Field

A block, weighted inside itself

Every resampling in this collection attenuates the dependence it is trying to keep, and until now every one of those attenuations was inherited rather than chosen. The construction the long-run-variance literature actually uses weights the residuals down towards each block's own ends — and what that buys turns out to be exactly the squared value of the window at the two ends and nothing else about its shape. It is an asymptotic gain that has not arrived at any block length a hundred and twenty rows can afford, and at the lengths that are available a tapered block is very nearly a shorter plain one.

Four blocks, and only the ends matter. The weights a block carries, normalised so that the resample keeps the residuals' variance and only their covariances are attenuated. What separates these shapes, for everything that follows, is the value at the two ends and nothing about the middle: the first-order attenuation is −(w(0)² + w(1)²)/(2∫w²), which is -1.0000 for the rectangle, -0.3896 for the trapezoid cut off at half height, and exactly zero for both windows that reach the axis. The half-height trapezoid is in the table to be the case that separates a shape from a boundary value: it is smooth, it is tapered, and it buys none of the order the other two buy.

A block weighted inside itself

The triangle every block resample attenuates by is not a fact about blocks. It is the self-convolution of a rectangle, and a block weighted down towards its own ends has a different one — whose leading term is the squared value at the two ends and nothing else about the shape.

How far each reference distribution's 95% point falls short. Seven constructions on rows that repeat each other, at a block length of 5 and 200 draws, against the statistic's own 95% point of 3.0224 computed from three thousand draws of the same world. Reading down: a multiplier on every row keeps no dependence at all and is 44% short; a multiplier shared along a block keeps the triangle; a fixed block keeps the same triangle and is 8% closer, which is the pair that says a taper is not what decides this; the stationary bootstrap; the two tapered blocks, both further short than the untapered one at this block length; and errors generated from a fitted model, which is the only construction here not bounded by what the residuals report.

A taper and a critical value

Two constructions whose tapers visibly differ give the same critical value, and two that share a taper exactly do not. Adding a construction whose taper is a decision rather than an accident says which half of that is true.

One t interval with its scale read three ways, 120 rows in blocks of 32. What a 95% t interval for the mean of a first-order autoregression at 0.7 covers, and how wide it is, when its block-means scale is read from the 3 disjoint blocks of 32 rows, from a block at every one of the 89 starting rows, and from those overlapping blocks tapered — over 8,000 draws. Disjoint: 94.4% at a width of 1.560, with a multiplier of 4.30. Every starting row: 94.8% at 1.054, multiplier 2.74. Tapered: 95.0% at 1.020, multiplier 2.57.

A block at every starting row

The t interval on the block count reads its scale from the disjoint blocks, three of them when 120 rows are cut into blocks of thirty-two, and throws away the other eighty-six blocks the series contains. Reading a block at every starting row leaves the scale's attenuation where it was — the two expectations agree to within a hundredth at every cell — and nearly doubles the degrees of freedom it realises, so the interval loses a third of its width at the same coverage. Where the interval falls short, at short blocks, it falls short by the same amount however the blocks are read.

Coverage against block length, 120 rows. The coverage of the t interval for the mean of a first-order autoregression at 0.7 on 120 rows, at every block length on a fine grid, with its scale read from the disjoint blocks and from a block at every starting row, over 8,000 draws. The squared-error length for disjoint blocks at the true persistence is 13.9 rows, and the same algebra for the overlapping scale gives 16.0, marked. The overlapping interval first reaches 95% — to within a tenth of a point — at blocks of 32 rows, where it covers 94.9% at a width of 1.056; at the overlapping squared-error length it covers about 92.5%.

A length chosen for the interval

The squared-error rule for a block length weighs the scale's bias against its variance. Rebuilt for a block at every starting row, whose variance is two thirds as large, it asks for blocks fourteen per cent longer and buys four tenths of a point of coverage. A t interval has already paid for the scale's variance in its multiplier, so its own coverage is predicted by the bias alone — within seven tenths of a point up to blocks of twenty-eight rows — and a rule that chooses the shortest length that prediction says will cover asks for blocks two and a half to three times longer and lands within about a point of 95% at 120, 240 and 480 rows. The price is width, and it is paid where the series is short.

A share read off the sample

A block length chosen for an interval's coverage needs the share of the mean's variance the scale will find, and the published rule reads it from a first-order autoregression at the sample's lag-one value. Reading it from the sample's own autocovariances instead was supposed to survive series nobody had modelled. At 120 rows it covers 92.5% to 93.5% on all four laws — worse than the wrong shape on every one — because a centred series' autocovariances sum to exactly zero and the share they imply is too generous; read to lag forty it covers 84%. The assumed shape costs width, not coverage: 17% too wide where dependence stops at lag four, and nothing visible under a break, where the interval covers for a reason the share knows nothing about.

A sequence that knows it was centred

A block length read off a sample's own autocovariances gets worse the further out it reads, because a centred series' autocovariances sum to exactly zero. To first order every lag loses the same amount, about the long-run variance over the sample size, and that can be added back. Corrected, a reading to lag forty goes from 84.5% to 89.2%, and to lag twelve from 93.3% to 94.5%. But the correction overshoots on the long lags, and on the rule that picks its own cut-off, which is short, it moves coverage by three tenths of a point. The identity was the reason the lag mattered. It was not the reason the rule falls short.

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