A cap that decides nothing

This field began with a charge nobody had derived. A regression on dependent errors can be fitted by generalised least squares through a covariance estimated as a band of sample autocovariances, tapered by a window, and the width of the band is then a tuning parameter chosen by a criterion that charges each lag of band. The convention charges a full log-likelihood unit a lag, which is the charge for a free coefficient; the charge nobody derived measured the optimism a lag of Bartlett band actually carries and found about three eighths of a unit.

What made that a curiosity rather than a defect was the next measurement. A width that moves and an error that does not found the derived charge choosing bands three times as wide as the convention and winning by half a per cent of error, because the error barely depends on the width at all: the surface of prediction error against band width is flat. The field’s whole subject therefore became conditional. A charge matters only if the error rises steeply away from the best width, and three essays went looking for a place where it does. The surface under every law found it flat under all four of the field’s laws at 120 rows. The charge at four times the sample found it flatter at 480.

That essay named the one place left. Long memory — a dependence whose autocorrelation is still 0.57 at the twentieth lag — wants a wide band, and at a large sample it might want one wider than any width on the list. Every list in the field stops at thirty lags. If the best band is at sixty, a criterion capped at thirty could be forced into a region where the surface is steep, and then the cap, not the charge, would decide the error. This essay lifts the cap.

The list, extended on the same draws #

The construction is the field’s own: a regression of a response on four predictors with errors from the field’s long-memory law at a fractional difference of 4/9, the full regression fitted through a Bartlett-tapered band of sample autocovariances, and prediction error scored on an independent replicate. Six hundred draws at each of 120, 480 and 960 rows. The list of band widths is the field’s eleven, from one lag to thirty, and then the same draws are run again with forty, forty-eight, sixty-four and eighty lags added. Because the draws are shared, every comparison between the capped and the extended list is paired on the draw: the same series, the same response, the same replicate, and a different list.

The surface answers the question before any charge is applied. At 120 rows its floor is at twelve lags and it rises past the cap: eighty lags is four thousandths above the floor, which is a band so wide that a sample of 120 cannot estimate it. At 480 rows the floor is at thirty, exactly at the cap, and every width from twenty-four to eighty is within a few ten-thousandths of it. At 960 rows the floor has moved past the cap, to forty-eight lags — but from thirty to eighty the whole surface lies within three hundred-thousandths of the floor. Long memory at a large sample does want a band wider than the list allows. It wants it by nothing.

How often the draw wants more than thirty #

The average surface is one statement; each draw has a best width of its own, the width whose band would have given that draw the smallest realised error. That width needs the truth to find, so no criterion can use it, but it says how often the cap stands in the way of something.

Often, and more often as the sample grows. With the list extended to eighty, the best width lies past thirty on 33.2% of draws at 120 rows, 45.5% at 480 and 48.7% at 960. With the list capped, the best width on the list is the cap itself on 27.3%, 40.2% and 44.8%. So the premise of the test holds: on nearly half the draws at the largest sample, the list stops short of where the draw’s error is lowest. If the surface had any slope behind the cap, this is where it would cost something.

Where each charge goes #

The four charges are the field’s: one unit a lag, the convention; half log n a lag, Schwarz’s; the window’s summed weights, half a unit for the Bartlett band; and the measured share, the charge the field derived from the optimism itself, about three eighths of a unit at 120 rows.

The two conventional charges never reach the cap and do not notice it is gone: one unit a lag chooses 4.1, 11.4 and 16.8 lags at the three sizes whichever list it is given, and half log n chooses 2.3, 5.7 and 8.0. They charge too much for any width near thirty to be worth its price, which is the field’s original finding at its most extreme.

The two smaller charges do reach it. At 480 rows the measured share chooses the cap on 44.8% of draws with the list capped, and with the list extended it goes past on 33.8%, its average width rising from 24.4 lags to 31.3. At 960 rows it chooses the cap on 80.0% of draws — a criterion pinned against the end of its list, which is exactly the situation the lead described — and with the cap lifted it goes past on 68.7%, from 28.4 lags on average to 45.8. The summed weights follow it part of the way, from 26.0 lags to 30.7. The charges the field derived are the ones that want the wider band, and when they are given it they take it.

