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

Covariance matrix — where it appears

The table of variances and covariances of a set of quantities, which for a series is fixed by its autocovariances at each lag. Truncating a sample autocovariance sequence at some lag does not leave one — the result has negative eigenvalues on most draws — and tapering the sequence instead restores the property for any sequence at all.

Named by 21 essays across 9 fields — each of them below, with the objects they name alongside it.

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 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 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 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 Most of the rise is the optimiser's, and under one law it is not. The rise in log-likelihood from the tapered plug-in to the maximum over the same eight-lag band, beside what the same optimiser produces on a sample generated from the plug-in's own covariance — where the family is correctly specified by construction and there is nothing to find. Under AR(1) at 0.8 the raw rise is 5.72 and the manufactured baseline is 4.79, leaving 0.93 at 1.8 standard errors; under long memory the excess is 0.14, at 0.2. Under the moving average it is 11.87 at 19.4 standard errors, on every draw. The taper is a shrinkage, and it costs nothing where the sequence decays smoothly and a great deal where it stops dead.

The plug-in and the maximum

A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.

family · Dependence 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 Four windows, one line, and one that is off it. The optimism measured for each window at a band of 30 lags, against what that window's weights sum to, on 2000 pairs of independent samples of 120 rows. The diagonal is where a window that spent exactly its summed weights would sit. Three of the points are one shape at three levels — the Bartlett window, its square and its cube, whose sums stand in the ratio 6 : 4 : 3 — and they lie on a line through the origin at 0.767 of the diagonal, with 0.033 between the highest and the lowest. Scaling the weights scales the charge by the factor the weights predict, which is what makes the weights the mechanism. The Parzen window has a comparable sum and a different shape, and it sits at 0.871: its weights stay near one over the first few lags, and the first few lags are where the information is. A weight sum treats every lag as equally informative and no sample does.

What a window leaves free

A Bartlett window's weights sum to exactly half its width, which is a candidate for what the band costs. Varying the weights without varying anything else says the weights are the mechanism; varying the shape at the same weight says they are not the arithmetic.

dimension · Criterion 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 One distribution, two effects. Three causal structures fitted to one covariance matrix over a treatment, a covariate and an outcome. Each reproduces it exactly — the largest entry-wise disagreement across all three is 4.4e-16 — so no sample of any size distinguishes them. The regression of the outcome on the treatment and the covariate returns 0.500 in all three, to within 4.4e-16, because that coefficient is a function of the covariance and of nothing else. The effect the three worlds hold is 0.500, 0.848 and 0.848: adjusting is exactly right in the first and off by −0.348 in the other two. The arithmetic cannot see the difference and the difference is the whole question.

The two worlds that look the same

Three causal structures were fitted to one covariance matrix and agree with it to 4.4·10⁻¹⁶. The regression returns 0.5000 under all three; the effect they hold is 0.5000, 0.8481 and 0.8481. What separates structures is a missing edge, and the signature of one is a correlation of exactly zero.

collider · Conditioning 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 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 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 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 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 Six intervals for the white predictor, a five-period moving average. Under a five-period moving average, over 200 regressions of 120 rows with an eight-lag band, how often each 95% interval for the white predictor's coefficient covers it: two-step band, as reported 95.5%; joint fit, as reported 72.0%; joint fit, White's sandwich 71.0%; joint fit, four-lag sandwich 71.0%; joint fit, band maximised again 85.5%; the law's own covariance 92.0%. The bootstrap redraws each regression 24 times from its fitted coefficients and fitted band, maximising the band afresh on every redraw.

An interval that knows the band was maximised

A regression whose coefficients and eight-lag error band are maximised together prints an interval that covers 72% to 83% where 95% was promised, because it reports nearly the precision of a known covariance and delivers less. Two repairs were named. A sandwich standard error read off the whitened residuals repairs nothing — 71% to 83% — because the maximisation has already made those residuals look independent. A parametric bootstrap that redraws from the fitted band and maximises the band again on every redraw repairs about half the gap, to 85.5% to 89%, by pricing the maximisation through repeating it. What it cannot price is that the band it redraws from is the maximised one.

family · Dependence 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 Six intervals, one of them honest by right. How often each rule's reported 95% interval covers the true coefficient under a five-period moving average, over 1000 draws of 120 rows, for the most persistent predictor (its own lag-one correlation 0.9) and the white one. The law's own covariance covers 94.9% and 94.1%. Least squares covers 65.0% on the persistent predictor and 94.4% on the white one; a fitted ρ̂ 93.0% and 96.3%; the two-step band 86.5% and 97.3%. The joint fit covers 73.2% and 71.4% — short on both, and short on the white predictor where every other rule is honest — and counting its eight band coefficients gives 76.1% and 73.5%.

The interval the joint fit reports

Maximising a regression's coefficients and an eight-lag covariance band together gives a better estimate than plugging the band in. It also prints an interval that covers 71% to 85% of the time where 95% was promised — on the white predictor too, where every other rule is honest — because it reports almost the precision of the law's own covariance and delivers less than half of that advantage. Counting the eight lags it estimated buys one to two points.

timeseries · Dependence

Named alongside it

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

Model selectionNuisance parameterWhiteningGeneralised least squaresInformation criterionTaperingAutocorrelationDegrees of freedomDependenceRegretLong memoryBandwidth

All concepts