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

Model selection — where it appears

Choosing which model to fit from the same data the model is then fitted to, which makes the choice part of the procedure. It has no error rate, because nothing is being rejected; what it has is a regret against the best candidate available, which is a different quantity entirely.

Named by 67 essays across 23 fields — each of them below, with the objects they name alongside it.

The profile a break point is chosen from. One sample of 120 rows under a break in the persistence, fitted as two first-order regimes at every admissible break point. The maximum is at row 78, where the true break is at 60. The shaded band is every break point within two log-likelihood units of the best one — 8 of the 73 positions searched, which is 11% of the range. The horizontal line is the one-regime fit the search is compared against; the whole profile is above it, at every position, which is the point: a maximum over 73 candidates is above the null by construction and not by evidence.

A break that was looked for

A two-regime whitening finds its change point by maximising a profile, and then reads a criterion that counts parameters. Under no break there is no parameter to count, because every position describes the same model.

charged · Break point 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 Four dependences a single parameter cannot tell apart. Every law here is standardised to a lag-one autocorrelation of 0.8, so a rule told the errors are a first-order autoregression finds the same number in all four and has no way of seeing what separates them. The geometric decay is the world in which estimating a covariance rather than naming it was priced, and found to cost. The five-period moving average has 0.200 at the fourth lag and exactly nothing past it, where the geometric law says 0.328 at the fifth. Long memory at d = 4/9 is still at 0.576 by the twentieth lag, where the geometric law has reached 0.012. The break has no autocorrelation function at all: what is drawn for it is the average over the pairs at each gap, which is what a stationary estimate converges to.

A dependence with a shape

Four ways for errors to repeat, all with the same first lag and nothing else in common. A rule told the errors are a first-order autoregression finds the same number in all four, and is right about one of them.

general · Dependence Where the general fit becomes the parametric one. The band family's objective at the autoregression's own geometric sequence, cut off at each width, on one sample of 60 rows. The horizontal line is the profile likelihood the parametric fit maximises, written independently through a different whitening. At the full width the two are the same number to 3e-14, which is what says the general construction contains the parametric one rather than resembling it. Below 10 lags there is no line at all: the geometric sequence cut off short is not a covariance matrix, so the objective has nothing to evaluate. Between the two the truncation is briefly above the parametric likelihood — a wrong covariance can fit one sample better than the right one, which is the whole reason a width has to be charged for rather than chosen.

A family before a fit

A regression's coefficients and one correlation can be maximised together. Replace the correlation with an estimated covariance and there is nothing left for "jointly" to mean — until a set of covariances is named, and the set turns out not to contain the truth.

family · Dependence The correction is not a property of the sample. tr(HΩ)/q for each of fifteen candidates, at ρ = 0.7. Two candidates that fit the same number of coefficients need corrections that differ by as much as 1.49, because one of them is fitting the persistent predictors and the other is not — so no single number can be right for both, and the scalar n/n_eff = 5.537 is above every one of them. The four predictors carry persistences 0.9, 0.6, 0.3, 0; at one persistence for every column the whole spread collapses and a scalar looks exactly as good as the trace.

A penalty is a trace

Akaike's 2q is not a count of coefficients. It is the answer a trace collapses to when the rows are independent — and once they are not, the trace is still the right object and is no longer the count.

effective · Order selection One factor moves and the other does not. The two factors of the same average, each drawn against its own largest value so that they share an axis. The rate at which the fifteen candidates disagree about the tuning parameter rises from 28.6% at 4 values on the list to 43.3% at 8, a factor of 1.52. What a disagreement costs, given that there was one, is 0.00975 ± 0.00224 and 0.00848 ± 0.00113 at the same two points — 0.5 standard errors apart, and the paired comparison on the draws that disagree under both lists puts it the other way. The guess this field was written to test was that a longer list makes disagreements commoner and each one smaller. The first half is right and there is no second half.

A rate times a size

A sweep reported what it costs to let every candidate choose its own tuning parameter and found it flat across the list. It was reporting a product, and the two things multiplied together do not behave the same way at all.

apiece · Order selection What each variant loses before anything has been searched for. The mean loss differential of each of the eight variants against the benchmark, over 600 tables of 60 origins, with every fit given 71 rows. The series is an AR(1) and every variant adds a lag whose coefficient is zero, so in population the two forecasts are the same forecast and the difference drawn here is estimation noise and nothing else. The marked line is σ²(q₁ − q₀)/n = -0.01408, which is an expression in how many coefficients each model has and how many rows it was fitted on — it knows nothing about the series, the persistence or which lag the variant added, and every bar is within a fifth of it. This is the amount a reference distribution recentred at each column's own sample mean believes the candidates are already behind by.

A table of nested models

A benchmark and eight variants of it, each adding one thing. Every variant is behind before the search begins, by an amount that can be written down before the data exists — and the two most natural ways of reading the table are wrong in opposite directions.

search · Forecast The area under the window is what the band actually costs. The three windows' weight sequences at a width of 30 lags, drawn against the lag as a share of the window. A truncated window applies a weight of one to every lag inside it and zero outside, which is why its sum is the width and why every conventional charge is right for it — and it is a covariance matrix on almost no sample, so it cannot be used. The Bartlett window falls linearly to zero and its weights sum to exactly 15.000000000000004, which is half the width, at every width: Σ(1 − k/(L+1)) over k = 1 … L is L − L/2. The Parzen window sums to 11.13 here, three eighths of the width, and it gets there by holding a weight near one over the first few lags and then falling faster. A plug-in estimate multiplied by a weight below one is a shrunk estimate, and a shrunk estimate is worth less than a free one — which is the whole of why a charge levied per lag is a charge for parameters the window has already spent.

The charge nobody derived

A band of lags is charged one log-likelihood unit apiece, because that is what a regression coefficient costs. A band's numbers are not regression coefficients, and measuring what they actually cost puts the convention out by a factor of nearly three.

dimension · Criterion The number the comparison was missing. What it costs to choose the tuning parameter for every candidate separately rather than once for the table, under AR(1) at 0.8, paired on the draw. The window's figure is the one the earlier field reported; the order's is the one it named and did not make. They are the same size — 0.00401 against 0.00360, at 2.30 and 1.72 paired standard errors — and matching the lists at eight values leaves them the same size again. The prediction that the longer list would make the order's cost the larger of the two is not what happens; what happens is that the two rules cost the same once they are scored by the same criterion, which took a missing term to arrange.

The comparison that was not made

Choosing a whitening's window separately for every candidate costs 0.00401 of regret. The same question about an order was named and left, because the two lists are different lengths. The order's answer is 0.00360, and matching the lists changes almost nothing.

lists · Order selection The eighth was not a constant. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations between the candidates, over 800 draws apiece. The earlier field reports this flat at about an eighth across list length, on a table and a world it never varies. Vary how much the omitted coefficients are worth — one multiplier, with the table, the list, the law and the sample size all held — and it runs from 17.6% to 1.5%, a factor of 11.75. The world in which every candidate is true is the world in which the tuning list decides most; the world in which one candidate dominates is the world in which it decides nothing.