What taking it is worth #

Nothing a criterion can collect. At 960 rows, the measured share’s error with the cap lifted is 0.00020 higher than with it in place, against a paired standard error of 0.00021; the summed weights’ is 0.00007 higher, against 0.00010. At 480 rows the measured share’s is 0.00024 higher, against 0.00018. The conventional charges move by exactly nothing, because they never choose a width the extension added. No charge’s error moves by as much as a thousandth, and the two that move furthest in width move, if anything, the wrong way.

The draw’s own best width does gain, by 0.0030 at 120 rows, 0.0031 at 480 and 0.0024 at 960, each more than ten standard errors from zero. So there is something past the cap, and it is worth about a quarter of a per cent of prediction error. But it is available only to a rule that reads the realised error on the draw, and the gain lives in the draw-to-draw noise of which width happens to be best rather than in the average surface any criterion can estimate. A feasible charge, choosing from the sample, cannot tell the draws on which sixty lags would have been better from the draws on which twenty would, because on the average surface they are the same.

A width that does not follow the draw #

There is a direct way to see why no charge can collect the quarter per cent. If a criterion were estimating the draw’s own best width, the width it chose would move with that best width from draw to draw: wider when the draw wanted a wide band, narrower when it wanted a narrow one. The correlation across draws between a charge’s chosen width and the best width is the measure of whether it is tracking anything, and a width that moves and an error that does not found it near zero for the field’s first law.

Under long memory with the list extended it is near zero for every charge at every size. At 960 rows the convention’s width correlates with the draw’s best width at 0.080, Schwarz’s at 0.051, the summed weights’ at 0.026 and the measured share’s at −0.015. At 480 rows all four lie between −0.057 and 0.011, and at 120 rows between −0.021 and 0.030. The measured share moves seventeen lags outward on average when the cap is lifted, and that move is the same on the draws whose best width is eighty as on the draws whose best width is twelve. It is a criterion responding to the average shape of the likelihood, which long memory tilts towards wide bands, and not to anything the particular draw says about its own error.

That is the condition under which a wider list can only add noise. A rule whose choice is unrelated to which width is best on the draw gains from a wider list only if the wider widths are better on average, and the surface says they are not. The oracle gains because it reads the realised error; it would gain from any list with more entries, whatever they were, because more entries are more chances for one of them to land on that draw’s luck. What a window leaves free traced the measured charge to the window’s own weights, a property of the estimator rather than of the data, and that is exactly why the charge has nothing per-draw to track with.

The slope the conditional needed #

The field’s conditional was about the slope of the surface, and the last figure measures it directly: how much worse the error is at eight lags, a band a conventional charge might plausibly choose, than at the surface’s floor, paired on the draw.

At 120 rows, eight lags is 0.00040 above the floor, within two standard errors of nothing. At 480 rows it is 0.00098, and at 960 rows 0.00087 — under a tenth of a per cent of an error of about one and a half, at the two sample sizes where the best band is widest. Thirty lags is level with the floor at both. For comparison, the best of the four charges sits 0.00015 above the floor at 960 rows. Shortening the band from its floor to eight lags, a factor of six, costs a thousandth; lengthening it from thirty to eighty, the move the cap prevents, costs nothing and gains nothing.

So the conditional fails here too, and here is where it had its best chance. Long memory is the one law in the field whose best band moves outward with the sample, the largest sample measured is where it has moved furthest, and the capped list is pinned against its end on four draws in five. If the error were going to rise steeply anywhere as the band was moved off its best width, it would be here. It rises by a thousandth.

