A share read off the sample

A length chosen for the interval found that a t interval on the overlapping block-means scale has already paid for the scale’s randomness in its multiplier, so its coverage at any block length is predicted by one number: the share $s$ of the mean’s variance the scale expects to find. The coverage is $P(|T_\nu| \le t_\nu\sqrt{s})$, and a rule that picks the shortest length at which that prediction reaches 94.5% landed within about a point of 95% at 120, 240 and 480 rows, where the squared-error rules all fell short. It could do so only on the overlapping scale, which makes long blocks affordable on a short series.

It reached that share by assumption. The rule reads the sample’s lag-one autocorrelation $\hat\rho$ and looks the share up in a table computed for a first-order autoregression at that value. The essay closed on what that costs: a series whose dependence decays faster than a geometric sequence needs a smaller correction than the table supplies, and one whose dependence lasts longer needs a larger one. It named the obvious repair — compute the share from the sample’s own autocovariances, which describe whatever shape the series actually has — and the price of it, a second choice of how far out to read them.

That repair is measured here, on two laws chosen because the geometric share is wrong about them in opposite directions, and the measurement goes the wrong way. At 120 rows the repair is worse than the assumption on every law, including the two the assumption describes badly. The reason is a single identity about centred series, and the assumption turns out to cost width rather than coverage, in the one place where it costs anything.

Four rules, four laws #

The share is a property of a covariance matrix, not of an autoregression. For any covariance $\Sigma$ of the rows, the overlapping scale is a sum of squared linear combinations of the series, so its expectation is a quadratic form in $\Sigma$, and the interval is aimed at $n\,\mathrm{Var}(\bar x)$, which is another one. Their ratio is the share, and it can be fed four ways:

• geometric — the published rule, unchanged: a first-order autoregression at the sample’s $\hat\rho$;
• read to the chosen lag — the share computed from the sample’s autocovariances, weighted by a flat-top window cut off at the lag Politis’s rule picks from the sample, the same cut-off a length read off the whole sequence used for its squared-error rule, where reading the sequence bought nothing at 120 rows and paid only off the autoregression at 960;
• read to lag six — the same, cut off at a lag stated in advance;
• known — the share computed from the law’s own covariance, which needs no sample and so chooses one length per law.

The four laws are the autoregression at 0.7 the rule was built on; a five-period moving average, whose dependence stops dead after the fourth lag; and a break in the persistence — 0.95 for the first sixty rows, then 0.65 — together with the same break turned over. The moving average decays faster than any geometric sequence with its lag-one value of 0.8; the break, read through its lag-one value alone, looks like a moderate autoregression and remembers far longer than one.

The known share does what the criterion promises: 96.5%, 95.3%, 95.2% and 95.0% under the autoregression, the moving average, the break and its mirror. The geometric rule covers 96.2%, 96.5%, 95.1% and 94.8%. Reading the share to the chosen lag covers 93.2%, 93.5%, 93.1% and 92.5%, and reading it to lag six 93.7%, 94.6%, 92.6% and 92.2%.

So the rule built for one shape is within a point and a half of the oracle on all four laws, and the rule that reads the shape from the data is short of its own 94.5% target on all four, by one to two points. That is not a result about the moving average or the break. It is a result about reading.

What the rules believe #

Coverage is the consequence; the cause is the share each rule believes. On the moving average the three can be drawn against block length.

At blocks of twenty-four rows the moving average’s scale expects 0.926 of the mean’s variance. The geometric rule believes 0.782: the lag-one value is about 0.8, a first-order autoregression with that value is still carrying a fifth of its correlation at lag seven, and the table charges the scale for every lag of it. The sample-read rule believes 0.935 — close to the truth here, and slightly above it.

The direction is the point. The geometric rule is wrong about the moving average in the cautious direction, so it asks for longer blocks than it needs, and a block that is too long costs width and nothing else. The sample-read rule errs a little the other way, and a block that is too short costs coverage. The same comparison on the autoregression — where the geometric shape is right — shows it more plainly: the law’s own share at twenty-four rows is 0.873, the geometric rule believes 0.865, and the sample-read rule 0.927. Six points too generous, on the one law where nothing needed estimating but a number.

The identity the long lags read #

That excess is not noise. It has a direction, and the direction comes from an identity that holds for every series of every length. A sample autocovariance is computed around the sample mean, and the deviations from a sample mean sum to zero, so

$$\sum_{|k|<n} \hat\gamma(k) = \frac{1}{n}\Big(\sum_t (x_t - \bar x)\Big)^2 = 0.$$

Summed over every lag on both sides, a centred series’ autocovariances describe no variance at all. The true sequence sums to $n\,\mathrm{Var}(\bar x)$, the very quantity the interval is aimed at; the sample’s sums to nothing, exactly, whatever the law.

