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

Normalisation — where it appears

Dividing a quantity by a chosen reference so that what is left is comparable across settings. The choice is not innocent: a ratio whose denominator moves with the thing being measured hides the movement, which is how a curvature in one variable can be an artefact of the variable it is read against.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 The same data, one regression per choice of left-hand side. The two-step procedure has to put one series on the left, and with 3 series there are 3 ways to do it. Each returns a relation and a residual test; the 5% point is -3.71, simulated. Here they do not agree: 2 of 3 reject, and the relations they report are written with a 1 in the position of whichever series was on the left, so they can be compared. Nothing in a printed output records which regression was run.

Which series goes on the left

The two-step procedure has to pick a series to regress the others on, and nothing in its output records which. With a pair that choice never changes the verdict. With three series and one relation between them, the three choices disagree about whether the system is cointegrated at all 98.0% of the time.

systems · Rank 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 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 One of these converges and the other does not. Two measurements on the same fits, against the sample length, for a system with 2 genuine relations. The distance from the fitted plane to the true plane falls from 0.1438 at 100 observations to 0.0075 at 1600 — halving with each doubling, which is the 1/n rate this field's estimates converge at. The angle between the leading fitted relation and the leading generating one reads 29.6° and 29.0° at those same lengths, and is flat in between. The plane is an estimate; the relation inside it is not.

A space is not a relation

The fitted plane approaches the true one at rate 1/n — 0.1438 at a hundred observations and 0.0075 at sixteen hundred. The angle between the leading fitted relation and the leading generating one reads 29.6° and 29.0° at those same lengths, and never moves.

systems · Rank How often a test of a supplied cointegrating relation rejects it, by how wrong it is and how long the sample is. 800 systems a point. At 100, 200, 400, 800 observations the right relation is rejected 9.4%, 5.4%, 5.4%, 6.4% of the time; one with c = 0.9, 19.9%, 37.8%, 78.5%, 98.5%; c = 1.1, 16.8%, 37.9%, 76.8%, 97.5%.

A restriction the data can check

A rank-two system of three series is identified by exactly two restrictions on its relations, and any two that describe the estimated plane fit it equally well — their forecasts agree with the unrestricted fit's to the last digit, so the data can neither confirm nor refuse them. Supply one relation in full instead, one restriction more than identification needs, and the surplus becomes a test: a relation a tenth wrong is rejected 37.8% of the time at two hundred observations and 98.5% at eight hundred, the error found half the time halving as the sample doubles. A right relation buys the forecast almost nothing; a fifth wrong costs 13% of it at two hundred observations and 25% at eight hundred.

systems · Rank

Named alongside it

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

Closed formEstimation errorMonte CarloBandwidth selectionCointegrationDegrees of freedomDependenceInformation criterionModel selectionOptimismReduced-rank regressionTapering

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