Normal Distribution
nothing is called 95% until it has been counted
Concept

Residual — where it appears

What is left of an observation after a fitted model's prediction is subtracted, carrying the error's size, its shape and its position. A resampling that keeps its size and loses its position repairs a different defect from one that keeps its position and loses its neighbours.

Named by 17 essays across 8 fields — each of them below, with the objects they name alongside it.

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.

taper · Bootstrap The price of each thing the rule is not told. What each rule gives up against the best model available, at a persistence of 0.85 on a fifteen-candidate table, over 400 draws. Reading down: least squares with the ordinary penalty; the whitening at the true ρ; the same at a ρ̂ estimated per candidate; that rule with the term the Gaussian likelihood carries and it omits; a Bartlett-tapered Ω̂ estimated once from the fullest candidate at L = 8; the same estimated per candidate; and the truncated Ω̂, which exists on only 45.0% of draws and is averaged over those. Knowing ρ recovers 89.9% of what counting rows gives up, estimating it 83.4%, and estimating a whole covariance 75.6%.

A covariance with no parameter in it

The whitening that repairs a criterion is told the dependence is a first-order autoregression and left to find one number. A real dependence is not one number, and the obvious estimate of it is not a covariance matrix.

banded · Dependence What each rule gives up against an oracle that is arithmetic. Expected squared error of the candidate each rule selects, minus the expected squared error of the best candidate in the table, over 500 draws of 120 rows. Both quantities are closed forms — σ_S²(1 + q/(n − q − 1)) — so the only Monte Carlo here is over which candidate got picked. The hold-out spends half its rows measuring what the criterion computes, and pays 1.8 times as much for it. Schwarz's criterion is worst because it is answering a different question: which candidate contains the truth, rather than which one forecasts best.

A criterion is a prediction of the hold-out

A rolling hold-out spends half the sample measuring what a criterion computes from all of it. Against an oracle that is arithmetic rather than an estimate, the criterion gives up 0.01701 and the hold-out 0.03200 — and the number the hold-out reports for its own winner is optimistic by more than either.

proxy · Order-selection The dependence, at four removes. Under AR(1) at 0.8, four different sequences all called the dependence. The top line is the law. The middle line is what a sample of 120 errors reports on average — computable exactly, because the expectation of a sample autocovariance is arithmetic once the covariance is known. The lower line is what a candidate's residuals report, which is what every two-step rule in this collection actually reads: a fit removes variance, and it removes more of the persistent part than of the rest. At the first lag the three are 0.800, 0.7773 and 0.7338. The dots are counted from draws and share no arithmetic with the line they sit on; the worst departure is 1.2 standard errors.

A dependence fitted with the line

Every whitening in this collection reads the dependence off a set of residuals, and residuals are not errors. Fitting the two together recovers most of what that costs, and changes almost nothing about the decision it feeds.

together · Dependence How much of one search the other has already found. Five pairs of searches on one sample, on a scale whose zero and one are both fixed by construction. Zero is two searches over disjoint sets of independent columns: they remove shares of the residual sum that add, at 0.8 standard errors from exactly additive, and they read 0.004. One is a break search paired with a step column it contains, which reads exactly one on every draw because the step adds nothing at all. Between them: two dictionaries of step columns cut a few rows apart read 0.125, and the pair the earlier field measured — a break and a whitening window, both reading the same residual series — reads 0.762, three quarters of the way to one search containing the other. And below zero, a break paired with a search over independent columns reads -0.306: the joint search finds configurations neither half of it contains, so charging the two separately under-charges.

Two searches that share nothing

Two searches over independent columns remove shares of the residual sum that add exactly. On the scale a chi-square point is quoted on they look super-additive by a fifth of a unit, and none of it is overlap.

apart · Criterion One true null, one table, five readings. every subset of four, fifteen models, at a null where nothing any candidate holds is worth anything, over 500 draws. Each bar is the share of draws on which that reading declares a difference at a nominal 5%. The reading is the whole of the difference between the bars: the data is identical. An open search over all 210 ordered pairs rejects 76.2%; the table's own 5% point is 3.163 against the 1.671 a single comparison uses. Bonferroni takes the open reading to 0.6% — and on the nested ladder the same correction does not reach the nominal level at all, because there the excess is a shift in the mean rather than a maximum over many.