The eighth that was not a constant

How often a per-candidate tuning list changes which candidate wins is reported flat at about an eighth across list length. Vary how far apart the candidates are instead and it runs from 17.6% to 1.5%.

turnover · Order selection The curvature is in the denominator. The optimism a Bartlett band of each width actually costs, divided by that width, on two ways of measuring the width, over 2000 draws at 120 rows. Measured in the weights the band spends — Σ w(k), which is what the earlier field levies its charges on — the reading falls from 0.9528 at two lags to 0.7486 at thirty, so a charge proportional to the summed weights is too dear at one end and too cheap at the other. Measured in the pairs the band uses — Σ w(k)(1 − k/n), because a lag of k is an average over n − k products — the same readings are flat from 4 lags up, at 0.0084 of χ² per width against 0.2359. The correction has no fitted parameter in it: it is a function of the window, the width and the sample size.

The width a band is measured in

A tapered covariance band spends 84% of its own weights at two lags and 74% at thirty. Every charge in the collection is a straight line through the origin in those weights, so it is too dear at one end and too cheap at the other.

curve · 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 sequence 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 The curvature is in the denominator. The optimism a Bartlett band of each width actually costs, divided by that width, on two ways of measuring the width, over 2000 draws at 120 rows. Measured in the weights the band spends — Σ w(k), which is what the earlier field levies its charges on — the reading falls from 0.9528 at two lags to 0.7486 at thirty, so a charge proportional to the summed weights is too dear at one end and too cheap at the other. Measured in the pairs the band uses — Σ w(k)(1 − k/n), because a lag of k is an average over n − k products — the same readings are flat from 4 lags up, at 0.0084 of χ² per width against 0.2359. The correction has no fitted parameter in it: it is a function of the window, the width and the sample size.

A lag the sample has less of

A sample autocovariance at lag k is an average over n − k products, not n. Count a band's width in the pairs it actually has and the curvature in its charge goes away, on a correction with nothing fitted in it.

curve · Criterion What a longer block buys and what it costs. A trapezoidal block at 120 rows, with the error split into the two things it is made of. The bias falls with the block length, because a longer block attenuates less, and it flattens at 23.2% because the sample's own autocovariances are short whatever window is applied to them. The spread rises with it, because a longer block means fewer of them. Their sum in quadrature has a minimum at ℓ = 16, which is not where either of the two has one. The faint line is the rectangle's total error, for scale: it is above the trapezoid's from ℓ = 12 onwards.

Bias is not the whole of it

A window that reaches zero at its ends attenuates less and uses less of each block. The block length that minimises its bias is not the one that minimises its error, and comparing two windows at one length compares one of them mis-tuned.

crossing · Bootstrap The one candidate an effective sample size is right about. n/n_eff with the finite-sample inflation Σ(1 − |k|/n)ρ^|k| is not an approximation to tr(HΩ) for a fit with only an intercept — it is that trace, to machine precision, because the hat matrix of a constant column is 1/n everywhere and its trace against Ω is the mean of Ω. The quoted limit form n(1 − ρ)/(1 + ρ) is not even right about that one. And the average correction the table's fifteen candidates actually need is 3.318 per parameter, well below the scalar, so applying it to all of them over-charges every one.

One number for a table of candidates

An effective sample size is a real quantity, it is exactly right about one thing, and that thing is a mean. Substituted into Akaike's criterion it changes nothing at all, because the penalty it is meant to fix has no sample size in it.

effective · Dependence Every candidate is behind by what its parameter count says. Each dot is one of the fifteen subsets of four predictors, fitted on a rolling window of 80 rows and scored against the benchmark out of sample over 60 origins, at a null where every one of them contains the truth. The line is σ²(q₀/(R − q₀ − 1) − q/(R − q − 1)), which is arithmetic on two integers and a window length. Most of these pairs are not nested — a subset of two predictors and a different subset of two share neither model — and the closed form does not care: the displacement is a statement about how many coefficients each side estimates. The candidates of the benchmark's own dimension sit at zero.

The displacement is a parameter count

A nested variant is behind its benchmark out of sample before anything is searched for. The closed form for how far turns out to have nothing about nesting in it — only two integers and a window length — and it prices a table where no candidate contains any other.

select · Multiplicity 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 What a disagreement costs, split on whether it decided anything. The regret from choosing the tuning parameter per candidate, on the draws where the candidates disagreed, split on whether the disagreement changed which candidate the table selects. Over 1200 draws at each list length: when the winner changes the regret is 0.02215, 0.03029, 0.03145; when it does not it is -0.00243, -0.00069, -0.00065 — negative, and small enough that it is inside two standard errors of nothing at every length. The whole of the cost lives in the first column, and the second column is not merely small but slightly the wrong sign: when the table's answer is unaffected, letting each candidate use its own window is a very slightly better rule than making them share one. So a disagreement about the tuning parameter is not a cost. A disagreement that changes the winner is.

The quarrel that changes the winner

A disagreement about the tuning parameter costs 0.031 when it changes which candidate the table selects and −0.0007 when it does not. The distance between the values disagreed about has nothing to do with it.

apiece · Order selection The term that cancels, and the term that does not. The volume each candidate's whitening moves — log|Ω̂| — for a sieve of order 4 on one sample of 120 rows. Estimated once from the fullest candidate and used for the whole table, it is the same number for every candidate, so it drops out of every difference the criterion reads: that is why nothing in this collection has ever needed to carry it. Estimated from each candidate's own residuals it ranges over 23.87, which is more than a parameter is worth, and the criteria being compared are then fits made under different error models with no term saying so. The window's rule has carried this term since the estimated-covariance field and the sieve's never had it.

The volume a whitening moves

A sieve's whitening has a determinant and this collection's criterion for it never carried one. Shared across a table the term cancels exactly, which is why nothing ever noticed; used per candidate it is worth more than a parameter and the whole comparison turns on it.

lists · Criterion A window with two wrong ends. The regret of a rule whitened by a Bartlett-tapered Ω̂, as the window widens, against three rules that need no window at all. At L = 0 the estimate is the identity and the rule is exactly least squares — 0.08556, the same number to five places. It falls to 0.01754 at L = 20 and rises again by L = 30, because a quarter of the sample's lags are then being estimated from it. The automatic bandwidth a practitioner would reach for, 4(n/100) to the power 2/9, which at this sample size is 4, gives 0.02963 — 69% above the best available window. The rule told the dependence is an AR(1) sits at 0.01440 throughout, which is the price of not knowing the form.

