Three series and a count

The cointegration field on this site stops at two series, and stopping there is not a simplification. It is a different question with a smaller answer set.

With two series there is a long-run relation or there is not. The inference is one test, the test has one answer, and the quantity underneath it is a slope — one number, with a standard error, converging at a rate the field spends an essay measuring. Everything an analyst does with a pair is a variation on estimating that number and deciding whether it is there.

With three, the same question has three answers. There may be no relation at all, in which case all three series wander independently. There may be one, tying two of them together and leaving the third free. There may be two, which leaves exactly one thing wandering and everything else pinned to it. The quantity being estimated is a count, and a count is not something a t statistic reports.

The object is one matrix #

Every claim in this field is a claim about a single matrix, and writing the system down is most of the work.

Δyₜ = Π yₜ₋₁ + εₜ

y is now a vector of three series rather than a number. Δy is the vector of this period’s changes. Π is a 3 × 3 matrix saying how last period’s levels push this period’s changes, and the whole of the field is the question of what Π looks like.

If the three series are unrelated random walks, Π is the zero matrix. Nothing about where a series was yesterday changes what it does today; that is exactly what a random walk is, said in matrix form.

If they are tied together, Π is not zero, and it is also not full rank. A full-rank Π would mean every combination of the levels pushes back on the changes, which would make every series stationary — and these are not stationary, they wander. So Π has a rank strictly between 0 and 3, and it factors:

Π = α β′

β holds the combinations of the levels that are stationary, one per column. α holds how hard each series is pulled back along each of those combinations. If β has one column, there is one long-run relation. If it has two, there are two.

The rank of Π is the number of long-run relations, and the case that matters — rank strictly between 0 and k — is precisely the case a pair does not have. For two series the only options are 0 and 1, and “1” is what the two-step procedure already reports. Three is the smallest number of series for which counting is a question.

What the three cases look like, and how little that helps #

The three panels below are the same generator at rank 0, 1 and 2. The upper halves are close to indistinguishable, which is the honest starting point for the field.

Put the two upper panels beside the rank-one panel at the top of this essay and there is nothing to choose between them. Three lines that wander and roughly track each other is what all three systems look like, because in all three cases every individual series is non-stationary. The difference is entirely in what is left when the right combinations are subtracted, and which combinations those are is not known in advance.

That last clause is the difficulty, and it is worth being precise about. With a pair, there is only one combination to consider up to scaling: y₁ − βy₂. With three series there is a two-dimensional space of combinations to search, and the answer is a subspace — the set of all combinations that come out stationary. Its dimension is the count.

Counting by looking at a spectrum #

The trick that turns this into arithmetic is to stop testing and start decomposing.

Regress the changes Δy on the lagged levels y₋₁, having first taken out a constant from both. That is three regressions, one per series, and what comes out is not directly interesting. What is interesting is the canonical correlation structure between the two blocks: how much of the variation in the changes can be explained by some combination of the levels, then how much of what is left can be explained by another combination, and so on.

That question has an exact answer for each of the three combinations in turn, and the answers are the eigenvalues of a matrix built from three cross-product matrices. Each eigenvalue is a squared canonical correlation, which is what makes it readable: an eigenvalue of 0.25 means there is a combination of the levels that accounts for a quarter of the variance of some combination of the changes. Nothing else in this field has a scale attached like that.

At rank one the average eigenvalues are 0.2519, 0.0276 and 0.0064. At rank zero they are 0.0446, 0.0212 and 0.0050 — small and unremarkable, none of them standing out. At rank two they are 0.3333, 0.1769 and 0.0101, with two large and one small.

The count is a count of how many eigenvalues are large. That sentence is doing a great deal of work and it needs a reason to be true, because nothing so far explains why the small ones should be small enough to tell apart from the large ones.

Why the gap opens rather than closes #

Here is the property that makes rank estimable at all, and it is the sort of claim this site would rather count than cite.

Take the same rank-one system at five lengths and average the eigenvalues over a hundred and twenty systems at each. If every eigenvalue shrank as the series got longer, no threshold could ever separate them, and the count would not be recoverable from data of any length.

