Which series does the moving

Two series drift apart, and then they come back together. Everything so far in this field and the one before it has been about establishing that they do come back — that a stationary combination exists, how many of them there are, and how to tell the case apart from two series that merely wandered in the same direction.

None of that says which series moves.

A pair whose gap closes because y falls back towards x, and a pair whose gap closes because x rises to meet y, have the same cointegrating relation, the same test statistic, the same everything an analyst normally reports. They are completely different mechanisms, and in most applications the difference is the substantive question — the one the analysis was commissioned to answer.

One equation is a choice, not a description #

The error-correction model as that essay writes it is a single equation:

Δy = c + α(y₋₁ − βx₋₁) + γΔx + ε

Today’s change in y depends on yesterday’s disagreement. α is negative, so a positive gap produces a negative change, and the gap is closed. That essay recovers α from data generated at a known value and gets it right.

The equation is correct. It is also only half of what is there, and the omission is not a matter of completeness. Nothing in that specification prevents x from also responding to the same disagreement. If it does, there is a second equation:

Δx = c′ + α₂(y₋₁ − βx₋₁) + γ′Δy + ε′

with its own coefficient. And the pair (α₁, α₂) is a different object from either entry: it says which series is doing the work.

Writing only the first equation does not assume α₂ is zero. It simply never forms the quantity, so the question cannot come up. The model is silent rather than wrong, which is a harder failure to notice, because a silent model produces output that looks complete.

Both speeds, recovered #

The repair is mechanical. Compute the disequilibrium from the estimated relation, then regress every series’ change on it — one regression per series, all with the same regressor.

On a pair generated with α₁ = −0.15 and α₂ = 0.1, averaged over four hundred systems at n = 300, the fitted values come back at −0.1536 and 0.1045. At n = 1000 they are −0.1508 and 0.1023. Neither number was anywhere in the data; both are recovered from it.

The signs are worth reading. α₁ is negative: when y is above where the relation says it should be, y comes down. α₂ is positive: when y is above the relation, x goes up. Both movements close the gap, and they close it from opposite ends.

On that second system the fitted speeds are −0.2040 and 0.0048. The second is not distinguishable from zero and it should not be — that series was generated with no adjustment at all.

The half-life a single equation gets wrong #

Here is why the vector is not merely more complete but arithmetically necessary.

The gap is y − x. Each series moves it by its own α. So the gap closes at α₁ − α₂ per step when the relation is (1, −1) — not at α₁, and not at either coefficient alone.

With α₁ = −0.15 and α₂ = 0.1, the gap closes at 25% of itself per step, which is a half-life of 2.41 steps. From the fitted values the same arithmetic gives 2.32 steps.

A single-equation model fitting only the first series reports α = −0.15 and, read the way such a coefficient is normally read, a gap closing at 15% a step and a half-life of 4.27 steps.

That is a half-life nearly twice too long, from a coefficient that is correct. Nothing about the single equation is mis-estimated. −0.15 is the speed at which y adjusts. The error is in the step from one coefficient to a statement about the gap, and that step is only valid when every other α is zero — a condition the single-equation model does not state, cannot check, and gives no indication of having assumed.

The three-series case behaves the same way. The system drawn above is generated with α = (−0.3, 0.1, 0) and the fitted vector over four hundred systems is −0.3023, 0.1042 and −0.0028. The gap between the first two closes at 40% a step; the equation for the first series alone says 30%.

What a zero in the vector means #

The third entry above is not a smaller version of the other two. It is a different kind of thing, and the difference is what makes α worth reading as a vector rather than as a list of coefficients.

A system of three series with one relation has two common trends — two independent sources of permanent movement, by the arithmetic of the first essay in this field. The α vector says how each series is connected to the stationary part of the system, so a zero entry says that series is only connected to the permanent part. It is a source rather than a follower.

That reading gives the count a second meaning. r relations and k − r trends was arithmetic; now it has a mechanism attached. Each column of α is one relation’s pull, each row is one series’ responsiveness, and a row of zeros identifies a series that pushes the system and is never pushed by it.

