Two searches, one sample

A break point that has been looked for costs more than a count of parameters says, because under the null the break point does not exist — every position describes the same model — and no count of restrictions describes the distribution of a supremum over a parameter that is not there. That field measures the charge for one search, on one profile, and says so.

A rule that also chooses a window is running two.

The construction #

The break here is a break in the regression, which is what the word usually means outside this collection and is not what the earlier field searches for: that one breaks the persistence of the errors. A structural break splits the sample at some row and fits separate coefficients on each side, so a split adds five coefficients and a count of parameters says a chi-square on five, whose 95% point is 11.07.

The window is the width of a tapered band estimated from the residuals and used to whiten the sample before any of that happens. It is chosen from a list of eight, it carries the volume it moves — which is not optional once widths are being compared — and it is charged its own dimension, without which the likelihood simply takes the widest band on every draw.

Three suprema, all measured against the same base of no break at no whitening:

A, the break alone at a fixed window. B, the window alone with no break in the model. C, both together.

What each search manufactures #

Under a first-order autoregression with no break anywhere, the break search alone reports a likelihood ratio of 34.70 on average. The chi-square point a count of coefficients would compare it against is 11.07, and its own 95% point is 63.5.

Two things are in that number and they are worth separating. Part of it is what the earlier field measures: a supremum over an unidentified location. The rest is the dependence — a Chow test on correlated errors is inflated even without any search, because the residuals on each side of any split are not independent of each other, and least squares has not been told.

The window search alone reports 84.00. That is not a manufactured quantity in the same sense: the errors really are correlated, so a whitening really does buy likelihood, and B mixes genuine gain with search luck. What matters here is not what B is made of but that it is large — nothing in this collection has ever charged for having chosen a window, and eighty-four is not a rounding.

The two together report 99.40.

The charges do not add #

34.70 plus 84.00 is 118.70. The pair reports 99.40, so a rule charging the two searches separately over-charges by 19.30, at 28 standard errors, on 99.7% of draws.

That is not a small correction to a bookkeeping question. The excess is 56% of what the break search manufactures on its own.

The same pattern holds on every law: 15.44 under the moving average, 14.24 under long memory and 26.64 under a break in the persistence, on 99.0%, 99.3% and 99.7% of draws. The direction is guaranteed — a supremum over a product of two sets is at least the supremum over either slice through it, so C is at least the larger of A and B, and A + B ≥ C is the claim that the two searches find some of the same luck. What is measured is the size.

Why they overlap #

The mechanism is the reason the next essay’s finding exists, so it is worth stating carefully.

A break search on unwhitened residuals is largely reporting the dependence. A run of correlated errors looks like a level shift, and the best split of a hundred and twenty correlated rows finds one almost every time — which is exactly why the earlier field’s charge is 5.08 against the 2 a parameter costs, and why this field’s is 34.70 against 11.07.

A window search removes that. Whitening at a band of eight lags leaves residuals with much less of the run structure in them, so the break search run afterwards has much less to find.

The two searches are therefore looking at the same feature of the sample from two directions, and whichever runs second finds what the first left. A + B − C is the size of the overlap, and it is about a fifth of the pair.

The window search is not free, and nothing had charged for it #

The measurement that has no precedent in this collection is B.

Every field before this one chooses a window from a list and then reads a criterion as though the window had been handed over. The window a whitening wants chooses it by a criterion on the residuals; the estimated-covariance field chooses it once from the fullest candidate; the per-candidate version chooses one for each row. None of them charges anything for the choosing, and every downstream statistic is read against a distribution computed as though a width had been given.

Eighty-four units of likelihood ratio is what that omission is worth on this construction. It is larger than the break search’s own charge, and the break search’s own charge is the thing an entire earlier field exists to measure.