The eigenvalue belonging to the real relation goes 0.3003, 0.2724, 0.2569, 0.2488, 0.2445 as n goes 80, 150, 300, 600, 1200. It is drifting down slightly and settling; it is not going anywhere near zero.

The second, which belongs to a common trend, goes 0.1012, 0.0548, 0.0267, 0.0131, 0.0066. Fifteen times the data, fifteen times smaller. That is the 1/n the theory gives, and it is the whole reason a threshold exists: the ratio λ₁/λ₂ goes from 3.0 at n = 80 to 36.8 at n = 1200.

The gap opens roughly in proportion to the length of the series. So a procedure that puts a cut somewhere in the middle of the spectrum becomes more certain of its answer as more data arrives, which is the behaviour one wants and is not automatic. It is worth saying how unusual it is in this neighbourhood. Two walks and a finding is about a t statistic on a regression between unrelated series that gets larger with more data, so the rejection rate rises towards one rather than settling at 5%; more data makes that analysis more confidently wrong. And more data is not monotone is about an interval whose coverage goes down when n goes up. Neither of those is a pathology of a badly written procedure. They are what happens when a quantity’s sampling behaviour is not the one the analysis assumes.

Here the behaviour is the good one, and the reason is the same reason the pair’s slope converges at rate 1/n rather than 1/√n: a genuine long-run relation is an enormously strong signal, because the series it constrains travel arbitrarily far while the combination does not travel at all. The eigenvalue is picking up that constraint, and it does not weaken. What weakens is everything else.

The rate, checked rather than described #

“Fifteen times the data, fifteen times smaller” is the right description and it can be made an exact one, which is worth doing because the whole procedure rests on the rate being 1/n and not something close to it.

Multiply the second eigenvalue by the length it was measured at: 8.10, 8.22, 8.01, 7.86 and 7.92 at n = 80, 150, 300, 600 and 1200. A fifteen-fold change in the sample and the product moves by two per cent either side of 8.0, with no trend in it.

So the nuisance eigenvalue is not approximately 1/n. It is 8.0/n over the whole range measured, and the third eigenvalue is 1.9/n by the same arithmetic at n = 300. Two constants, and the count is read between them and the one that does not move.

The top eigenvalue’s own approach is the same rate seen from the other side. Fitting a constant plus c/n to 0.2488 and 0.2445 at the two longest lengths gives a limit of 0.240 approached from above with c ≈ 5.2, and that fit predicts 0.2574, 0.2746 and 0.3047 at the three shorter lengths against 0.2569, 0.2724 and 0.3003 measured. The real relation’s eigenvalue converges to a population number at rate 1/n; the spurious ones converge to zero at rate 1/n. Same rate, different destinations, and the difference between the destinations is the entire estimand.

The gap in closed form, and where the threshold has to sit #

Those two constants give the separation without any further simulation. The ratio is

$$\frac{\lambda_1}{\lambda_2} \approx \frac{0.240}{8.0/n} = \frac{n}{33},$$

which returns 2.4 at n = 80, 9.1 at 300 and 36 at 1200 against the 3.0, 9.6 and 36.8 measured. The whole of the field’s headline movement is one division.

It also says where a cut can live. A procedure separating the two eigenvalues by a fixed number needs $8.0/n < c < 0.240$, so the window is empty below n = 33 and is (0.10, 0.24) at eighty observations, widening to (0.007, 0.24) at twelve hundred. The upper end never moves and the lower end falls like 1/n, which is why more data helps here and why it helps asymmetrically: it buys room against under-counting and none at all against over-counting.

That asymmetry is worth carrying into the next section, which is about the two errors not being interchangeable. They are not symmetric in the arithmetic either. The distance from a threshold at 0.15 down to λ₂ grows without limit as the sample grows; the distance up to λ₁ is fixed at 0.09 for ever.

What a count being the estimate actually changes #

Three things follow, and each is a reason this field is not the cointegration field with an extra series.

There is no standard error. An estimate of 1.7 relations is not a thing. The output of the procedure is an integer, its error is a probability of returning the wrong integer, and the two kinds of mistake are not symmetric or interchangeable. Under-counting throws away a relation that exists and leaves a model that differences away information it had. Over-counting claims a combination is stationary when it is a random walk, which is the spurious-regression failure re-entering through a door marked “model selection”.