A share needs that sum as its denominator, cut off somewhere. Close to the origin the cut-off sum is roughly right; the further out it reaches, the more of the identity it inherits, and the smaller the variance it reports. A small denominator makes the scale look as though it already captures most of the variance, so the rule believes the blocks need not be long, and chooses short ones.

Read to lag six, the four laws cover 93.7%, 94.6%, 92.6% and 92.2%. Read to lag forty they cover 83.9%, 85.9%, 84.0% and 84.4%, and the median lengths chosen fall to four, three, twenty-three and twenty-four rows. The autoregression and the moving average collapse first, because their sequences are nearly spent by lag twelve and every lag read past that point contributes only the centring’s share of nothing. The breaks hold on longer, because the persistent half still has something real to report at lag twenty-four, and then collapse too.

So the second choice the earlier essay expected to resemble the first is not a nuisance parameter beside the estimate. On 120 rows it is most of the estimate. A cut-off lag that is longer is not more general; it is more of the identity. The length nobody has found the best block length for a variance estimate wandering by sixteen rows from draw to draw; the length chosen from a sequence read to lag twelve wanders as well — from 4 to 40 rows between the tenth and ninetieth percentiles on the autoregression — and every row of that wander comes from a different share.

Where the shape assumption costs something #

If reading the sequence is the wrong repair at 120 rows, the next question is how much there was to repair. The answer is in widths rather than coverages.

On the moving average the known share asks for blocks of 31 rows and gives intervals 1.013 wide. The geometric rule asks for the cap, 40 rows on every draw, and its intervals are 1.184 wide — 16.9% wider — for 1.2 points of coverage beyond the target nobody asked for. That is the cost of the assumed shape on a law whose dependence stops sooner, and it is a cost in width alone: the interval keeps its promise and pays too much for it.

This is the same trade bias is not the whole of it drew between windows: a rule tuned to the wrong quantity does not usually miss its promise, it overpays for it, and the overpayment is invisible to anybody who only counts coverage.

On the autoregression there is nothing to pay. The known share and the geometric share both run out at the cap, the known one because even the true share does not lift the predicted coverage to 94.5% before forty rows, the geometric one because it is right. Both intervals are about 1.20 wide.

And under the break there should have been a great deal to pay, because the geometric rule is badly wrong about the break’s share. At twenty-four rows the break’s scale expects 0.509 of the mean’s variance; the geometric rule believes 0.806, the sample-read rule 0.830. Both are far too generous, in the direction that ought to cost coverage. Yet the geometric rule covers 95.1% and the known share 95.2%, at the same length, the cap, and nearly the same width.

A short series hides the break #

The first half of the explanation is the cap. At 120 rows the break’s true share at forty rows is 0.626, and the coverage predicted from it is 91.4% — so the known share, honestly computed, says no length a third of the series allows will reach the target, and asks for the longest one. The geometric rule asks for the longest one too, for a different reason, and once two rules choose the same length on every draw their errors about the share no longer matter. The two rules agree because the series is too short for them to disagree.

The second half is the gap between that prediction and the count. The interval was predicted to cover 91.4% and covers 95%. The autoregression over-covers its own prediction at blocks this long too, by two points — the caution the earlier essay found in the prediction once a scale rests on three blocks’ worth of degrees of freedom — but the break over-covers by nearly four, and the extra has a different source.

The prediction treats the scale as a share of the variance times a chi-square independent of the mean, which is what the t multiplier assumes. Under the autoregression the independence holds: the size of the sample mean and the size of the scale are uncorrelated (0.007), and the interval covers 96.5% with each mean scored against its own scale and 96.2% scored against twenty other draws’ — the same, so the two points over the prediction belong to the prediction rather than to the pairing. Under the break the two are correlated at 0.409. The persistent sixty rows drive both: when they wander far from zero they move the mean and make the blocks inside them disagree, so the interval is wide on exactly the draws that need width. Scored against its own scale the break’s mean is covered 95.2% of the time; scored against other draws’ scales, which keeps both distributions and removes only their pairing, 90.8% — four and a half points lost, and six tenths below the prediction.

It is the same mechanism that makes the interval with no resampling in it work at all: a studentised ratio is protected by its denominator moving with its numerator, and here the break supplies that movement in quantity. So the break is covered at 120 rows by a property of the t ratio that no reading of the share contains. That is real coverage and it is not an accident of the seeds; it is also not something the length rule did. A protocol that reports a length chosen for coverage under a break is reporting a length that was chosen for a reason that does not apply and works for one it does not mention.

At four times the length #

A short series makes every rule ask for the cap, and the cap hides differences. At 480 rows the cap is 160 rows and nobody reaches it.

On the autoregression the geometric rule asks for a median of 62 rows and covers 95.2% at a width of 0.462; the sample-read rule asks for 44 and covers 94.9% at 0.449; the known share asks for 58 and covers 95.0% at 0.455. Reading the sequence now costs three tenths of a point and buys three per cent of width — on the one law where it has nothing to discover.

