The charge at four times the sample

The charge nobody derived measured what a lag of covariance band actually costs a criterion. A band’s numbers are shrunk by the window rather than fitted freely, and the optimism one more lag of Bartlett band adds — how much better the fit looks on its own sample than on a fresh one — came to about three eighths of the one log-likelihood unit the convention charges. What a window leaves free traced the mechanism to the window’s weights, whose sum is exactly half the band’s width, and found the measured charge sitting a quarter below that sum.

A width that moves and an error that does not then priced the mistake. The derived charge picks bands three times as wide as the convention and wins, at six paired standard errors, by half a per cent of error — a fifth of what was on offer. Its last section stated the condition under which that would stop being harmless: the error surface across band widths is flat, so a charge that moves the width by a factor of four costs almost nothing, and a setting where the error rose steeply past twenty lags would have the same three-fold error in the charge and a real cost attached to it.

A longer sample is the first place to look for that setting. Every autocovariance is estimated on more pairs, a wide band’s noise costs less, and the criterion has more to discriminate with. This essay repeats the field’s two measurements at 240, 480 and 960 rows.

What a lag of band costs as the sample grows #

The optimism is measured as before: estimate the band on one sample, evaluate the concentrated likelihood on that sample and on an independent one from the same law, and difference, for band widths from one lag to thirty. The rise from one lag to thirty, divided by the twenty-nine lags added, is the charge a lag deserves. Two thousand four hundred pairs of samples at each size, under the first-order autoregression the field uses.

At 120 rows a lag carries 0.3745 of a unit, with a standard error of 0.0088. At 240 rows it is 0.3842; at 480 0.4523; at 960 0.4022, with standard errors of 0.0133, 0.0196 and 0.0282. The rise from 120 rows to 480 is 3.6 of their combined standard errors, and every reading sits below the half a unit the window’s weights give.

So the field’s measured share of the weights — three quarters at 120 rows — is a property of that sample size, not of the window. With more rows the shrunk band’s numbers behave more like the free parameters the summed weights count, and the charge climbs towards the weights’ figure. The 960-row reading falls back, but its standard error is three times the 120-row one, and the four readings together say only that the charge rises from three eighths towards a half and has not reached it by 480 rows. The convention’s full unit is not approached at any size.

The standard error grows with the sample because the optimism is a difference between two likelihoods that both grow with $n$; the difference stays of order one while each term’s noise grows, so measuring the charge precisely gets harder exactly where it changes.

Where on the band the extra charge comes from #

The rise is not spread evenly along the band, and reading it in two parts separates the trend from the noise. The first few lags carry the least optimism and the most noise: at 960 rows the second lag adds only 0.17 of a unit where at 480 rows it added 0.71, a difference no mechanism predicts and every draw-to-draw fluctuation allows. From eight lags to thirty the curve is steadier, and its slope there is 0.368 a lag at 120 rows, 0.404 at 240, 0.443 at 480 and 0.445 at 960.

Read that way the charge rises monotonically and levels off a little under nine tenths of the weights’ half. The 960-row dip in the whole-range figure is in the first few lags, where the measurement is weakest; along the wide part of the band, which is what a criterion choosing between eight and thirty lags is actually charging for, a lag costs about forty-four hundredths of a unit once the sample has a few hundred rows.

That number is the one a criterion should use at large samples if it wants a derived charge, and the field’s 120-row figure is the one to use at small ones. Neither is the convention’s one. Whether the long-lag slope eventually reaches the weights’ half is not settled by four sample sizes; it has stopped moving between 480 and 960 rows, which is evidence that it may not, and the reason could be the same downward bias in a sample autocovariance that held it under at small $n$, shrinking more slowly than the rest.

The surface flattens #

The more important question is whether a longer sample makes the band width matter, and the answer is the opposite of the condition the field named.