The window that has to be chosen, and the term that was dropped

An estimated covariance has a bandwidth in it, and both ends of the dial are wrong for different reasons. The rule a practitioner would reach for is two thirds worse than the best window there is.

banded · Order selection Two factors, opposite directions. The two factors the cost of a per-candidate tuning parameter is a product of, as the candidates are pulled apart, over 800 draws at each of 5 separations. How often the candidates disagree about the tuning parameter rises from 31.8% to 88.8%; the share of those disagreements that change which candidate the table selects falls from 51.6% to 1.7%. So the setting where the candidates quarrel most about the tuning parameter is the setting where the quarrel matters least, and a sweep that reads the rate and stops has read the factor pointing the wrong way.

Two factors pointing opposite ways

As the candidates on a table are pulled apart, they quarrel about the tuning parameter three times as often and the quarrel decides the winner thirty times less often. A sweep that reads the first factor has read the one pointing the wrong way.

turnover · Order selection What a search manufactures, law by law. The average likelihood ratio a search over 120 rows reports, on each of the four laws, over 400 draws. Three of them have no break at all and report 5.080, 4.839 and 4.748; the fourth has one and reports 8.757, so the real break is worth only 3.677 beyond what the search would have found anyway. The dashed line is 2, which is what an information criterion charges for one extra parameter. A search costs about two and a half of them, and the number is a measurement rather than a count.

What a search costs in parameters

An information criterion's penalty is an estimate of the optimism a fit carries. For a break point the optimism can be measured and cannot be counted, and it comes to about two and a half parameters.

charged · Break point A test between nested models, under a null that is true. 1000 comparisons: an AR(1) truth, forecast by a fitted AR(1) and by a fitted AR4 whose extra coefficients are zero. In population the two forecasts are identical, so every rejection is false. The larger model's mean squared error is 1.1663 against 1.0583 — worse, by exactly the noise in estimating coefficients that are not there — and the ordinary test therefore declares the smaller model significantly better 67.2% of the time. Read one-sided in the direction anybody asks about, it finds the larger model better 0.0% of the time. Adding the squared difference between the two forecasts back into the loss differential puts the level at 4.9%.

When one model contains the other

The comparison a forecaster most often wants is between a model and the same model with one more term. That is exactly the comparison the standard test cannot make — and it fails by declaring the smaller model significantly better, more confidently the more data it is given.

evaluation · Forecast Generality in the wrong direction buys nothing. Regret on a sample whose persistence changes from 0.95 to 0.65 at row 60, over 200 draws. The three stationary rules — told one number, told a window, told an order — are within 0.4 standard errors of each other, and all three stop in the same place: they are general in the lag direction, and the departure is in the other one. Letting the model change once, at a point estimated from the same residuals, is worth 0.05021 more at 4.5 paired standard errors — about as much again as the whole of the first repair. Being told where the break is adds 0.01926, and being told the entire covariance adds 0.02465.

Where the generality runs out

A covariance that changes half way through a sample is not one a window can estimate. One number, a window and an order are worth the same as each other on it — and letting the model change once, at a point nobody can locate, is worth as much again as all three.

general · Dependence The crossing is in the dependence, not in the split. Regret of each rule as the design and the errors are made persistent at the same coefficient, scored on fresh rows because the closed form assumes exactly what is being taken away. An optimism theorem counts rows; when the rows repeat each other there are fewer of them than there are rows, the penalty is too small for the fit it is correcting, and the criterion starts buying coefficients it should not — its average winner grows from 3.31 coefficients to 3.90. The hold-out never used the theorem and overtakes at ρ ≈ 0.81. Schwarz's criterion, worst of the three on independent rows, is best on repeating ones — its heavier penalty is right for the wrong reason.

Where the two searches cross

The obvious dial between a criterion and a hold-out is how much of the sample to hold out, and moving it never changes the answer. The dial that does is one nobody chooses — how much each row repeats the one before it — and the two rules change places at about 0.81.

proxy · Forecast What a search costs is not a property of that search. The likelihood ratio a searched break in the regression reports, two ways, on every law. On its own — the whole rule being a split of the sample, at no whitening — it averages 34.70 under AR(1) at 0.8, against the 11.07 a chi-square on the five coefficients a split adds would use as a threshold. Inside a rule that also chooses a window from a list of eight, the same search adds only 15.40 — less than half. Most of what a break search finds under correlated errors is the correlation, and a whitening chosen from the same sample has taken it already. A charge measured for one search, carried into a rule that makes two, is not conservative in some harmless direction: it is measuring a different quantity.

A charge that depends on the rule

The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.

twice · Break point A line in the right width beats two curves. How far each candidate charge sits from the measured optimism across the plateau, in units of each width's own standard error, over 2000 draws. The straight line through the origin in the band's summed weights — which is what the earlier field levies — misses by 0.2382 per width. The same straight line in the pairs the band actually uses, Σ w(k)(1 − k/n), misses by 0.0095. A fitted power law misses by 0.0293 and a fitted decaying rate by 0.0172, both on one fitted constant more. The deferral this field answers asked for a curve; the answer is a line, in a variable with nothing fitted in it.

A line that beats two curves

A deferral asked for a curve. Fitted against the same measurements, a straight line in a variable nobody had to fit describes the plateau better than either curve does with a constant more — and for three windows out of four it does not.

curve · Criterion What a longer list actually changes. How often the fifteen candidates choose different tuning parameters, at a true null where every one of them contains the truth, so a disagreement is manufactured rather than discovered. The order's list is an interval of integers, and thinning it moves the rate smoothly from 0% at two values to 39% at thirteen. The window's is not an interval — it runs 0, 1, 2, 4, 8, 12, 20, 30 — so a thinned window list jumps depending on whether it happens to keep the width the criterion wants, between 0% and 42% with no order to it. So "the same length" was never quite the same thing for the two rules, and it is a smaller effect than the field it was invoked to explain.

A list is not a rule

How often fifteen candidates disagree about a tuning parameter runs from nothing at two values on the list to two draws in five at thirteen. What the disagreement costs does not move at all.

lists · Order selection Flat along a row, apart between them. The probability that a per-candidate tuning list changes the winner, at three list lengths on three candidate tables, over 800 draws in each of the nine cells. Along a row — the reading the earlier field takes — it moves by a factor of at most 1.21, so that field's invariant survives on every table. Down a column it moves by up to 1.98. The list length is the dial that does not move this number and the table is one that does, and the earlier field varied only the first.

A table and a list

A nested ladder of candidates differing by one coefficient was predicted to turn over more often at every list length. It turns over less at every one, and its list changes the winner half as often.

turnover · Order selection Cut the charge and the width follows it. The band width each charge picks, averaged over 400 draws of 120 rows under AR(1) at 0.8, with the standard deviation across draws beside it. Schwarz's charge — half a log n a lag, which is 2.39 here — picks 3.67. Akaike's picks 6.02. Charging the numbers the window actually leaves free, which is half the width, picks 10.12; charging what the optimism measures, 0.767 of that, picks 14.15. A charge and the width it buys are very nearly reciprocal, which is what a likelihood rising at a fixed rate a lag implies and is why the four answers span a factor of 3.86. The width that was actually best on the draw averages 13.90 and moves by 10.30 from draw to draw — three times as much as any rule's answer does.