On the moving average the difference is large. The geometric rule still believes the dependence is long and asks for a median of 102 rows — nearly three times the 36 the known share asks for — and gives intervals 0.485 wide that cover 95.6%. The sample-read rule asks for 35 rows and gives intervals 0.412 wide that cover 94.4%, the same width and within two tenths of a point of the known share’s 94.6%. At 480 rows the repair the earlier essay named does what it was supposed to: it removes the shape’s cost, 15% of width, at almost no cost in coverage.

The break at 480 rows, with the change moved to the middle, is where the cap stops hiding anything, and the three readings come apart in an order nobody would have guessed. The geometric rule asks for a median of 92 rows; it believes the scale at a fifth of the series expects 0.944 of the mean’s variance where the law’s scale expects 0.806, too generous by fourteen points, and its intervals cover 94.6% at a width of 0.867. The known share, trusting the prediction, finds that no length reaches the target and asks for the cap of 160 rows on every draw; it covers 97.8% at 1.111, 28% wider. The sample-read rule, whose chosen cut-off now reaches far enough into the persistent half to see it — a median lag of 32 — asks for 127 rows and covers 95.2% at 0.995. The mirror break gives the same order: 95.0%, 98.4% and 96.0%.

So under a break the share known exactly is the worst of the three readings, because the prediction it feeds leaves out the pairing that rescues the interval; and the geometric rule’s error about the share and the prediction’s error about the t ratio point in opposite directions and very nearly cancel. On this law the rule that is wrong about the share delivers what the rule that is right about it promised.

What changed between the two sizes is the identity’s reach. To first order every sample autocovariance loses the same amount to the centring, about the variance of the sample mean, which is the long-run variance divided by $n$. A sum read over $2K + 1$ lags therefore loses about $(2K + 1)/n$ of what it is estimating — a ninth at lag six on 120 rows, a thirty-seventh on 480 — and the sample-read rule’s generosity falls with it. At 480 rows its believed share at a fifth of the series is within seven thousandths of the truth on both stationary laws, where at 120 it was five points high on the autoregression.

Four thousand draws, exact shares, and where they stop #

Every coverage and width is over four thousand draws per law and size, with all four rules applied to the same series. Every true share is exact, a quadratic form in the law’s covariance with nothing simulated; the oracle route reproduces the published autoregressive table to $10^{-9}$ once that table’s denominator — the long-run variance rather than $n\,\mathrm{Var}(\bar x)$, a choice that makes it 2.3% cautious at 120 rows — is converted. The zero-sum identity is checked on drawn series to machine precision rather than taken from the algebra. The breaks are drawn exactly, through the Cholesky factor of their covariance, with the change at row sixty at 120 rows and at the middle at 480.

The sample-read share uses the flat-top window because that is the window the length rule for the variance uses; a Bartlett window would shrink the long lags harder and soften the collapse, and would not remove the identity. The cut-off lags of six, twelve, twenty-four and forty were stated in advance. Not measured: long memory, under which no block a short series allows is long enough and every rule asks for the cap — the observations that repeat each other do so for longer than any block can hold; persistences other than the four laws here; and target coverages other than 94.5%.

Reading loses to assuming at 120 rows, and long memory is not measured #

The claims are that reading the share from the sample is worse than assuming a geometric one at 120 rows under every law here, by one to four points of coverage; that it is worse because a centred series’ autocovariances sum to zero, so the share they imply is too generous and grows more so the further out the sequence is read; that the geometric assumption costs 17% of width on a moving average at both sizes and no coverage; that the break is covered at 120 rows by a correlation between the mean and the scale, worth four and a half points, which pairing them apart removes; and that at 480 rows the sample-read rule matches the known share on both stationary laws.

What it does not claim is that the geometric rule is safe, nor that what the interval covers on a first-order autoregression is what it covers anywhere else. It is safe here because the two laws that defeat it most are a law it over-corrects and a law the cap and the t ratio rescue. A law that decays more slowly than its lag-one value suggests and has no persistent stretch to drive the scale — a smooth long memory, say — would be under-corrected with nothing to save it.

Still open: a sequence that knows it was centred #

The identity is exact, so its effect on each sample autocovariance is computable: to first order every lag loses the same amount, about the long-run variance divided by $n$. A sequence corrected by adding that amount back — iterated, since the long-run variance is what is being estimated — would no longer sum to zero, and its share would no longer drift towards one as the lags are read further out. Whether that correction restores the sample-read rule to the geometric rule’s coverage at 120 rows on the autoregression, keeps its advantage on the moving average at 480, and makes the cut-off lag the minor choice the earlier essay expected it to be, is the measurement that would say whether a share read off a short series can be trusted at all. The counter evidence is already in the error no window repairs: the sample’s attenuation of the long-run variance is eight points at 120 rows, and a correction of that size has to be estimated from the same rows it corrects.