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

Cointegration — where it appears

That two or more series wander individually but some combination of them does not, so they are tied together in the long run. Testing for it is a different question from testing whether either series is stationary, and regressing one on the other without it produces a relationship out of nothing.

Named by 17 essays across 3 fields — each of them below, with the objects they name alongside it.

A pair pulled back at 20% of the gap per step. Above, the two series. Below, the difference between them. The gap is pulled back towards zero by 20% of itself each step, so it stays inside a band of 14.3 while the series themselves travel much further. Nothing here is stationary except the difference. The faint line below is the gap for two free walks from the same seed, drawn for comparison.

The regression that is not spurious

Two random walks regressed on each other are called significantly related three times in four, so the time-series field ends in a warning. The exception it names and does not measure is here — and when the pair is genuinely tied, the fitted relation converges at rate 1/n rather than the usual 1/√n.

cointegration · Spurious Three series and one relation between them. Above, three series generated from Δy = Πy₋₁ + ε with Π of rank 1. Below, the combination y1 −y2. It stays inside a band of 9.5 while the series themselves travel 28.4. The count of combinations that behave this way is the rank of Π, and it is what every method in the field sets out to estimate.

Three series and a count

A pair of series is either tied together or it is not, so its whole inference is one test with one answer. Three can carry none, one or two relations at once — and the thing being estimated stops being a slope and becomes an integer, read off the gap in a spectrum whose top eigenvalue holds at 0.25 while the rest fall like 1/n.

systems · Rank 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 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 The correction, at a generating α of -0.2. Each point is one step: the gap at the end of yesterday against the change in y today. The fitted slope is -0.202 against the -0.2 the data was generated from, which means 20% of any disagreement between y and its long-run relation with x is undone in a single step. A shock therefore has a half-life of 3.1 steps. Neither series is stationary; the relation between them is.

The model that corrects its error

A cointegrated pair can always be written as a mechanism — today's change in y depends on yesterday's disagreement between y and its long-run relation with x. The coefficient of that disagreement is recovered from data that never saw it — and on unrelated series the same fit produces one a t table would call real 41% of the time.

cointegration · Dependence What differencing fixes, and what it costs, 100 steps. The first pair is the false-positive rate for two independent random walks: 77% on the levels, 4.9% on the differences. The second pair is how much of a real relationship survives: R² falls from 0.91 to 0.33. The same operation does both.

What differencing costs

Differencing takes the false-positive rate between two unrelated walks from 76.7% to 4.9%, and takes a genuine relationship's R² from 0.91 to 0.33. Applied to a series that did not need it, it doubles the variance and installs a correlation of −0.5 that the data never had.

timeseries · Spurious What the long-run relation is worth, at α = -0.2. Root mean squared one-step forecast error of the error-correction model divided by that of the model fitted on differences alone; below one means the levels helped. With the equilibrium known the ratio is 0.929 at 100 observations and settles on 0.905 by 3,200, against a closed form of 0.905 that mentions no sample size at all; the excess at short series is the cost of fitting three coefficients on fifty observations. With the equilibrium estimated as well it is 1.127 at 100 — worse than differencing — and 0.914 at 3,200. The gap between the two curves is the cost of not knowing β.

The cost of differencing a pair

Differencing two cointegrated series makes every standard error honest and throws away the one thing known about where they are going. The error-correction model forecasts better by exactly what a closed form says — and at four hundred observations it is better on four series in five and worse on average.

cointegration · Dependence Every equation's adjustment speed, and the one number they make together. Each series gets its own equation, each is regressed on the same lagged disequilibrium, and what comes back is the whole vector α. Averaged over 400 systems at n = 300: α₁ = -0.154 against -0.15 generated, α₂ = 0.104 against 0.1 generated. The gap closes at the combination of them rather than at any one entry — 25% of any disagreement per step, a half-life of 2.41 steps, where the single equation that fits only the first series reports 4.27.

Which series does the moving

“y adjusts towards x” and “x adjusts towards y” are different mechanisms with identical long-run relations, and a single-equation model cannot tell them apart because it only writes one equation. Writing all of them recovers a vector — and a gap that closes at 25% a step where one equation alone reports 15%.

systems · Adjustment The bounded error and the unbounded one. How the sequential trace procedure's answer is distributed, against the sample length, for a three-series system with 2 genuine relations. Over-counting — claiming a stationary combination that is a random walk — reads 4.9%, 7.2%, 5.7%, 6.2%, 5.9%, 4.2% across the six lengths, never far from the 5% of a single test. Under-counting reads 69.5%, 40.2%, 14.0%, 0.5%, 0.0%, 0.0%. The procedure is described as a 5% rule and the 5% applies to one of those columns.

The rank is a decision

The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.

systems · Rank 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 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 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 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 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 Four treatments of a shock sixteen pairs share: false alarms at the independence point, and power at each one's own point. Sixteen pairs, two hundred dates, nine tenths of each series' variation common. every pair loads the shock equally: nothing removed, false alarms 31.6% and power 22.3%; the cross-sectional average removed, false alarms 4.8% and power 99.8%; an estimated factor removed, false alarms 9.2% and power 76.3%; the observed shocks removed, false alarms 4.5% and power 6.5%. loadings run from a half to one and a half: nothing removed, false alarms 30.8% and power 25.8%; the cross-sectional average removed, false alarms 13.3% and power 82.5%; an estimated factor removed, false alarms 8.9% and power 73.8%; the observed shocks removed, false alarms 4.5% and power 7.5%.

A factor estimated from the panel

When sixteen pairs feel a common shock with loadings from a half to one and a half, removing the cross-sectional average leaves the pooled test rejecting an unrelated panel 13.3% of the time. Estimating the factor and each pair's loading instead brings that to 8.9% — and no further, because each loading comes from the same two hundred dates however many pairs there are — and it costs power against the average at the right critical value, 73.8% against 82.5%. Removing the shocks as observed restores 4.5% exactly and finds a shared relation 7.5% of the time: a series tied to its partner carries the partner's factor in its level, not its own shock.

cointegration · Spurious 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.

Random walkThe error-correction modelCritical valueSpurious regressionStationarityStatistical powerCointegrating rankCommon trendEngle–GrangerReduced-rank regressionUnit rootDifferencing

All concepts