at a persistence of 0.7, on a hundred and twenty rows, which is the world the crossing field and the feasible field both work in, and the world the field this one re-reads works in.

Keeping it is the point. This field’s job is to change what is done with a set of resamples, so changing the world at the same time would make the two differences inseparable.

What it means is that every number here is one persistence and one sample size. The earlier fields’ own sweeps over sample size have no counterpart, and the shortfall this field measures — nothing reaching ninety-five per cent — is a statement about a hundred and twenty rows at a persistence of 0.7 and not about block bootstraps in general.

The block lengths are a grid of seven: 2, 4, 8, 12, 20, 32 and 48. It is the grid the earlier field uses, and the protocol length of eight and the rule of thumb’s four are both on it exactly, so the two fixed rules are measured at the lengths they actually choose.

The identity the construction rests on #

There is one relation the three intervals have to satisfy, and it is asserted rather than assumed.

The studentised interval contains zero exactly when the observed t-statistic — √n x̄ over σ̂ — falls inside the resampled t band. That is arithmetic: the interval is x̄ − q σ̂/√n with q running over the band, so zero is inside it when √n x̄/σ̂ is inside the band.

It is checked directly, on both windows, at four thousand resamples: the inequality on the interval and the inequality on the statistic agree, draw for draw.

The percentile band is checked the same way — that it is the resampled means’ own quantile, taken off the same sorted array the t-statistics were built beside. Two intervals that are supposed to come out of one loop, checked to come out of one loop.

Why the normal interval is on the table #

Three intervals rather than two, and the third is there for a reason that is easy to skip.

A normal interval is x̄ ± 1.96 σ̂/√n. It uses the block-means variance and nothing else — no resampling, no quantiles, no blocks laid end to end. It is what somebody would compute if they had heard of a long-run variance and had never heard of a bootstrap.

Putting it on the table costs nothing, because σ̂ is already computed on every draw to studentise with. What it buys is the comparison the whole exercise is implicitly making: a bootstrap that does not beat x̄ ± 1.96 σ̂/√n has bought nothing at all, and neither the earlier field nor the ones before it ever put a non-resampled interval beside their resampled ones.

It also gives the two windows a control. A normal interval built from the block-means variance does not use the resamples, so the only way the window can affect it is through the block length the rule picked. If the two windows’ normal intervals differ, the difference is the rules picking different lengths; if they do not, then any difference between the windows on the other two intervals is manufactured by what is done with the resamples.

They do not differ. The two windows’ normal intervals are within a third of a point of each other at every rule.

The repair’s sign is decided by the window #

The eight cells where the studentised and percentile intervals can be differenced split perfectly on the window, and the split is the field’s finding before any of its later essays.

Under the rectangular window studentising is worth +5.0, +2.7, +7.6 and +3.7 points at the four rules. Under the tapered window it is worth −4.7, −6.0, −1.7 and −3.0. Eight cells, eight signs, and the sign is the window in every one.

A split that clean is not sampling noise: eight coin flips agreeing costs one chance in a hundred and thirty. The mean effect is +4.75 under one window and −3.85 under the other, so the interaction between the repair and the window is 8.6 points — larger than either main effect and larger than the whole spread the earlier field’s three essays were choosing inside.

Read from the other side it is the same number. The tapered window beats the rectangular one by an average of 4.0 points on percentile intervals and loses to it by 4.6 on studentised ones, so the window’s ordering reverses completely and by the same 8.6 points. Which window is better is decided by what is done with the resamples afterwards, and the two questions were being asked as though they were separate.

The repair widened the spread rather than lifting the floor #

Set the new cells against the ones they were added to and the effect on the table as a whole is not an improvement in level.

The earlier field’s eight percentile cells run from 80.8% to 91.0%, a spread of 10.2 points. The studentised cells run from 75.7% to 92.3%, a spread of 16.6. The best cell rose by 1.3 points and the worst fell by 5.1.

So the standard repair, applied across the table rather than at one cell, increases the dispersion of outcomes by two thirds and moves the centre almost not at all. That is a worse property than doing nothing, because a practitioner does not know which cell they are in: the choice of window and rule is made before the coverage is knowable, and a repair that widens the range of what those choices lead to has made the choice matter more without making any of it better.

