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.

The 3 eigenvalues of the reduced-rank regression, averaged over 200 systems at n = 300, with each one's trace statistic and the 5% point it is read against. An eigenvalue is a squared canonical correlation between the changes and the levels, so 0.252 means a combination of levels explaining 25.2% of the variance of a combination of changes. The first clears its critical value and the rest do not, and the count of the ones that do is the estimate.

500 systems of 3 series at n = 300, each put through the sequential trace procedure at the 5% level with a critical value simulated for every null it tests. The true rank is 1 and it is returned 96.2% of the time. The errors are not symmetric: 0.0% under-count, which throws away a relation that exists, and 3.8% over-count, which claims a stationary combination that is a random walk.

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.

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.

Squared forecast error 4 steps ahead at each imposed rank, relative to the correctly specified fit, at 200 observations. 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.

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.