When the benchmark is a candidate

A specification search with a benchmark nailed down is the case with a closed form. Take the nail out — let the model that would have been reported be one of sixteen, chosen by the same data as its rivals — and the same true null is read three ways, at 2.0%, 7.8% and 76.2%.

select · Forecast What the second search finds, alone and afterwards. For four of the pairs, what the second search removes on its own and what it removes once the first has already run. The gap between the two is the overlap in absolute terms. Where the searches share nothing the two readings are the same: an independent column removes 0.0261 alone and 0.0260 afterwards. Where one contains the other they are 0.1387 and exactly zero. The pair the earlier field measured sits between: a whitening window removes 0.5033 alone and 0.3120 after a break search has run. This is the earlier field's own reading of its pair, on the share scale rather than in log-likelihood units, and it is the number a rule that runs both searches actually has to charge for.

A search that is already the other

A break search shifts every coefficient after a row, so a step column is one of the directions it can move in. Paired with a dictionary of them it reads exactly one, on every draw, and that fixes the top of the scale.

apart · Criterion 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.

taper · Reference How much memory a fit takes out, candidate by candidate. Under AR(1) at 0.8, the lag-one autocorrelation a candidate's residuals report, computed exactly for each candidate on 200 draws. The upper line is the law at 0.8000. A candidate that is an intercept alone reports 0.7773 — which is exactly what a sample of 120 errors reports, because an intercept annihilates the sample mean and nothing else, and the two arithmetics agree to the last bit. Every predictor after that takes more out, down to 0.7341 at the fullest candidate. That is the collision this field is about: the rule every whitening here uses estimates its nuisance once, from the fullest candidate, so that the criteria stay comparable — and the fullest candidate is the one whose residuals report the least.

The fit that takes the memory out

A candidate's residuals report less dependence than its errors do, and how much less is arithmetic rather than noise. The rule used for a good reason reads the series that has lost the most.

together · Dependence Which repair goes with which defect. The share of true nulls rejected at a nominal 5% by a reference distribution generated from the fitted benchmark, over 120 draws with 59 resamples each. Where the errors are well behaved every resampling is fine and all four are conservative. Where the variance is a function of the design, the two that detach a residual from its own row reject 5.8% and 8.3% — and the block bootstrap, which is the resampling three earlier fields on this site reach for, repairs nothing at all, because the dependence it is built for is between origins and the rolling scheme reproduces that on its own. Where the errors are skewed the symmetric multiplier is the one that is wrong, and Mammen's two-point version is the only one of the four that is right in both columns.

Residuals that keep their own variance

A reference distribution for a search has to be generated from a fitted model, and the generator draws residuals. Four ways of drawing them keep four different things — and the one this site has reached for three times repairs nothing at all here.

select · Bootstrap Two constructions on one triangle, and a third that is not. Three resamplings that all keep runs of neighbours, on the same residuals at a block length of 5, with the lags running past ℓ so that the tapers separate. A blocked multiplier never moves a residual; a fixed-length moving block moves every one; and they attenuate identically, worst gap 1.4 standard errors, both sitting on γ_resid(k)(1 − k/ℓ)⁺ and both exactly zero past ℓ — so the attenuation is the block boundary rather than the multiplier. The third is the stationary bootstrap, whose runs are geometric rather than fixed: its taper is γ_resid(k)(1 − 1/ℓ)^k, it agrees with the other two at the first lag and at no other, and at lag 6 it still carries 0.0081 where they carry -0.0005.

The triangle that was not the multiplier's

A resampling that leaves each residual on its own row can keep only what the residuals have, times a triangle. A construction that moves every one of them has the same triangle — and the one in this collection's own table has a different taper entirely.

banded · Bootstrap Each repair is for its own defect, and one is for both. The 95% point of the statistic's own distribution in each world, against the mean 95% point of five reference distributions built from one sample. Where the error variance is a function of the design, the two resamplings that detach a residual from its row fall short and the two multipliers that keep it there do not; where the rows repeat each other it is the other way round. With both defects at once the blocked multiplier — drawn once per run of 5 rows, so the residual never moves and its neighbours share a sign — is the closest of the five, at 2.999 against a truth of 3.803. It is still short by 0.804, and that shortfall is the next figure.