Between six lags and thirty, the prediction error varies by 0.00203 at 120 rows, 0.00123 at 240 and 0.00047 at 480. Quadrupling the sample makes the surface more than four times flatter. Every width on the list delivers nearly the same error, and the differences between widths shrink at least as fast as the errors themselves approach their floor.

The reason is that the criterion is choosing a whitening for a regression, and the regression’s error at a given width is made of the band’s bias, which a wider band reduces, and the band’s estimation noise, which more rows reduce. Both shrink as the sample grows, and the difference between two widths is a difference of two small things. Long samples do not make the choice of width harder to get right; they make it harder to get wrong, and the band width joins the list of tuning choices in this collection whose stakes shrink as the data grow.

The widths each charge picks #

Every charge’s width grows with the sample, because a fixed charge per lag becomes relatively cheaper as the likelihood’s gains grow. The convention picks 6.04 lags at 120 rows, 8.17 at 240 and 11.17 at 480; the summed weights 10.19, 13.90 and 17.93; the measured charge 14.01, 18.39 and 23.47. The best width on each draw grows least: 13.95, 14.40 and 15.22.

So the widths diverge from each other and from the best width as the sample grows — at 480 rows the convention is four lags short of the best and the measured charge eight lags long — while the errors they deliver converge. That is the field’s finding, sharpened: the width a charge picks is not an estimate of anything, and a longer sample makes that more true, not less. The measured charge, which at 120 rows picked nearly the best width on average, overshoots it at 480 because its per-lag charge is fixed at the 120-row measurement while the true optimism has climbed.

The rules that read no data #

Two rules in the field’s table choose a width without a criterion at all, and they are worth reading beside the charges because they bound the question from below. Schwarz’s charge, half the log of the sample size a lag, picks 3.63 lags at 120 rows, 5.14 at 240 and 6.46 at 480 — the narrowest bands of any rule, because its charge grows with the sample while the optimism does not. Newey and West’s automatic width, a fixed function of the sample size, rounds to four lags on the field’s list at all three sizes.

Their errors converge with everyone else’s. At 120 rows Schwarz’s charge delivers 1.11414 and the automatic width 1.11302, against 1.10807 for the summed weights; at 480 rows they deliver 1.02825 and 1.02904 against 1.02783. The worst rule in the table is about a ninth of a per cent from the best feasible one at 480 rows, where it was half a per cent at 120.

So at four times the sample the whole question of how to charge a band — convention, Schwarz, the window’s weights, the measured optimism, or no criterion at all — is worth about a tenth of a per cent of error between its best and worst answers. A field that began by finding the convention wrong by a factor of three ends, at this sample size, by finding that almost nothing a practitioner could choose is wrong by more than a rounding error in the result.

A gain that fades into its own noise #

The summed-weights charge beats the convention by 5.49 paired standard errors at 120 rows, 2.71 at 240 and 1.73 at 480. The measured charge beats it by 4.10, 1.97 and 0.79. In absolute terms the summed-weights charge’s advantage is 0.00179 at 120 rows, 0.00038 at 240 and 0.00010 at 480, against errors of about 1.11, 1.05 and 1.03.

That disposes of the condition the field named, at least where it was expected. The convention over-charges a band by a factor of two to three at every sample size measured, and the derived charge’s advantage over it falls from 0.00179 of error, clearly measurable, to 0.00010, under two of its own standard errors. A longer sample is not where the convention’s error acquires a price.

Why the charge moves and the price does not #

The two halves of the result point different ways, and both follow from the same arithmetic.

The charge a lag deserves is the optimism, which is a statement about how much the fit adapts to its own sample. At small $n$ the window’s shrinkage and the sample autocovariances’ own downward bias hold that adaptation under what the weights count; with more rows the autocovariances are estimated nearly without bias and the shrunk numbers adapt nearly as much as their weights allow. So the charge rises towards the weights’ half.