The honest qualification is the one made above: B is not all manufactured, because the whitening buys real accuracy on a law whose errors really are correlated. Separating those two parts would need a world with no dependence in it at all, where the window search has nothing to find and the whole of B is luck — and that world is not in this field’s four laws, because a field about whitening under dependence has nothing to say at independence.

What the profile looks like at four widths #

The overlap has a picture, and the picture says something the three numbers do not.

Take one sample and draw the searched break profile at four widths of the whitening. Four curves, each measured against its own no-break likelihood so what is compared is shape rather than level. On the sample drawn above the four peaks are at rows 49, 80, 92 and 96 — four different break points from four widths of a nuisance nobody is interested in.

So a rule that chooses the window and then searches for a break is not choosing between four readings of one profile. It is choosing between four profiles, and the one it takes is the one whose peak is highest after the volume term and the width’s charge are paid. That is the sense in which the two searches are not separable: neither one has a well-defined answer until the other has been settled.

It also explains the sub-additivity without any order statistics. The unwhitened profile is the tallest of the four, because it is the one still carrying the dependence, and the joint search’s answer is a compromise between a tall profile and a well-charged width.

Under a law with a break #

The fourth law breaks its own persistence half way through, and its column is the largest in the table: A of 46.38 and an excess of 26.64.

There is no break in the mean under that law — the coefficients are constant, as they are under all four — so the searched structural break is still finding nothing that is there. What it is finding is a change in the error structure, read through a test that has no way to express one, and the answer is to report a break in the regression instead.

That is a specific and useful warning. A searched Chow test on a series whose variance structure changes will find a break in the mean, at 46 units against a threshold of 11, and nothing about the output says which of the two it found.

The base the three suprema are measured from #

One choice in the construction is worth defending because it decides what the numbers mean.

All three suprema are measured against the same base: no break, no whitening, least squares on the whole sample. That is why A and B are directly comparable and why C can be compared with their sum. An alternative would be to measure B from no-break-at-the-chosen-window and A from break-at-no-window, which is what a practitioner reading two separate literatures would implicitly do — and that arrangement has no common origin, so the sum of the two is a sum of quantities measured from two different places.

Having one base is also what makes C − B meaningful, which is the next-but-one essay’s whole subject. It is the ordinary discipline this collection applies to every regret measurement: hold the benchmark fixed while the rules move, or a sweep over a rule becomes a sweep over its own benchmark.

What a count of parameters says #

It is worth stating what would happen to somebody who charged in the ordinary way.

A split adds five coefficients, so a chi-square on five, whose 95% point is 11.07. On a sample with no break in the mean under a first-order autoregression, that threshold declares a break on 73.6% of draws. Under long memory, 87.6%.

Three in four is not a test with a poor size; it is not a test. The two fields’ conclusions coincide here and the sizes differ: the earlier field’s persistence break at a naive threshold rejects far too often, and this one’s regression break at a naive threshold rejects almost always.

What comes next #

Two essays follow, and the second one is the reason the field exists rather than being a paragraph.

The charge that is not a sum takes the arithmetic above into a decision: what happens to size and power when a practitioner charges each search what its own literature says, and the answer is that the test stops firing at all.

A charge that depends on the rule takes the sharper half. The quantity a rule doing both searches actually reads is not A and not C; it is C − B, the likelihood the break search adds once the window search has run — and it is 15.40 where A is 34.70. What a search costs is not a property of that search.

The log scale understates the overlap it is measuring #

The excess of 19.30 is 56% of what the break search manufactures on its own, and the same pair read on the calibrated share scale of the field built to give overlaps an origin comes out at 0.762. Those are two readings of one relation and the gap between them is not noise.

The share scale’s zero is two searches that share nothing; the log scale’s is not, because a log of a ratio of residual sums is super-additive before any overlap — two searches each removing a tenth of the residual sum remove a fifth together, and the logarithm makes the second one’s gain look larger for having gone second. That artefact pushes A + B − C down, so it partially cancels a genuine overlap.