That first mistake is worth dwelling on, because the obvious safe move makes it. A reader who has absorbed the spurious-regression result knows that regressions between wandering series are untrustworthy and reaches for the standard repair: difference everything, and work with the changes. That repair does work on its own terms — the differenced regression holds its size where the levels regression rejects three times in four — and it is exactly wrong here, because differencing removes the levels and the relation is a statement about the levels. An analyst who differences a cointegrated system has not made their analysis safe. They have made it blind, and it will report nothing while looking like it worked.

The count and the common trends are the same fact twice. A system of three with two relations has one thing wandering freely; a system with one relation has two. r relations and k − r common trends are the same number read from opposite ends, and that equivalence is what makes rank the right object rather than a technical device. “How many independent sources of permanent movement are there?” and “how many long-run relations tie the series together?” are one question.

And the procedure has to be sequential. Testing “is the rank at least 1?” and “is the rank at least 2?” are different hypotheses with different null distributions, so the count comes from a chain of tests rather than from one, and each link needs its own critical value. That is the subject of the next essay but one, and the critical values turn out to be in no table this site can point at: 8.12, 18.64 and 31.74 at the 5% level, depending only on how many things are still wandering under the null being tested. That is the same situation the pair’s residual test is in — its 5% point is −3.38 where a t table says −1.65 — one dimension worse, because there is now a different distribution for each link in the chain rather than one distribution for one test.

The relation is recovered, not assumed #

One more thing the decomposition gives away, and it is the reason the machinery is worth its complexity.

The eigenvector belonging to the largest eigenvalue, mapped back into the original coordinates, is the estimated cointegrating relation. It is not proposed by the analyst and then tested. It is the combination the data says is most stationary, and on a rank-one system built from y₁ − y₂ it comes back at that, normalised so the first entry is one.

So the procedure answers two questions with one computation: how many relations there are, and what they are. The single-equation approach answers the second by being told the first, and being told the first is precisely what a system of three does not permit — which is why an essay on choosing a left-hand side is the next thing this field has to do.

At n = 300 the procedure returns the true rank 96.2% of the time, and at n = 80 it returns it 77.2% of the time, with 18.4% under-counts — the series are too short for the relation to be visible. By n = 150 the under-counting is 0.2% and it never comes back.

What does not move is the over-counting: 4.4% at n = 80 and 3.8% at n = 300. That is not an error that more data fixes, because it is not an error in the usual sense. It is the 5% level, doing what a 5% level does. A procedure that stops when a test fails to reject will over-shoot about one time in twenty however long the series are, and the only way to change that number is to change the level.

What this field takes on, and what it does not #

The claim staked here is systems of three or more series where the number of long-run relations is the object being estimated. That covers the rank, the spectrum it is read from, the critical values that turn it into a decision, and the adjustment vector — which is what makes “which series does the moving” a question with an answer, and is the second half of this field.

What is not claimed, and is named here so a later phase knows it was left deliberately: forecasting as a craft, model selection over lag lengths, deterministic trends inside the cointegrating space, and any system large enough that the arithmetic stops being three-by-three. Each of those is a real subject and none of them is measured here, so none of them is asserted here either.

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

The claim this essay rests on is that the spectrum separates: the eigenvalues belonging to relations stay up as n grows, the ones belonging to trends fall, and the gap opens roughly in proportion to n. All three halves are asserted in this field’s library, at two lengths four times apart, with the growth bracketed on both sides rather than merely required to be positive — because “the gap opens” is satisfied by any growth at all, and the claim being made is about the rate.

An assertion that has never rejected anything proves nothing, so the field’s machinery is fed input it must refuse. Three unrelated walks must not produce a relation: the rank procedure returns zero 95.0% of the time on them, and the 5.0% it does not is again the level rather than a defect. And a trace statistic read against the wrong row of the simulated table — the row for one common trend, applied to a three-series system — calls unrelated walks cointegrated most of the time, which is the mistake this field is most likely to produce and the reason its critical values are indexed by the thing they depend on rather than by the statistic’s name.