The one cell that is nearly acceptable — 92.3%, studentised, rectangular, the plug-in rule — is reachable only by three decisions that the field’s own measurements give no way to make in advance. Its neighbour on the other window, at 88.0%, is the same repair and the same rule.

The failure mode a bootstrap-t is known for #

A resample whose blocks happen to agree has a near-zero σ̂* and an enormous t*, and enough of those would make the studentised quantiles unusable.

It is measured rather than assumed away. Across the whole sweep — three hundred draws, seven block lengths, two windows, two hundred and fifty resamples each — the share of resamples with a scale too small to studentise is 0.1996%.

One in five hundred. Those resamples are dropped from the t-array and counted, so the quantiles are taken over the rest.

Dropping rather than clamping is the right treatment and it is worth saying why. A resample whose blocks agree is a resample that says the series has almost no variance, which is a statement the data did not make; keeping it with a huge t would let one resample in five hundred set a 2.5% quantile taken over two hundred and fifty. Clamping it at some large value would be a fitted parameter nobody declared. Dropping it and reporting the share is the only option that changes no number a reader has to trust.

That number matters for what comes next: it says the studentised interval’s width, which the next essay finds to be more than double the percentile interval’s, is not a degenerate tail. It is the ordinary behaviour of a t whose denominator is estimated from few blocks.

Why the scale is not the implied variance #

There is a more natural-looking estimator available and it is deliberately not used, which is worth explaining because the earlier fields all use it.

The instrument those fields compare on is an implied long-run variance: Σκ(k)γ̂(k), computed in closed form from the residuals’ own autocovariances with no resampling in it. It is the cheapest accurate estimate of the quantity available and it is what makes their sweeps affordable.

It is the wrong thing here for two reasons.

It costs too much per resample. Studentising needs a scale for every resample, and an implied variance is O(n·ℓ) where the block-means variance is O(n). At two hundred and fifty resamples, seven lengths, two windows and three hundred draws, the difference is the difference between a sweep that runs in seconds and one that does not run.

And it is not the scheme’s own variance. The implied variance estimates the long-run variance of the original series. What a resample’s mean actually has is the variance implied by how it was built — k independent blocks — and studentising by anything else divides by a quantity that is not the standard deviation of the numerator. The t-statistic would then not be a ratio of an estimate to its own standard error, and its resampled distribution would not be the distribution of the thing being read off it.

The estimator has to be the resampling scheme’s, and the resampling scheme’s is the block-means one.

What the rules and the windows are #

Nothing about the block length or the windows changes from the earlier field, and that is deliberate: this field changes what is done with the resamples and nothing about how the length is picked, so any difference is attributable.

Two windows. A rectangular moving-block bootstrap, and a tapered one that weights each block down at its ends.

Four rules for the block length: a length written into a protocol, the rule of thumb n^(1/3), a plug-in from the sample’s own lag-one correlation, and an oracle.

And the oracle is deliberately not an oracle on coverage. A coverage on a single draw is a coin and has no per-draw argmin, so a rule that picked the length whose interval happened to cover would report a coverage it had selected rather than one it had achieved. The oracle here is the earlier field’s: the length whose resampled 95% point is nearest the finite-sample truth on that draw, which is 4.6326 at a hundred and twenty rows and a persistence of 0.7.

Keeping that definition is what makes the two fields’ oracle rows comparable.

What the three intervals actually cover #

The result, before the two essays that read it:

Under the rectangular window the studentised interval covers 88.7%, 82.7%, 92.3% and 85.0% at the four rules, against the percentile interval’s 83.7%, 80.0%, 84.7% and 81.3%.

Under the tapered window it covers 84.3%, 75.7%, 88.0% and 82.3%, against the percentile interval’s 89.0%, 81.7%, 89.7% and 85.3%.

Under both windows, at every rule, the studentised interval’s coverage is a different number from the percentile interval’s — moved up by as much as 7.7 points and down by as much as 6.0 — so this is not a repair that does nothing. It is a repair that does something, at every cell, and the something is not what was wanted.

Not one of the twenty-four cells reaches ninety-five per cent. Not on the studentised interval, not on the percentile one, and not on a normal interval built from the same scale.

The standard repair for a percentile interval’s shortfall does not repair it. What it does instead — double the width, and reverse the ordering between the two windows at every rule — is the rest of the field.