So 56% and 76% are the same finding with one of them net of an artefact, and the log-scale figure is the conservative one. That is worth knowing before the number is carried anywhere: a rule charging the two searches separately over-charges by 19.30 units of likelihood, and the share of the smaller search that the two genuinely have in common is three quarters rather than a half.

Six point nine by its mean, four by its tail #

The naive threshold’s failure can be read two ways and they do not agree, which is the same signature the single-search field reports.

By its mean, the searched break statistic is 34.70 against a chi-square-on-five’s 5 — so the search is worth 6.9 chi-squares. By its rejection rate, a chi-square on five scaled by k rejects at 11.07 with probability 0.736 when 11.07/k sits at the 26th percentile of χ²₅, which is about 2.75 — so k is 4.0.

Mean-equivalent 6.9, tail-equivalent 4.0, and no single scaled chi-square matches both. That is the same disagreement, in the same direction and at nearly the same ratio, that the persistence-break search shows between five degrees of freedom by its mean and three and a half by its 95% point. Two different searches, two different laws, one shape of failure: the manufactured statistic is less dispersed than any chi-square with its mean and more dispersed than any with its tail, so choosing a bigger number of degrees of freedom repairs one end and breaks the other.

The four profile peaks say the same thing about the window in the other direction. Rows 49, 80, 92 and 96 span two thirds of the searched range, and the steps between them are 31, 12 and 4 — so almost all of the movement is between no whitening and some, and the choice among widths past that moves the answer by a few rows. A rule that has decided to whiten at all has made most of the decision the break search is sensitive to, which is why C − B at 15.40 is so much smaller than A at 34.70.

Where this generalises #

Two searches over one sample is not a special situation; it is the ordinary one, and the pattern of findings here should transfer wherever two of them overlap.

Any pair of nuisance searches that look at the same feature of the data will be sub-additive, and the overlap will be large exactly when the two are looking at the same thing from different directions. Here the shared feature is the run structure of the errors: a break search reads it as a level shift and a window search reads it as autocorrelation, and each is diminished by the other having run.

Where the two searches look at genuinely different features the overlap should be small, and this field cannot show that because both of its searches read the same residual series. It is worth naming as the limit: the size of the overlap is a property of the pair, and 19.30 is this pair’s.

The one thing that does transport without qualification is the sign. Any two searches over one sample are sub-additive, always, on every draw, because a supremum over a product set is at most the sum of what the two slices manufacture. A rule that charges separately is therefore conservative — and what conservative costs is not what the word suggests.

Everything here is measured on stationary designs with a constant mean, so the null is a null in the strong sense and every unit of likelihood a search reports is manufactured.

What is claimed here, and what is not #

This essay takes what two searches over one sample manufacture together. The claims are that a searched break in the regression reports an average likelihood ratio of 34.70 under a first-order autoregression with no break in it, against a chi-square-on-five point of 11.07; that a searched window reports 84.00 and that nothing in this collection has charged for it; that the two together report 99.40 rather than 118.70, so a rule charging them separately over-charges by 19.30 at 28 standard errors and on 99.7% of draws; that the same sub-additivity holds on all four laws, by 15.44, 14.24 and 26.64; and that a count of coefficients declares a break on 73.6% of samples that have none under the autoregression and 87.6% under long memory.

What stays out, and is named as a decision: what B is made of. The window search’s eighty-four units mix genuine gain from whitening a genuinely correlated series with luck from having chosen the width. Separating them needs a world with no dependence, where the whole of B is luck, and a field about whitening has no such world in it. The consequence is that B should be read as what the window search adds, never as what the window search manufactures.

Also out: a break in the errors and a break in the mean at once. The earlier field searches for the first and this one for the second, and a rule that searched for both would be running three searches. That is a real construction and it is not measured here; what is measured is that the second search picks up the first when the first is present, at 46.38 against 34.70.

The boundary against the searched break field is that it prices one search exactly and this one asks what happens when a second is running beside it.