A width that moves and an error that does not

Four charges give four widths a factor of four apart and four errors half a per cent apart. The derived charge wins, significantly, by a quarter of what was on offer — and none of the four is an estimate of anything.

dimension · Criterion What each criterion selects, at 50 observations. 700 series from an AR(2) with coefficients 0.6 and -0.3, every order from 0 to 8 fitted to the same 42 responses so the log-likelihoods are comparable. AIC finds the true order 55.1% of the time and lands above it 25.7%; BIC finds it 54.1% and lands above it 4.0%. The closed form for one extra lag is P(χ²₁ > 2) = 15.73% for AIC, which does not depend on n at all, and P(χ²₁ > ln n) = 4.79% for BIC at this size, which falls to zero. Under the true order is the other failure and it is BIC's: 41.9% against 19.1%.

Choosing the order

One criterion is consistent and one is not, which is the whole of what gets said about them. At two hundred observations the consistent one is right 95% of the time and the other 70%; at fifty they are both right 54% of the time and wrong in opposite directions, and consistency has not started to mean anything yet.

forecast · Order selection How often the split is taken, and by which rule. Over 400 draws on each of five laws. The first two rows have no break in them at all, the last two have one at row 60, and the middle one is a moving average. A criterion that counts a fitted two-regime model's parameters and nothing else takes the split on 99% of draws where there is no break. Counting the break point as one more parameter brings that to 67%. Charging what the search actually manufactures — 5.16 units, measured on a law with no break — brings it to 16%, and still takes the split on 61% of draws where there is one.

Choosing whether to break

Charging what the search manufactures takes a rule from splitting a stationary sample on 99% of draws to 16%. It also costs regret, because the two mistakes a rule can make are not the same size.

charged · Break point The quantity that does not depend on the list. The probability that letting each candidate choose its own tuning parameter changes which candidate the table selects — the product of the two moving shares — against the length of the list, over 1200 draws apiece. It is 14.2%, 11.9%, 12.3%: a spread of 2.2% across a list length that moves the disagreement rate by a factor of 1.52. This is the invariant the whole field turns on. Everything downstream of the winner — the coefficients, the regret, whatever a reader is going to quote — is a function of whether the winner changed, and how often that happens is not something the list controls. A longer list changes how often the candidates quarrel and not how often the quarrel matters.

How often it matters

The disagreement rate rises by half across the list and the share of disagreements that decide anything falls by nearly the same factor. Their product — how often the tuning list changes which candidate wins — sits at an eighth and does not move.

apiece · Order selection A wider band is always a better fit. The likelihood maximised over the band, at five widths, averaged over 30 samples. A band at L lags is a band at L + 1 with the last entry held at zero, so the families are nested and the maximised likelihood cannot fall — it does not, on any draw. What it does is rise at 0.984 of log-likelihood a lag. A parameter that is doing nothing buys half a unit in expectation and Akaike's criterion charges one, so this is a criterion very nearly indifferent between every width on offer. The dashed line is what a charge of one unit a lag would exactly cancel. Nothing in the fit chooses a width, and what does choose one is a charge somebody has to pick.

Nothing in the fit picks the width

A wider band is always a better fit, and it is better by about one unit of log-likelihood a lag — which is the order of what a criterion charges for a parameter. Three defensible rules choose widths a factor of three apart.

family · Order selection R² against the number of useless predictors, n = 30. The response is pure noise and so is every predictor, so the true relationship is nothing at all. R² rises from 0.000 to 0.648 anyway, following k/(n − 1) — which is what a criterion that rewards higher R² is actually rewarding.

R² is not a measure of fit

Adding a predictor with no relationship to anything cannot reduce R², and in expectation raises it by 1/(n − 1). Twenty useless predictors on thirty points give an R² of 0.69 from pure noise.

regression · Summary The row count entered twice, and a penalty is one place. Regret against the best available model as the errors are made persistent. Counting rows more than quadruples; both penalty repairs — the trace, and the scalar effective sample size — are worse than it at every persistence measured; and whitening the sample and keeping the ordinary penalty falls, recovering 86.9% of what counting rows gives up at ρ = 0.85. Doing it at an estimated ρ recovers 80.6%, so having to estimate the dependence from the rows being selected on costs 7.3% of what knowing it is worth. Mallows' forms are drawn beside the logarithmic ones and behave the same, which is what rules the linearisation out.

The repair that was exact and made it worse

A penalty computed from the trace is exactly the optimism it estimates, and selecting with it gives up a fifth more than not correcting anything. The row count entered the criterion twice, and a penalty is the second place.

effective · Forecast The ranking on the left, the weights on the right. Eight moving-average forecasts of an AR(1) at φ = 0.4895, the persistence at which the best of them exactly ties the 60-observation benchmark. On the left, each candidate's expected squared error in units of the series' own variance: the smallest belongs to L = 2, at 1.0156. On the right, the weight each carries in the variance-minimising combination of all eight — and the best of them carries 0.00000. The two ends of the family carry 1.0172 of the weight between them, and the combination they make is worth 0.7817, which is 23.0% below the best single forecast. Both columns are closed forms in φ. Which forecast to keep and which forecasts to use are different questions, and this is a set where the answers share nothing.

The weight that is a vector

Two forecasts have a best combination and one number describes it. Eight have a best combination too, and the vector describing it puts nothing at all on the forecast with the smallest mean squared error.

search · Rank The window a whitening wants is not the memory of the errors. Regret under a five-period moving average as the tapered estimate is given more lags, over 120 draws at n = 120. The best window is L = 30; the automatic bandwidth is 4 and the error model's own likelihood chooses 9.7 on average. Both land in the same place and both are short, and the reason is the taper: a Bartlett weight at lag k is 1 − k/(L + 1), so a window of 8 keeps 0.556 of whatever the fourth lag carries and a window of 30 keeps 0.871. A window has to be several times the memory before it stops removing the memory. The dashed line is the rule told the errors are a first-order autoregression, which needs no window at all.

The window a whitening wants

Every law here is best whitened by a window several times longer than its own memory, including the one whose memory ends at the fourth lag. The three ways of choosing it from the sample all land in the same place, and it is the wrong one.

general · Order selection What a second break adds. Over 200 draws, the likelihood ratio a search over one break point reports, and how much more a search over an ordered pair adds on top of it. Under AR(1) at 0.8, which has no break at all, the first search manufactures 5.697 and the second adds 4.278. Under a law with exactly one break — where a second one is as absent as the first was in the row above — the first search reports 9.442 and the second still adds 5.800. Searching for something that is not there costs the same whether or not something else was there to find.