The price of charging wrongly is the error the chosen width delivers, against the error the best width delivers, and that is governed by how much the error changes across widths — the surface’s flatness — which falls with $n$ faster than the charge rises. A charge that is wrong by a factor of two moves the width by a few lags, and a few lags of band at 480 rows are worth a few hundredths of a per cent.

What a practitioner should charge #

The recommendation the field could make at 120 rows was that the convention is wrong, a derived charge is right, and the difference is small. At larger samples the first two halves sharpen and the third dominates.

The convention remains wrong at every size, by a factor that falls from about two and two thirds at 120 rows to about two and a quarter at 480 on the long lags. Any derivation of a band’s charge that starts from a regression coefficient’s unit overstates it.

The window’s weights are the better guide as the sample grows. At 120 rows the measured optimism sat a quarter below them; on the long lags at 480 and 960 rows it sits about a tenth below. A charge of half a lag of Bartlett band — the summed weights, which need no measurement at all — is within a tenth of right at a few hundred rows, and it is what the plug-in and the maximum would have needed had its band been chosen by a criterion.

And none of it matters much. The point of nothing in the fit picks the width was that the likelihood cannot choose a band; this essay adds that at larger samples it no longer needs to, because any band in the useful range delivers the same error to within a few hundredths of a per cent.

How the two measurements were sized #

The two halves of the result needed very different amounts of simulation, and the difference is itself informative.

The charge is a difference of two likelihoods that each grow with the sample, so its noise grows with the sample too; at 960 rows its standard error over the whole band is more than three times its 120-row value. Two thousand four hundred independent pairs of samples at each size were enough to separate 120 rows from 480 by three and a half standard errors and not enough to place 960 rows on the trend with the whole-band figure, which is why the long-lag slope is reported beside it.

The price is a difference of two errors that both approach the same floor, so its noise shrinks with the sample; three hundred draws at each size were enough to measure the summed-weights charge’s gain at 120 rows to five and a half paired standard errors and to see it fall inside two by 480. The trials were not chosen to make the gain disappear. They were chosen as the field chose them, and at 480 rows the gain itself is a twentieth of what it was.

A measurement whose precision falls with the sample and a quantity that becomes negligible with it is the combination that makes a charge academic: the more data there is, the harder it is to know the right charge and the less it matters to have it. A covariance with no parameter met a version of the same combination in the banded estimate’s own variance, and it is a general feature of tuning quantities whose effect is second-order.

Still open: a dependence the list cannot reach #

The steep surface the field anticipated may still exist, and the measurement above says where not to look for it rather than where to find it. It is not at larger samples under a first-order autoregression, and the surface under every law found it under none of four laws at 120 rows. The remaining candidate is a dependence whose useful band is wider than the list allows — long memory at a large sample, where the best width would want to pass thirty lags and every charge on the list is capped below it. There the cap, not the charge, would decide the error, and the comparison would be between charges that agree on the list’s last value. Whether the error then rises steeply as the band is shortened from the cap is the measurement that would test the field’s conditional one last time.

The claim, and where it stops #

This essay takes the field’s charge and its price at longer samples. The claims are that the optimism a lag of Bartlett band carries under a first-order autoregression is 0.3745 at 120 rows and 0.4523 at 480, 3.6 combined standard errors apart, with 960 rows too noisy to extend the trend, and below the window’s half a unit throughout; that the prediction error’s spread between six and thirty lags falls from 0.00203 at 120 rows to 0.00047 at 480; that every charge’s chosen width grows with the sample while the best width grows least; and that the summed-weights charge’s paired advantage over the convention falls from 5.49 standard errors at 120 rows to 1.73 at 480.

What stays out, named as a decision: other windows and laws, which the field measured at 120 rows only; a charge recalibrated at each sample size, since the measured charge here is fixed at the field’s 120-row share and its overshoot at 480 rows is partly that; and the long-memory case, deferred above. The boundary against a width that moves and an error that does not is that it found the error flat at one sample size and named a longer sample as where it might not be; this essay finds it flatter there.