Adjustment — the series · Normal Distribution

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%.

part 1 · systems

Two failures that cancel

A mildly skewed covariate under a lower-tail copula leaks 0.002% of an interaction where each failure alone leaks eight and seven per cent. Turn the copula over and the same pair compounds.

part 12 · compound

The same copula, turned over. A Clayton copula and its reflection, at the same Spearman correlation of 0.40 and the same Kendall tau of 0.275, against the covariate's marginal. With a symmetric covariate the two are the same number to nine decimals — 7.707% apiece — because the leak then depends on how much asymmetry the copula has and not on which way it points. Skew the covariate and they come apart: at a skewness of 2.26 the lower-tail copula leaves 3.431% and the upper-tail one 36.213%, a factor of 10.6. Both halves of the dependence are asymmetries and an asymmetry has a direction; a lower-tail copula concentrates the dependence where a right-skewed marginal is compressed and the two distortions partly undo each other, and an upper-tail one concentrates it where the marginal is stretched.

A symmetry that was not enough

A heavy-tailed symmetric covariate has a skewness of zero and leaks exactly nothing under three copulas. Under the two asymmetric ones it doubles the leak, from 7.707% to 14.229%.

part 13 · compound

Three copulas that break nothing, and a factor of two between them. The three radially symmetric copulas, at a matched Spearman correlation of 0.40, against the covariate's marginal. All three leave exactly nothing with a symmetric covariate — that is the guarantee, and it holds to twenty decimal places. What they do to a skewed covariate is not the same at all: at a skewness of 2.26 a Frank copula leaves 12.118% where a Gaussian leaves 21.539% and a t on four degrees of freedom leaves 23.640%. A factor of 2.0 between two copulas that are both symmetric, both matched on rank correlation, and both harmless on their own. So the copula matters to the marginal's leak without breaking any symmetry of its own, which is a milder version of the same finding and applies to every trial rather than to the asymmetric ones.

A copula that halves a marginal

Three copulas break nothing on their own and put a factor of two between the same skewed covariate's leaks — 12.118% under a Frank against 23.640% under a t, at the same rank correlation.

part 14 · compound

The one zero neither half of the dependence can touch. A median split's interaction leak at all 30 combinations of copula and marginal, on a log scale. Every one is under 10⁻¹⁶ and the largest is 1.74e-20, which is the quadrature's own noise rather than a leak. The reason is arithmetic and it is short: a centred median split takes the values ±½, so its square is a quarter identically — for every unit, on every draw, whatever the covariate's scale is and whatever joint law the ranks have. The interaction is then orthogonal to both main effects by construction, and there is nothing for either half of the dependence to break. Both of the fields this one joins report this zero holding under their own variation; running both variations at once is what establishes that it is not two coincidences.

The zero that survives both

A median split's interaction leak is under 10⁻¹⁶ at all thirty combinations of copula and marginal. It is the only guarantee in the collection that neither half of the dependence can touch.

part 15 · compound

One margin rises; the other turns over. The two halves of the table's margin across the sweep: a lower-tail copula's own leak with a symmetric covariate, and a covariate skewed at 0.95 under a Gaussian copula. The marginal's leak rises at every step, from 3.727% to 36.056%. The copula's does not: it rises to 9.064% at a Spearman of 0.6 and falls to 8.219% by 0.7. It has to turn over, because at a rank correlation of one the two variables are a deterministic function of each other and there is no interaction left for a split to leak. So the margin of the table turns over before any cell in it does.

A margin that turns over

A skewed covariate's leak grows without limit as the dependence strengthens. A copula's own leak does not — it peaks at a rank correlation of 0.6 and falls. The margin of the table turns over before any cell in it does.

part 16 · stronger

A median with ties

The median split's interaction zero survives every marginal and every copula, because a split that is on for exactly half the units squares to a constant. A covariate measured on a few levels cannot be split exactly in half: on a five-point scale three in eight units sit on the median, and a rule balancing 'above the median' removes 6.39% of the interaction under a Gaussian copula and 10.06% under one of the two Claytons. The size is set by the share of units on the median, which is the formula's own factor. And a fair coin for the tied units restores the zero exactly, on every copula.

part 16 · compound

A near-perfect cancellation, at one correlation. A covariate skewed at 0.75 under a lower-tail copula, at each of 7 rank correlations. The earlier field measures this cell at a Spearman of 0.4 and reads 0.002% where adding the two halves' own leaks gives 16.579% — a cancellation so near exact that it is that field's headline. Across the sweep the same cell reads 0.0084%, 0.0182%, 0.0097%, 0.0015%, 0.0864%, 0.4693%, 1.5327%. Its smallest value is at 0.4, in the interior, and by 0.7 it is 1007.40 times larger. The near-zero is where two curves cross, and they cross beside the one correlation that was measured.