Why the surface is flat, which is the field’s actual finding #

Every essay in the field has now found the same surface, and the reason can be stated rather than inferred from five measurements. A band width is a choice about how much of the error covariance to estimate. Estimated lags beyond the dependence that matters add noise to the whitening, and omitted lags leave dependence in the residuals; both enter the prediction error through the estimated regression coefficients, and the coefficients of a regression on four predictors are insensitive to small errors in the weights that whiten it — a generalised least squares estimator is consistent under any positive-definite weighting, and its efficiency changes only to second order in the error in the weights. The surface is flat because the object it scores is robust to the parameter being tuned.

That is why a charge derived for the parameter could be right and still not matter: the field’s first essay was correct that the convention over-charges a lag of band by a factor of two or three, and every later essay was correct that this costs almost nothing. The sample size, the window, the law and now the list’s reach have each been moved, and the over-charge has never once become expensive.

How the numbers were made #

Six hundred draws at each sample size, the field’s long-memory law generated exactly from its autocovariance sequence by a dense Cholesky factor, the full four-predictor regression fitted through the Bartlett plug-in band at each width, and prediction error measured on an independent replicate drawn from the same law. The capped and extended runs share every draw, so each paired difference is between two lists on one series; the standard errors quoted are those of the paired differences across draws. The best width on each draw is the argmin of its realised error over whichever list is in force. The charges are evaluated at each width as the log-likelihood minus the charge, the field’s construction throughout.

What is checked on the measurement: at 960 rows the best width lies past the cap on more than two draws in five and the measured charge moves more than ten lags outward when the cap is lifted; extending the list moves every charge’s error by under a thousandth while the oracle gains more than that; and from thirty lags to eighty the 960-row surface stays within two ten-thousandths of its floor. The refusal is the sentence the lead proposed — that long memory at a large sample makes the error rise by a hundredth from the floor to eight lags — and at 480 and 960 rows it fails by an order of magnitude.

Still open: a regression the weights do matter to #

The surface is flat because the coefficients of this regression are robust to the whitening. That suggests where a charge for a band could finally matter, and it is not a different dependence. It is a different target. An interval for a coefficient, rather than a prediction, depends on the whitening at first order through its standard error, and a test of a coefficient depends on it through its size. The field on what the interval covers found block-bootstrap intervals short by four to fourteen points under a dependence far milder than this one. A band-width charge read on the coverage of a generalised least squares interval — the same four charges, the same laws, the interval for one coefficient instead of the prediction error — is the measurement that could find a steep surface, because coverage is the quantity the whitening enters first. It has not been made.

Short of that, the field’s line of argument about prediction error is finished: the conditional has been tested at every setting the field can vary, and the next essay on this target would restate this one.

What is claimed here, and what is not #

This essay takes the field’s band-width charges under long memory with the list’s cap lifted. The claims are that the error surface’s floor is at twelve lags at 120 rows, at thirty at 480 and at forty-eight at 960, and that from thirty lags to eighty the 960-row surface lies within three hundred-thousandths of its floor; that the draw’s own best width lies past thirty on 33.2%, 45.5% and 48.7% of draws at the three sizes; that the two conventional charges never reach the cap, while the measured share is pinned at it on 80.0% of draws at 960 rows and moves from 28.4 lags to 45.8 when it is lifted; that lifting the cap changes no charge’s error by as much as a thousandth, the measured share’s by +0.00020 ± 0.00021 at 960 rows, while the best width on each draw gains 0.0024 to 0.0031; that no charge’s width correlates with the draw’s own best width by more than 0.08 at any size; and that the error at eight lags is under a thousandth above the floor at 480 and 960 rows.

What stays out, named as a decision: other windows, since every band here is Bartlett; other regressions, whose coefficients might be less robust to the whitening; and a target other than prediction, deferred above as the one place a charge could still matter.

The boundary against the charge at four times the sample is that it found the surface flatter at 480 rows under a first-order autoregression and named long memory as the case it had not tested; this essay tests it, at a larger sample still. Against the surface under every law, which held the list fixed and moved the law, this essay holds the law fixed at the one that strains the list and moves the list.