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.

Median absolute error in the fitted slope, over 300 pairs at each length, on log axes so a power law is a straight line. The pulled-back pair has exponent -0.96 — the error falls by a factor of ten when the series get ten times longer — against the −0.5 that every ordinary estimator obeys, drawn as the dashed reference. Two free walks give exponent -0.01: the error at 1,600 observations is 1.01 against 1.05 at 50.

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 same statistic read against two critical values, on unrelated walks and against a real error-correction mechanism at α = -0.2. Read against a t table the test calls unrelated walks cointegrated 70.5% of the time, against the 5% it claims. Read against the simulated value it fires on 4.8% of unrelated pairs and still finds 99.7% of real ones. The rule that holds its size loses almost nothing.

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.

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 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.

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.

On unrelated walks, the Engle–Granger statistic's 5% point is −3.38; the Dickey–Fuller statistic on the gap a supplied coefficient implies has its 5% point at −2.89; the t table says −1.65. The distance between the first two is the charge for having chosen the coefficient to make the residual look stationary.

With a gap that halves in three steps, a supplied coefficient of 1.2 is confirmed 99.1% of the time, 1.5 81.9% and 2 48.5%, though every one of those gaps contains a random walk. With a gap that halves in eight, the true coefficient is confirmed 68.3% of the time and 1.5 45.9%.

A pair whose gap halves in 3 steps. At the true coefficient the gap's Dickey–Fuller statistic is −5.11; at 1.5 it is −4.86, against a 5% point near −2.9. The second gap is the first minus 0.5 times the wandering series, drawn dashed: a random walk that the fast return around it hides from the test.

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.

The average statistic rejects "no pair is tied" 5.1% of the time with none tied, 7.8% with one, 55.4% with eight and 99.8% with all 16. A Bonferroni reading of the 16 pairs finds any at 13.8% with eight tied and 22.6% with all. A pair-by-pair test at 5% finds a tied pair 16.6% of the time.

With all 16 pairs sharing a gap that halves in 12 steps, a pair-by-pair test finds 2.65 of them on average; with none tied it finds 0.78. The two distributions of the count share 42.4% of their mass: no count of individual findings separates the panels.

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%.

Over 1,200 panels each. The 5% point computed for independent pairs is -2.380; the independent panels' own 5% point is -2.374 and the correlated panels' is -2.860, so 27.3% of correlated panels fall past the independence point.

Every pair tied with a half-life of 12 steps, 200 observations, 400 panels a point, each test read against the 5% point its own null has at that common share. Sixteen pairs, raw: 99.3% at 0, 93.0% at 0.25, 71.8% at 0.5, 43.5% at 0.75, 22.3% at 0.9; averages removed: 99.3%, 98.8%, 98.0%, 100.0%, 99.8%. Four pairs, raw: 53.8%, 49.8%, 53.0%, 41.3%, 21.8%; averages removed: 46.8%, 44.5%, 49.8%, 56.0%, 45.3%. One pair alone: 20.0%.

Sixteen pairs, none tied, against the independence 5% point. Equal loadings, averages removed: 5.3%, 5.1%, 5.8%, 5.3%, 4.8% at common shares 0, 0.25, 0.5, 0.75, 0.9. Loadings from 0.5 to 1.5, averages removed: 5.3%, 6.1%, 6.5%, 8.0%, 13.3%; raw: 4.8%, 7.2%, 14.3%, 26.8%, 30.8%.

One pair's null variance over the variance of the 16-pair average, over 1,200 panels a point. Raw: 15.96 at a common share of 0, 12.25 at a common share of 0.25, 5.37 at a common share of 0.5, 2.61 at a common share of 0.75, 1.44 at a common share of 0.9. With the cross-sectional averages removed: 14.97, 15.99, 15.24, 15.39, 15.72.

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%.

Nine tenths of the variation common, loadings from a half to one and a half; at 4, 8, 16, 32 pairs: nothing removed, 15.9%, 20.5%, 30.8%, 34.4%; the cross-sectional average removed, 8.0%, 8.9%, 13.3%, 14.7%; an estimated factor removed, 9.5%, 6.1%, 8.9%, 11.2%; the observed shocks removed, 5.4%, 3.6%, 4.5%, 5.3%.

Nine tenths common, loadings from a half to one and a half. At 4, 8, 16, 32 pairs: nothing removed, 1.43, 1.64, 1.58, 1.75; the cross-sectional average removed, 2.55, 4.78, 7.49, 11.16; an estimated factor removed, 2.78, 5.48, 8.27, 11.57. The dashed line is a panel of independent pairs.

A half-life of twelve in every pair, nine tenths common, loadings from a half to one and a half. At 4, 8, 16, 32 pairs: nothing removed, 24.5%, 28.7%, 25.8%, 33.0%; the cross-sectional average removed, 41.0%, 64.5%, 82.5%, 93.8%; an estimated factor removed, 34.8%, 67.8%, 73.8%, 85.0%; the observed shocks removed, 3.5%, 10.0%, 7.5%, 9.5%.