It also explains why the question “which series does the moving” has no answer for a pair where both coefficients are non-zero. Both series move. Neither is the source. The gap closes because they converge on each other, and asking which one is responsible is asking for a decomposition the data does not contain — as opposed to the case where one α is genuinely zero, where the data does contain it and the test below finds it.

Weak exogeneity, which is a hypothesis with a test #

A series whose α is zero is a particular thing. It is pushed by its own shocks and never pulled back towards the relation, so it carries the common trend and the other series follow it. The word for it is weakly exogenous, and it is the closest thing this field has to a statement about direction.

The test is a t on one coefficient, which makes it sound routine, and it is worth saying plainly what a rejection means and does not mean. It does not mean one series causes the other. It means that series does not respond to disequilibrium, so conditioning on it loses no information about the long-run relation — which is a statement about what a valid model may leave out, not about mechanism.

On the pair where the second series genuinely does not adjust, the test rejects 3.8% of the time with β known and 4.5% with β estimated from the same data, both against a nominal 5%. On the three-series system, testing the third series — the one outside the relation entirely — gives 5.3%. And where there is something to find, it finds it: on the pair where the second series adjusts at 0.1, the test rejects 99.3% of the time.

When that test is not a test #

The third bar is where this essay’s finding is, and it came out of a refusal rather than a hypothesis.

The check that was written was the ordinary one: three unrelated random walks must not produce an adjustment speed. They do not — the fitted α comes back at 0.0098 against a spread of 0.0096, which is nothing.

The t on it is a different matter. On systems with no relation at all, the weak-exogeneity test rejects 18.8% of the time at a nominal 5%.

The reason is one field back. The regressor in each equation is the lagged disequilibrium — the levels projected onto a combination — and when the combination is not cointegrating, that projection is a random walk. Every equation in the system is then a regression of a stationary change on a wandering regressor, which is the spurious regression this site measures at several times its nominal size elsewhere.

So the weak-exogeneity test is conditional on something that has to be said out loud: it is a t only where the relation it conditions on is stationary. Where it is, the size is between 3.8% and 5.3% across the three situations measured. Where it is not, it is 18.8%, and the printed output is identical.

This matters more than it looks, because of the order in which the two questions get asked. A rank procedure that returns 1 has established that some combination is stationary. It has not established that the combination an analyst then writes down by hand — a difference, a ratio, a theoretically motivated relation — is that one. Testing adjustment towards a combination the data has not certified is exactly the failing case, and there is no diagnostic on the exogeneity output that reports it.

The two estimates are not on the same footing #

One asymmetry deserves stating, because it is easy to read the α vector as though every entry were the same kind of number.

β is estimated at rate 1/n — the superconsistency the pair’s field measures at an exponent of −0.96 — which is why the second step of a two-step procedure can treat it as known. α is estimated at the ordinary 1/√n, because the regression that produces it has a stationary regressor once β is settled, and stationary regressions converge at the ordinary rate.

The practical consequence is visible in the intervals in this essay’s figures. On the pair where both series adjust, the spread of the fitted α₁ across four hundred systems is 0.0262 at n = 300 and 0.0149 at n = 1000. That is a factor of 1.76 for a factor of 3.33 in the sample, against the √3.33 = 1.83 that the ordinary rate predicts — close enough to be the ordinary rate and nowhere near the factor of 3.33 that β’s rate would give. The relation is pinned down far faster than the speeds at which it is enforced.

So a study long enough to establish a long-run relation beyond argument may still be much too short to say which series adjusts, and the two conclusions get reported in the same paragraph as though they carried the same weight. The relation converging fast is what makes everything downstream tractable; it also means the downstream numbers are the ones with the real uncertainty in them, and they are the ones being interpreted.

What the direction is and is not #

Three things this field’s α can support, and one it cannot.

It supports a claim about adjustment. A series with α = 0 does not respond to disequilibrium; one with α large responds fast. That is a measurable property with a recovered value and a half-life.

It supports a claim about which series can be conditioned on. Weak exogeneity is precisely the condition under which a single-equation model of the other series loses nothing, so the test says when the simpler analysis is safe — which is the practical use, and the reason the concept has a name.

It supports a decomposition into trends. k − r series carry the permanent movement and the rest follow; the α vector is what identifies which.

