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

Engle–Granger — where it appears

A two-step test for whether two wandering series are tied together: fit the levels, then test whether what is left over returns to a level. Its statistic is not a t and not an ordinary unit-root statistic, because the first step chose the combination to look as stationary as it could.

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

Where the residual test's statistic actually falls, at n = 200. Four thousand pairs of unrelated random walks, each regressed on the other and each residual tested for a unit root. The statistic is computed as a t and its distribution is not a t: five per cent of it falls below -3.38, where the ordinary one-sided 5% point of a t on 198 degrees of freedom is -1.65. Everything left of -1.65 — 70.2% of the whole distribution — is a pair of unrelated walks that a t table calls cointegrated.

The test with no table

The statistic that separates a real long-run relation from a spurious one is computed as a t and is not a t. At two hundred observations its 5% point is −3.38 where the t table says −1.65, and reading it against the table calls two unrelated random walks cointegrated 70.5% of the time.

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

Named alongside it

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

CointegrationMonte CarloSpurious regressionComposite nullConvergence rateCritical valueThe error-correction modelError rateFalse positiveHalf-lifeNull hypothesisp-value

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