Two defects and one resampling

Four resamplings, each the repair for one defect and wrong about the other. Put both defects in the same world and the statistic's 5% point is 3.8028, where the best of the four reaches 2.8326 — until a multiplier that stays on its own row and shares a sign with its neighbours reaches 2.9988.

proxy · Bootstrap What each construction carries, against what there was. The autocorrelation of a resampled error series at five lags, averaged over 60 samples of 40 resamples each. Three facts are in the picture. The residuals lie below the errors at every lag, which is the ceiling a multiplier cannot exceed. The blocked multiplier and the fixed-length block lie on top of each other below it — they attenuate identically, because the attenuation is the join — while the stationary bootstrap, whose runs are geometric rather than fixed, sits above them both. And the sieve is the exception in kind rather than in degree: at lag six it carries 0.0638 where the residuals have 0.0300 and the multiplier has -0.0011, because a fitted model extrapolates past the lags it was told about and a truncated sample sequence cannot.

Errors generated from a fitted model

The one construction that is not bounded by the residuals, because a model extrapolates past the lags it was told about and a truncated sample sequence cannot. It is nearly exact where the only defect is dependence, and it pays for it where there are two.

banded · Reference A bias against a variance, with the answer in between. How wrong one sample's reference distribution is, split into the two things it is wrong by. Sharing the multiplier over more rows keeps more of the dependence and closes the bias from 1.688 to 0.835; every row it is shared over also removes an independent sign from the 101 the sample started with, and the spread of the resulting quantile rises from 1.307 to 2.172. The distance a practitioner with one sample is actually exposed to is the two together, and it is smallest at ℓ = 5.

How long a block a multiplier shares

Sharing a sign over more rows keeps more of the dependence and leaves fewer independent signs to build a distribution from. The bias falls from 1.6885 to 0.8479 and the spread rises from 1.3073 to 2.1716, and the rejection rate walks straight through its nominal level on the way from 11.3% to 1.3%.

proxy · Reference A fit takes the low frequencies out of what it leaves behind. The autocorrelation of the errors, of the residuals of a fitted benchmark, and of those residuals rescaled by their own leverage. (I − H) removes the component of the errors lying in a column space that is itself slow-moving, so the residuals are less persistent at every lag — by 5.9% at the first and 26.6% by the fourth. The leverage correction is the standard repair for what a fit does to a residual's size; drawn here against what it does to a residual's dependence, it does nothing.

The residuals are not the errors

A fit removes the part of the errors lying in its own column space, and a persistent design's column space is itself slow — so what is left behind is smoother than what went in, at every lag, by an amount that grows with the lag.

effective · Bootstrap What each correction is worth, exactly. Each variance estimate's expectation under a constant error variance, divided by the variance the slope actually has, at 20 rows on an even design, by two routes: the closed form E[eᵢ²] = σ²(1 − hᵢᵢ) carried through each correction's own weight, and the mean of 20000 counted estimates. The maximum leverage here is 0.1857 and the design's fourth-moment share Σu⁴/(Σu²)² is 0.0897, which is the only thing the closed form reads. HC0 comes out at 0.8603 — short by construction, since its factor is exactly 1 − 1/n − Σu⁴/(Σu²)². HC1 reaches 0.9559, HC2 is exactly 1.0000 at every design and every sample size, and HC3 overshoots to 1.1647. On an even design the four are within a fifth of each other and the choice barely matters.

Three corrections and a leverage

On an even design of twenty rows the four robust corrections read 0.8603, 0.9559, 1.0000 and 1.1647 of the truth and the choice barely matters. Add one point at x = 8 and they read 0.3191, 0.3419, 1.0000 and 5.1127.

sandwich · Misspecification The ceiling a multiplier cannot reach past. A wild-type resampling forms e*_t = e_t·w_t with the multiplier independent of the residual, so what comes out has autocovariance γ_resid(k)·γ_w(k) — the residuals' own, multiplied by the multiplier's. Since |γ_w| ≤ 1 the reference distribution's dependence is bounded above by the residuals', and the residuals' is already below the errors'. The two shortfalls compose. For a block of ℓ the multiplier's autocorrelation is exactly the triangle (1 − k/ℓ)⁺, drawn here as the dashed prediction against the realised resamples at ℓ = 5; the bound is attained only at ℓ = n, where the reference distribution is built from one sign.

What a multiplier cannot keep

Two reasons were named for the quarter a blocked resampling falls short, and taking either away makes the gap larger. What is left is a bound — a multiplier can only take dependence out, and the residuals' own is already below the errors'.

effective · Reference

Named alongside it

The objects these essays reach for when they reach for this one.

Reference distributionAutocorrelationBlock bootstrapWild bootstrapDependenceMonte CarloResamplingSpecification searchCritical valueError rateHeteroskedasticityPersistence

All concepts