A second break on a flat profile

Searching a hundred and twenty rows for one change point manufactures five units of likelihood. Searching for a second manufactures four more, on a series that has at most one — and on a profile whose whole range is under seven.

charged · Break point One window for the table, or one each. The regret of the same fifteen-candidate table under AR(1) at 0.8 over 150 draws, with the window attached three ways. Chosen once from the fullest candidate's residuals it gives up 0.02518. Chosen from each candidate's own residuals, with the covariance estimate still shared, it gives up 0.02799 — a paired cost of 0.00281 at 2.0 standard errors for the tuning parameter alone. Estimating the covariance per candidate as well costs 0.01087, so the objection already on record is about 3.9 times the size of the one that was not.

A window for every candidate

The window and the order a whitening needs are chosen once, from the fullest candidate, on an argument that was made about an estimated covariance. A tuning parameter is not a covariance, and the two cost different amounts.

together · Order selection What the interval covers once the order is chosen as well. 1200 series of 40 observations from an AR(1) with φ = 0.7, at each horizon, on one set of seeds. The upper line is the interval computed at the true parameters — it covers 95.5% on average, which is the check that σ²Σψ² is the right formula rather than a claim about anything a forecaster can do. The lower line is the same formula fed σ̂² and φ̂: 93.6% at one step and 90.8% at 6. The third line chooses the order by AIC from the same data before computing the interval, which costs a further 0.8 points at h = 6.

The interval after the choice

Estimating the coefficients of a known model costs a 95% forecast interval about two points of coverage. Choosing which coefficients to estimate, from the same forty observations, costs another four and a half — so the step nobody records in the output is the more expensive of the two.

forecast · Order selection Four sequences, and the rule only ever sees the last one. Under long memory at d = 4/9, four things that are all called the dependence. The law itself is the top line. What a sample of 120 rows reports on average is the second, computed exactly: subtracting a sample mean takes the first lag from 0.800 to 0.538. What a candidate's residuals report is the third, lower again at 0.472, because a fit removes dependence along with signal. The autoregressions are fitted to that third sequence and reproduce it exactly out to their own order — the Yule–Walker equations are solved to make it so — so everything they say past that is extrapolation. At the twentieth lag the law has 0.576, the residuals report 0.006, and an AR(8) extrapolates 0.028.

The order the tail is drawn at

A fitted autoregression reproduces the sample exactly at the lags it was fitted on, so everything it says past them is extrapolation — and the order is the dial that decides how much of it there is.

general · Order selection The scale moves the width; the curve does not. The band width each charge picks, averaged over 150 draws of 120 rows. The two conventions — a unit a lag and half a log n a lag — pick 6.08 and 3.65 lags. The four charges derived from the measured optimism pick 15.05, 15.60, 15.47 and 15.44, against a best width on the draw of 14.13. So the scale a charge is levied on moves the width by a factor of 4.27 and the shape of the charge moves it by 3.6%. None of the six is an estimate of the draw's own best width: the correlations are -0.006, -0.003, 0.017, -0.001, 0.016, -0.004.

What a better charge buys

Four charges derived from the same measurements pick band widths within six per cent of each other and deliver errors within two per cent of the gap any of them leaves. The scale a charge is levied on decides the width; the shape of the charge decides nothing.

curve · Criterion Fitting them together is worth something under one law. Four fits of the same regression under four dependences: least squares, the two-step plug-in every whitened rule in this collection runs, the coefficients and the band maximised together, and a whitening at the law's own covariance that nobody has. Under the moving average — the one law the band family contains — the joint fit beats the two-step by 0.0077 at 3.3 paired standard errors. Under the autoregression, long memory and the break it is a tie: 0.4, 1.0, 0.3 standard errors. That is the same ordering the likelihood gap gave, arrived at through the coefficients rather than through the objective.

What fitting them together buys

Maximising over the coefficients and the covariance together beats the two-step under one of four dependences and ties under the other three. It is the one the band family contains, and the likelihood said so before any coefficient was compared.

family · Dependence The same dial, on a list that steps by one. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations, for both tuning parameters at a matched list length of 8, with errors from a first-order autoregression. The sieve order runs from 13.1% to 2.6%, a factor of 5.00; the whitening window, from 16.4% to 1.5%, a factor of 11.75. What is held is the number of options, the table, the law, the sample size and the seeds; what cannot be held is the size of a step, since an integer step and a geometric step are different amounts of change. The dial moves both, and it moves them by 2.35 times as much on one as on the other.

A step that is not a ratio

Run the separation sweep on a tuning list of integers rather than a geometric ladder and the two factors still point opposite ways. The invariant does not survive: along a row of integers the probability moves by 2.163 where along the geometric ladder it moves by 1.208.

turnover · Order selection Every pair of sieve orders. Each cell is a tuning list of exactly two sieve orders, the row's and the column's, and its shade is how often fifteen candidate regressions choose different values from it, at a true null over 240 draws. Every pair holding order zero disagrees on no draw at all, because the criterion never picks zero under this law. Every adjacent pair of positive orders disagrees on between 17% and 25% of draws — 22.1% for one and two, 25.0% for eleven and twelve — and the rate falls as the two orders move apart, to 5.0% for one and twelve. The largest cell is 25.0%.

Two values the criterion cannot tell apart

Fifteen candidates disagree about a whitening's order on two draws in five when the list holds all thirteen orders — and on more than one in five when it holds only one and two. Every adjacent pair of positive orders disagrees about as often as that pair does, and every pair holding an order the criterion never picks disagrees on none. A list's disagreement is set by the pairs of values on it that the criterion cannot tell apart, not by how many values it has, and one candidate's own doubt about its pick predicts all fifteen candidates' disagreement with a rank correlation of 0.96.

lists · Order selection An overlap that depends on where it is read. Each pair's overlap on the doubled log-likelihood scale — the sum of the two searches' own quantiles less the joint search's quantile, as a share of the smaller — at five quantile levels, over 1200 draws under the first-order autoregression. two dictionaries of independent columns: -0.176 at 0.5, -0.000 at 0.75, 0.238 at 0.9, 0.320 at 0.95, 0.538 at 0.99; a break, and an independent column: -1.211 at 0.5, -0.406 at 0.75, -0.029 at 0.9, 0.205 at 0.95, 0.161 at 0.99; two dictionaries of step columns: -0.397 at 0.5, 0.145 at 0.75, 0.438 at 0.9, 0.541 at 0.95, 0.682 at 0.99; a break, and a step column it contains: 1.000 at 0.5, 1.000 at 0.75, 1.000 at 0.9, 1.000 at 0.95, 1.000 at 0.99; a break, and a whitening window: 0.512 at 0.5, 0.640 at 0.75, 0.716 at 0.9, 0.732 at 0.95, 0.754 at 0.99. Every pair but the containment overlaps more at the 95% point than at the median, the control included.