The zero that was a crossing

A cell that leaks 0.002% where adding its two halves gives 16.6% is a field's headline. On a finer grid it passes through zero at a rank correlation of 0.38 — two hundredths from where it was measured.

part 17 · stronger

Seven products of three splits. The correlation of every product of three centred median splits with every other, at a Spearman rank correlation of 0.4 between each pair of covariates. Under a Gaussian copula a main effect correlates with the other mains and with the triple (0.273), and with no pair at all, because the third moment is 0.000. Under a Frank copula, whose pairs are radially symmetric, the third moment is 0.078 and every other cell fills in: a main correlates 0.082 with the pair it is not in. In both panels a main is exactly uncorrelated with a pair it belongs to, which is the known zero.

The zero that needed two

A rule balancing two median splits removes nothing of their interaction, on every copula, because a centred split squares to a constant. Add a third covariate and the same arithmetic stops protecting the thing it looked as if it protected. The product of three splits is not orthogonal to the mains: at a rank correlation of 0.4 a rule on the mains removes 14.47% of it under a Gaussian copula. And the symmetry that decides the rest belongs to the triple rather than the pair, so Frank's copula, whose pairs are radially symmetric, leaks where Clayton's, whose pairs are not, barely does.

part 17 · compound

Four cells change their answer. The four cells of the twenty whose excess changes sign as the dependence strengthens, over 7 recalibrations. Above the line the two failures compound — the cell leaks more than adding the copula's own leak and the marginal's — and below it they cancel. All four start above and end below, and all four are at the two most skewed covariates: skew 0.90 under heavy-tailed, skew 0.95 under heavy-tailed, skew 0.90 under upper tail, skew 0.95 under upper tail. Whether two failures of a dependence compound or cancel is therefore not a property of the pair. It is a property of the pair at a strength of dependence, and a fifth of the table changes its answer inside the range measured here.

An answer that changes

Eleven of twenty cells cancel and nine compound, at one rank correlation. Sweep the correlation and four of the twenty change sides — all four from compounding to cancelling, all four at the most skewed covariates.

part 18 · stronger

The table swept along the covariate instead. What a rule holding a mean of each covariate fails to remove of their interaction, at each of 4 copulas, as the covariate is skewed further and the rank correlation is held at 0.4. The sweep runs from a symmetric covariate at g = 0 to a skewness of 11.16, and the three settings the earlier table names — g = 0.3, 0.6 and 0.9 — are on it, where this sweep reproduces that table to the last digit. Every row rises and then falls: the lower-tail copula from 7.707% through 0.0015% and back to 5.587%, the upper-tail one to a maximum of 36.213%. So the quantity a trial is exposed to is not monotone in how skewed its covariate is.

The other dial

The table is swept along the strength of the dependence and never along the shape of the covariate. Swept along the shape at a fixed correlation, the same two copulas cross, the same way — and the near-zero cell turns out to be a minimum in both directions at once.

part 19 · stronger

A threshold read along the skew. What a rule balancing a threshold at the value 1 on each covariate's own scale removes of the interaction between the thresholds, against the covariate's skewness, on five copulas at a Spearman rank correlation of 0.4. Gaussian: 16.41% with no skew, 10.07% at a skewness of 11.16; t on four: 22.18% with no skew, 12.11% at a skewness of 11.16; Frank: 13.19% with no skew, 10.14% at a skewness of 11.16; Clayton: 5.33% with no skew, 4.82% at a skewness of 11.16; Clayton turned over: 33.34% with no skew, 17.61% at a skewness of 11.16. Every leak falls, because the value moves towards the median.

Where the skew puts a threshold

The field read the mean-balancing rule along a sequence of skewness and left the threshold rule — a cut at a fixed value on the covariate's own scale — unread. Read, it turns out to be no new curve. Skewing the scale moves the value 1 from the 84th percentile of the latent covariate to the 74th, and the rule's leak is the copula field's cut sweep read at that quantile: it falls on every copula, from 16.41% to 10.07% under a Gaussian. Whether skew and dependence compound or cancel is then a fact about the slope of each copula's curve, and unlike the mean rule's, no cell changes its answer along the sequence.

part 20 · stronger

A cut on each side

Every table of the threshold's leak put one quantile on both covariates, which is the diagonal of a surface nobody had drawn. Off the diagonal the copulas swap places: a cut at the 5th percentile on one covariate and the 95th on the other removes 54.8% of the interaction under a Gaussian copula and 1.1% under a t copula, the reverse of their order at the near corner. Walked further out, the far corner has two opposite limits, and which one a Gaussian reaches is decided by whether its correlation is above or below exactly one third.

part 21 · stronger