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

Half-life — where it appears

How long a disturbance takes to decay to half its size, which is a persistence stated as a length of time rather than as a coefficient. It is the form in which the persistence of a series can be compared directly with the length of the data.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The damage and the warning, against the same dial. Two readings at each persistence. In the darker colour, how often a regression between two independent series of 200 steps is called significant at 5%: 4.9% at φ = 0, 34.2% at 0.8, 52.4% at 0.9, 83.4% at a unit root. In the lighter, how often the standard unit-root test refuses a unit root on one of those series — the chance the analyst is told the series is stationary and may be regressed: 87.2% at φ = 0.9 and 31.9% at 0.95. At φ = 0.9 both are high at once, which is a correct diagnostic licensing a regression that is wrong half the time.

The cliff that is a slope

A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.

timeseries · Spurious How slow a return each test can see at 200 observations: the relation estimated against the relation supplied. The estimated relation's test reaches 80% power up to a half-life of 5.05 steps and the supplied relation's up to 6.88. At a half-life of 8 the two find the pair 36.3% and 71.9% of the time; at 12, 16.9% and 39.3%. Their 5% points are −3.38 and −2.89.

A relation supplied in advance

When theory supplies the long-run coefficient, the gap between two series can be tested directly and the search the Engle–Granger test pays for disappears: its 5% point moves from −3.38 to −2.89, and at two hundred observations the slowest return found four times in five goes from a half-life of 5.05 steps to 6.88. The price is in what a rejection means. With a gap that halves in three steps, a coefficient a fifth too large is confirmed 99.1% of the time and one half again too large 81.9%, though every such gap contains a random walk — and at a thousand observations the second is confirmed more often, not less.

cointegration · Spurious How slow a shared return a panel of pairs can see at 200 observations each. The average of the pairs' Engle–Granger statistics, against its own 5% point. One pair reaches 80% power up to a half-life of 5.1 steps, four pairs 9.2, sixteen 15.0. At a half-life of 12 the three find the relation 20.0%, 54.2% and 99.8% of the time.

A slow return across many pairs

One pair of two hundred observations is found tied four times in five only if its gap halves within 5.1 steps. Sixteen independent pairs sharing the same half-life, tested by the average of their Engle–Granger statistics, are found four times in five up to a half-life of 15.0 — and at a half-life of twelve, 99.8% of the time against one pair's 20.0%. What the widened test rejects is a statement about the panel: with half the pairs tied it rejects 55.4% of the time, with a quarter 21.2%, and a pair-by-pair reading of a panel in which all sixteen are tied finds 2.65 of them, against 0.78 when none are.

cointegration · Spurious How fast a gap has to close before a sample can see it close. The power of the test against the half-life of a disagreement, at 100, 200, 400 observations, each read against its own simulated critical value. Every pair in every reading is genuinely tied together, so a non-rejection is a miss. At 200 observations a gap that halves in 3 steps is found 99.9% of the time and one that halves in 12 steps is found 15.3% of the time — and by 35 steps the reading is 6.1%, which is the test's own size. Beyond that the curves are flat because there is nothing left to detect with.

How slow a return a sample can see

At two hundred observations the test finds a gap that halves in five steps four times in five, one that halves in eight 37.3% of the time, and one that halves in fifty 4.95% of the time — which is the rate at which it finds pairs with no mechanism at all. The boundary moves with the sample, not with its square root.

timeseries · Spurious How often the pooled test finds a relation in a panel with none, when the pairs share a shock. Against the 5% point computed for independent pairs, over 1,200 panels of pairs of 200 observations with no pair tied. Sixteen pairs, raw: 4.6% at a common share of 0, 7.2% at a common share of 0.25, 15.5% at a common share of 0.5, 27.3% at a common share of 0.75, 31.6% at a common share of 0.9; after removing the cross-sectional averages: 5.3%, 5.1%, 5.8%, 5.3%, 4.8%. Four pairs, raw: 5.4%, 6.5%, 6.8%, 12.6%, 16.9%.

A shock every pair shares

Sixteen pairs of series whose shocks are half common are worth 5.37 independent pairs to the panel's average statistic, and sixteen that share nine tenths are worth 1.44. Read against the 5% point computed for independent pairs, the pooled test then calls a panel with no relation tied 15.5% and 31.6% of the time; read against its own point, it finds a relation shared by every pair 22.3% of the time at nine tenths common, against one pair's 20.0%. Removing each date's cross-sectional average gives back the level and all sixteen pairs when every pair feels the shock equally, and leaves 13.3% false alarms when the loadings run from a half to one and a half.

cointegration · Spurious

Named alongside it

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

Critical valueStatistical powerCointegrationEngle–GrangerComposite nullRandom walkAugmented dickey fullerEffective sample sizeMonte CarloSpurious regressionUnit rootAutocorrelation

All concepts