The overlap at the critical value

Two searches over independent noise columns are additive on average: the joint search finds what the two find separately, to within a fortieth of the smaller. At the point a test actually uses — the joint statistic's 95% quantile — the same pair overlaps by 0.320, and a rule that adds the two searches' own critical values rejects 1.4% of true nulls instead of 5%. Nothing about the searches causes it. Quantiles of a sum do not add, and on every pair in the field the overlap at the critical value is larger than the overlap the field measured.

apart · Criterion Which factor the sample settles. The field's decomposition at the full tuning list, by sample size, over 1,200 draws at each: how often fifteen candidates choose different values, what share of those disagreements change which candidate the table selects, and the product — how often the list changes the winner. The window: 120 rows 43.3%, 28.5%, 12.33%; 240 rows 17.5%, 14.8%, 2.58%; 480 rows 38.3%, 2.6%, 1.00%; 960 rows 11.6%, 2.2%, 0.25%. The order: 120 rows 48.9%, 21.6%, 10.58%; 240 rows 47.5%, 13.2%, 6.25%; 480 rows 46.1%, 5.1%, 2.33%.

What a longer sample settles

At a hundred and twenty rows, letting each candidate choose its own whitening changes which of fifteen candidates wins on about an eighth of draws, whatever the list, the table or the law. At four hundred and eighty rows it changes the winner on 1.0% of draws for a window and 2.3% for an order, and at nine hundred and sixty on a quarter of one per cent. The sample settles it through one factor only: how often the candidates disagree about the tuning value hardly moves, and the share of their disagreements that change the winner collapses.

apiece · Order selection What a lag of band costs, as the sample grows. The optimism one more lag of a Bartlett band adds, measured by fitting on one sample and scoring on an independent one, 2,400 pairs at each size, with two standard errors: 0.3745 ± 0.0176 at 120 rows, 0.3842 ± 0.0266 at 240 rows, 0.4523 ± 0.0392 at 480 rows, 0.4022 ± 0.0563 at 960 rows. The window's summed weights give one half a lag and the convention charges one.

The charge at four times the sample

A band of lags is charged a full unit apiece and is worth about three eighths of one at a hundred and twenty rows. The field's last essay said the error in that convention was harmless only because the error surface is flat, and that a steeper surface would give it a cost. A longer sample was the obvious place for one. It is not there: from 120 rows to 480 the error's spread across band widths falls by three quarters, and the derived charge's advantage over the convention falls from 5.5 standard errors to 1.7. What the sample does move is the charge itself, from 0.37 of a unit a lag towards the half a unit the window's weights predict.

dimension · Criterion Largest where least is needed. What the pairs correction supplies against what each window's measured profile needs, across this field's plateau, over 2000 draws at 120 rows. Both are stated as the multiplicative rise the charge per unit of width has to take between four lags and thirty. What the correction supplies is arithmetic — (1 − μ(4)/n)/(1 − μ(30)/n), where μ is the mean lag of the weight the band adds — and it runs 1.0795, 1.0580, 1.0456, 1.0539 for the four windows. What the measurement needs runs 1.1076, 1.2928, 1.2296, 1.6550. The two orderings are opposite: the plain Bartlett window has the longest mean lag, so it gets the biggest correction, and the flattest profile, so it needs the smallest. They coincide to 0.9746 of each other, and nowhere else does the correction account for more than 85.0% of the fall.

What the correction assumes

A correction with nothing fitted in it repairs one window of four. The reason is that its size is set by where a window puts its weight and the curvature it must repair is set by something else — and for one window at one sample size the two happen to agree.

curve · Criterion The displacement per parameter, under five predictor laws. The out-of-sample differential one extra parameter costs a correct candidate, at a window of 80 rows, by two routes that share no arithmetic: counted from rolling forecasts over 1,500 draws of the fifteen-candidate table, and implied by the expected leverage of a fresh row over 80,000 draws of the predictors alone. The Gaussian closed form is 0.01386. normal: 0.01382 counted, 0.01386 from the leverage; Student's t, 8: 0.01379 counted, 0.01407 from the leverage; Student's t, 5: 0.01419 counted, 0.01448 from the leverage; Student's t, 3: 0.01751 counted, 0.01780 from the leverage; exponential: 0.01465 counted, 0.01487 from the leverage.

The displacement on a heavy tail

A correct candidate with one more parameter is behind out of sample by a closed-form amount, and the closed form rests on one Gaussian fact: the expected leverage of a fresh row is q/(R − q − 1). On predictors that are not Gaussian it becomes whatever the leverage is, and two routes that share no arithmetic agree on what that is. With a finite fourth moment it barely moves — 1.5% more per parameter at eight degrees of freedom, 4.5% at five, 7.3% on an exponential. At three degrees of freedom it is 28% more, and the estimate of it will not settle: four batches of twenty thousand draws disagree by more than a quarter.

select · Multiplicity What the two-regime residue was made of, persistent first. Under a break in the persistence at row 60 of 120 (0.95 then 0.65), the selection regret of six whitenings over 200 draws: restart, unscaled, residual persistences 0.04043; restart, unscaled, persistences given 0.03488; restart, unscaled, fitted with the line 0.03466; the law's recursion, residual persistences 0.03856; the law's recursion, fitted with the line 0.01640; the law's own covariance 0.01579. The whitening that restarts at the break and does not rescale its two regimes gives up 0.01910 against the oracle even when it is given the true persistences; the law's recursion, which divides each row by its regime's innovation scale and continues across the break, is the oracle when given them and within 0.00061 of it when they are fitted with the regression.

The residue was the weights

A regression whose errors change persistence at a known row was whitened as two autoregressions and still gave up 0.0246 against the law's own covariance. That residue was put down to estimating the two persistences from residuals, and the repair named was to fit them with the line. Fitting them with the line moves the persistent regime's estimate from 0.81 to 0.89 and recovers a fifth of the residue — exactly what being told the true persistences recovers. The rest was the transform: it never divided the two regimes by their own innovation scales, which differ sixfold. The law's own recursion, told the persistences, is the oracle to machine precision; fitted with the line, it estimates the persistent one at 0.945 and comes within 0.0006 of the oracle.

general · Dependence The same dial, on a list that steps by one. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations, for both tuning parameters at a matched list length of 8, with errors from a five-period moving average. The sieve order runs from 29.5% to 5.8%, a factor of 5.13; the whitening window, from 26.3% to 4.8%, a factor of 5.53. What is held is the number of options, the table, the law, the sample size and the seeds; what cannot be held is the size of a step, since an integer step and a geometric step are different amounts of change. The dial moves both, and it moves them by 1.08 times as much on one as on the other.