It does not support a claim about cause. Adjustment is a statement about the conditional mean of a change given a lagged combination, in a model the analyst wrote. Two series can both adjust to a third thing not in the system at all, and every number in this essay would look the same.

How wrong the single equation can get #

The factor of 1.77 between 4.27 steps and 2.41 is quoted for one system, and it is worth knowing whether that is a typical size or an extreme one, because the answer decides how much the omission matters in general.

Hold the total closing speed at 0.25 a step and vary how it is shared. If a fraction f of it belongs to the second series, the single equation reads (1 − f)·0.25 and reports a half-life of log(0.5)/log(1 − (1 − f)·0.25) against the truth’s 2.41 steps. At f = 0 — one series adjusting and the other inert — the single equation is exactly right. At f = 0.4, which is this system, it is 1.77 times too long. At f = 0.5, the two series sharing the work evenly, it is 2.16 times too long.

So the error runs from nothing to a factor of about two, and it is largest exactly where the substantive question is most interesting. A pair in which one series clearly leads is the pair the single equation describes correctly; a pair in which both adjust — which is the case somebody commissioned the analysis to detect — is where the reported half-life is worst, and the reporting carries no indication of which case is in hand.

The three-series system prices the same defect at a different sharing: 40% a step against 30%, so 1.36 steps against 1.94, a factor of 1.43 at f = 0.25. Three systems, three sharings, and a distortion that tracks the sharing exactly because it is arithmetic rather than estimation.

Two rates, told apart by a factor of nineteen tenths #

The claim that α converges at the ordinary rate and β at twice it is worth reading as a discrimination rather than as a consistency check, since the two candidate rates are far apart.

The spread of the fitted α₁ falls from 0.0262 to 0.0149 as the sample goes from 300 to 1000. The ordinary rate predicts a factor of √3.33 = 1.83; superconsistency would predict 3.33. The counted factor is 1.76, which is 4% from the first and a factor of 1.9 from the second. As an exponent that is −0.47 against a −0.5 and a −1.

Two lengths are enough to settle it, which is worth knowing because measuring a convergence rate normally takes a sweep. Here the two candidate exponents differ by a whole order, so any two sample sizes a factor of three apart separate them decisively, and the reading is not close.

The same inversion prices the spurious case. A test rejecting 18.8% of the time at a nominal 5% is a statistic whose spread is 1.96/Φ⁻¹(0.906) = 1.49 times what its own standard error claims — so the t is not slightly mis-sized, it is running at half again its assumed scale. That is a milder failure than the levels regression’s 76.7%, and it is the same mechanism at a lower amplitude: a stationary response regressed on a wandering regressor, where the response’s own stationarity limits how far the statistic can drift.

What to report #

The vector is cheap to produce and it is not what gets printed, so the practical residue is short.

Report every α, not the one whose equation happened to be written. It is k regressions on the same regressor rather than one, the extra k − 1 cost nothing, and the entries that come back near zero are as informative as the ones that do not.

Convert to a half-life using the whole vector. β′α is the speed the disequilibrium closes at, and any statement about how long a shock persists needs that number rather than a single coefficient. The difference on the system in this essay is 2.41 steps against 4.27 — a factor that would change any narrative built on it.

And check that the relation was certified before testing adjustment towards it. The exogeneity test’s level depends on it, the dependence is invisible in the output, and the cost of getting it wrong is a test running at 18.8% instead of 5%. A rank procedure returning at least 1 is not sufficient on its own: it says a stationary combination exists, and the one an analyst typed in by hand may not be it.

The check, and the refusal that makes it mean something #

The claims gated in this field’s library are the two that a figure cannot make because they compare settings. Both adjustment speeds are recovered at their generated values, and the half-life computed from the vector matches the mechanism’s while the half-life from one equation alone does not — asserted as an inequality rather than as two numbers, so it fails if the single-equation reading ever stops being wrong.

The refusal is the one that turned into a finding. A system with nothing in it must not produce an adjustment, and it does not. What it does produce is a test that rejects nearly four times too often, and the assertion now carries both halves: the size is held where the combination is stationary, and it is several times too large where it is not. An assertion that had only checked the point estimate would have passed, and the essay would have recommended a test that does not work.