The same dial under a second law

On a first-order autoregression, separating the candidates moved the chance that a tuning list changes the winner 2.35 times as far on a whitening window as on a sieve order, and the difference was put down to the step: an integer step is worth a third of a geometric one. Rerun the whole sweep with errors from a five-period moving average and the difference is gone — the probability spans a factor of 5.53 on the window and 5.13 on the order, a ratio of 1.08 — and near the null both lists leave about five and a half candidates within reach, where the autoregression left 3.6 and 1.9. The two factors still point opposite ways under both laws. The step's size was a property of the law the lists were tuned on.

turnover · Order selection A penalty that counts independent rows. Regret of five selection rules over fifteen candidates as the design and the errors are made persistent at the same coefficient, 300 draws at each of 7 settings, scored on fresh rows. Akaike's criterion gives up 0.0193, 0.0196, 0.0294, 0.0428, 0.0666, 0.1075, 0.1281; charging each coefficient its own variance inflation, with the persistences read off the sample, gives up 0.0188, 0.0231, 0.0339, 0.0375, 0.0496, 0.0656, 0.0652, and with them given 0.0193, 0.0236, 0.0348, 0.0374, 0.0426, 0.0469, 0.0318. The hold-out gives up 0.0358, 0.0365, 0.0393, 0.0453, 0.0594, 0.0913, 0.1087 and Schwarz's criterion 0.0355, 0.0295, 0.0331, 0.0397, 0.0575, 0.0849, 0.1047.

A penalty that counts independent rows

Akaike's penalty charges a fit two error variances a coefficient because each coefficient is fitted from n independent rows. When the design and the errors both persist, a row is worth less than a row, and the honest charge is twice the coefficient's variance inflation. On a hundred and twenty rows at persistence 0.7 that charge is 34.7 against an optimism counted at 34.5, where Akaike's charges 10. Charged it, the criterion's regret at 0.95 falls from 0.128 to 0.032 with the persistences given and to 0.065 when they are read off the sample — and at 0.5, where the charge is right and Akaike's is half of it, Akaike's still selects better.

proxy · Forecast Where two profile likelihoods put the break. Under a break in the persistence at row 60 of 120, and the same break turned over, over 300 draws: how often each search's estimate lands within twelve rows of the truth, with its average and spread beside it. persistent first · one scale for the whole series: 46.7%, at 74.8 with a spread of 11.8; persistent first · a scale per regime: 94.3%, at 61.8 with a spread of 5.8; persistent last · one scale for the whole series: 49.3%, at 45.3 with a spread of 11.7; persistent last · a scale per regime: 93.0%, at 56.7 with a spread of 5.9.

The break a shared scale hid

The break searches in this field all maximised a likelihood with one innovation variance for the whole series, and on a break from 0.95 to 0.65 they found it badly: an average of 74.8 where the truth is 60, within twelve rows on 47% of draws, from a profile everyone called flat. Give each regime its own innovation variance — one more parameter, and the one the law actually has — and the same search lands at 61.8 with a spread of 5.8, within twelve rows on 94% of draws, reporting four times the likelihood. The break was never barely there; the likelihood was blind to the half of its evidence that is in the scale. Under no break the extra parameter costs about one and a half units of manufactured likelihood.

charged · Break point Past the cap, long memory's surface is flat. The prediction error at every band width from four lags to eighty, above the lowest on the extended list, under long memory at d = 4/9 at three sample sizes, 600 draws each. At 120 rows the floor is at 12 lags and eighty is 0.0039 above it. At 480 the floor is at 30 lags and every width from twenty-four to eighty is within 0.0003 of it. At 960 it is at 48 and thirty to eighty lags are within 0.0000. The dashed line is the list's cap: the surface does not rise behind it, it levels off.

A cap that decides nothing

Every band-width list this field priced stopped at thirty lags, and long memory at a large sample is the one dependence whose best band wants to be wider. At 960 rows it does: the draw's own best width lies past the cap on 48.7% of draws. Lift the cap to eighty and the measured charge follows, from 28 lags to 46 — and the error moves by two ten-thousandths, in the wrong direction and inside its standard error, because the surface behind the cap is flat. The field's last conditional fails where it had its best chance.

dimension · Criterion When a longer sample settles the tuning list. How often the tuning list changes which candidate the table selects, against the sample size, for the whitening window (solid) and the sieve order (dashed), on three tables of candidates, twelve hundred draws at each size. Every subset of four predictors: window 12.33% at 120, 2.58% at 240, 1.00% at 480, 0.25% at 960; order 10.58% at 120, 6.25% at 240, 2.33% at 480. The nested sequence of four: window 5.83% at 120, 1.25% at 240, 0.67% at 480, 0.25% at 960; order 5.83% at 120, 4.00% at 240, 2.25% at 480. Six subsets of two predictors each: window 11.75% at 120, 3.75% at 240, 5.75% at 480, 2.33% at 960; order 9.42% at 120, 6.67% at 240, 4.42% at 480. The nested table starts at half the full table's rate and reaches a small one a sample size sooner; the table whose candidates all have the same dimension is still changing its winner on one draw in forty-three at 960 rows.

The table decides when the sample settles it

Letting each candidate regression choose its own whitening changes which candidate wins on about an eighth of draws at 120 rows, and a longer sample settles it — for fifteen nested-and-crossed subsets, at 0.25% by 960 rows. A nested sequence of four candidates gets there a sample size sooner, from half the rate. Six candidates that all hold two predictors never get there: at 960 rows their tuning list still changes the winner on 2.3% of draws, nine times the full table's, because a disagreement among candidates of one size goes on deciding the winner as often at 960 rows as at 240.

apiece · Order selection What a quarrel between two orders costs. Every two-value list of autoregressive orders from one to twelve, read on the same 2400 draws of a first-order autoregression at a hundred and twenty rows: each cell is the average regret of choosing the order per candidate rather than once for the table, on the draws where the fifteen candidates disagree about it. The cost grows away from the diagonal and stays level along it. Next to the diagonal, a quarrel between neighbouring orders costs 0.00126 for one and two and 0.00088 for eleven and twelve; four orders apart it costs 0.00803 and 0.00802; eleven apart, 0.02436. Every list holding order zero is left off because the candidates never disagree about one.

A quarrel's cost is its width

When fifteen candidate regressions disagree about a whitening's autoregressive order, the disagreement costs in proportion to how far apart the two orders are — 0.0027 of regret for every order of distance, a line that explains 96% of the cost across sixty-six pairs — and not at all by which orders they are: a quarrel between eleven and twelve costs what one between one and two costs, though one of those holds the truth. Fitted on pairs, the line predicts every longer list's cost. The window's quarrels follow no such line.

lists · Order selection The floor a repair has to choose. Regret of the truncated covariance repaired by flooring its eigenvalues, at a persistence of 0.85 on 120 rows, 120 draws, by floor, on a log scale. Window 4: 1.5649 at 0.001, 0.0945 at 0.01, 0.0385 at 0.05, 0.0285 at 0.1, 0.0226 at 0.2, 0.0235 at 0.4, 0.0292 at 0.7; the taper at the same window 0.0275. Window 8: 1.1919 at 0.001, 0.0980 at 0.01, 0.0334 at 0.05, 0.0258 at 0.1, 0.0197 at 0.2, 0.0209 at 0.4, 0.0249 at 0.7; the taper at the same window 0.0178. Window 12: 0.8692 at 0.001, 0.0621 at 0.01, 0.0239 at 0.05, 0.0209 at 0.1, 0.0212 at 0.2, 0.0239 at 0.4, 0.0282 at 0.7; the taper at the same window 0.0150. Window 20: 1.7110 at 0.001, 0.2495 at 0.01, 0.0399 at 0.05, 0.0238 at 0.1, 0.0202 at 0.2, 0.0219 at 0.4, 0.0259 at 0.7; the taper at the same window 0.0135. Least squares gives up 0.0890 and the whitening told the truth 0.0075.

A covariance repaired

A truncated autocovariance sequence fails to be a covariance matrix on most draws of a persistent series, and this line of essays has always scored it on the draws where it survives. The textbook repair is to floor its eigenvalues. At a floor of 0.001 that moves the matrix by four per cent and gives up 1.56 of regret, seventeen times what least squares gives up, because a whitening reads an inverse and an inverse reads the smallest eigenvalues. At a floor of 0.2 it beats the taper at the shortest windows. So the repair is not a repair: it is a second dial.

banded · Dependence Reading the draw changes what is charged, not what is tracked. The correlation between the band width each rule picks and the best band width on the same draw, over 400 draws. The three fixed charges read -0.069, -0.066, 0.012. The three that read the sample read -0.012, -0.019, -0.041. None of the six is distinguishable from nothing. The statistic the first plug-in reads does vary — the draw's own summed squared autocorrelation runs from 2.06 to 10.19 with a mean of 4.36 — so the failure is not that the charge stopped moving. It is that what it moves with carries no information about which width this draw wanted.

A charge that reads the draw

Three charges built to read the sample track the best band width on their own draw at −0.012, −0.019 and −0.041, deliver more error than the fixed rule they are calibrated to, and pick a width half again as variable. The statistic moves; the answer does not.

curve · Criterion What each wrong count costs, 4 steps ahead. Squared forecast error 4 steps ahead at each imposed rank, relative to the correctly specified fit, at 200 observations. With 1 genuine relations, imposing 0 costs 13.3% and imposing 2 costs 4.8%. With 2 genuine relations, imposing 1 costs 15.6% and imposing 3 costs 2.5%. Under-counting is the more expensive mistake in both systems, and it is the one the procedure's level does not bound.

Which mistake about the rank costs

On a system with two relations, imposing none costs 29.2% of squared forecast error and imposing three costs 2.5%. The expensive mistake is under-counting, which is the error the procedure's 5% does not bound — so the guarantee protects the cheap side.

systems · Rank Three charges and two counts. The optimism of an in-sample fit of the widest candidate on 120 rows, in units of its own error variance, over 1500 draws, by persistence. Counted on the 40 rows straight after the sample: 11.1, 21.5, 35.6, 51.5, 65.5, 87.8, 119.1. Counted on rows 600 further on: 9.3, 18.3, 33.0, 50.6, 67.0, 96.1, 159.0. Charged by the long-series inflation: 10.0, 19.3, 34.7, 54.4, 74.3, 114.2, 234.1. Charged by the same inflation summed over the sample's rows: 10.0, 19.2, 34.2, 53.1, 71.8, 108.2, 208.8. Charged by twice the trace of the hat matrix against the errors' correlation on the sample's own design: 10.0, 18.7, 32.0, 47.5, 61.5, 85.2, 132.2.

A charge the sample can afford

The inflation charge for dependent rows runs to twice the counted optimism at persistence 0.95, and the reason offered was that a series of n rows cannot be optimistic about movements slower than n rows. Summing the inflation over n rows does almost nothing: 234 becomes 209, against 119 counted. Evaluating the same variance on the sample's own design gives 132, and moving the fresh rows away from the sample raises the count to 159. So the charge can be made nearly right, and when it is, the regret doubles from 0.032 to 0.061. A selection wants a much larger charge than the optimism, and past about fifty per predictor it is no longer selecting.

proxy · Forecast The excess over the Gaussian displacement, by window. By how much the expected leverage per parameter exceeds the Gaussian closed form, with the fresh row integrated out, at windows 20, 40, 80, 160, 320, 640. normal: -0.90%, -0.61%, -0.04%, -0.06%, -0.02%, 0.00%. t on eight: 4.14%, 2.78%, 1.79%, 0.73%, 0.28%, 0.18%. t on five: 10.70%, 7.50%, 3.93%, 2.01%, 1.26%, 0.70%. t on three: 44.44%, 32.67%, 22.62%, 16.43%, 11.70%, 7.63%. exponential: 27.91%, 14.01%, 7.21%, 3.53%, 1.93%, 0.85%. On normal predictors the closed form is exact from a window of 80.

An excess and its window

The parameter-count displacement is a Gaussian fact, and on heavy-tailed predictors it runs a few per cent over. Swept across windows from 20 to 640 rows, the excess is about the predictors' excess kurtosis divided by the window: a law with a fourth moment of nine on 80 rows is 7.5% over, and on 640 rows under one. At three degrees of freedom it falls only as the root of the window, from 44% to 7.6%. And the instability once blamed on that law was not there: with the fresh row integrated out, the batches agree to a point and a half, and the mean exists.

select · Multiplicity How often a league table's own data choose a bent centre over a straight one. Over 400 tables of a hundred groups each. The quadratic's curvature coefficient is significant at 5% on 5.3% of tables whose relation is straight, 14.5% of tables whose relation is flattening below the median, 29.3% of tables whose relation is steepening below the median; its average AIC weight is 0.374, 0.456, 0.558.

A centre chosen by its own table

A league table that picks its centre's shape from its own data — a line unless a curvature test says otherwise — picks the line on 85.5% of tables whose relation flattens below the median, because the bend lives where the groups are smallest and noisiest. Its bottom ten then flags like the line's: 2.61 innocent small groups a table against 2.19 for a quadratic told the shape. Always fitting the quadratic costs 0.07 of a correctly flagged group when the relation is straight and matches the told answer when it bends, and a flag set by probability carries the choice through a mixture of the two.

borrowed · Shrinkage How often a quantile plot sees a skewed error when the fitted columns were chosen from the data. Forty rows, 3,000 samples a point, read against the band for forty independent normals. With the intercept alone the test flags 82.5%. With 1, 2, 5, 10 columns fixed in advance: 79.9%, 76.1%, 65.4%, 47.0%; with the same number picked by a forward search from twenty candidates unrelated to the outcome: 66.6%, 55.5%, 36.8%, 25.8%.

Residuals of a model the data chose

Five columns picked by a forward search from twenty candidates that have nothing to do with the outcome leave residuals that show a skewed error 36.8% of the time, where five columns fixed in advance show it 65.4% — the power of a fit of 13.4 fixed columns. Against light-tailed errors the chosen fit's plot is blind, 8.5% against 34.5%. Simulating the search on normal errors gives a band that holds 5%, which the ordinary band already nearly did; the level was never what broke. A reader told the final model has five columns is reading a plot with the power of thirteen.

lineup · Qq

Named alongside it

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

Information criterionMonte CarloRegretDependenceWhiteningSelection effectOverfittingAutocorrelationDegrees of freedomBandwidth selectionOptimismTapering

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