What is 'co-kurtosis' precisely?

Co-kurtosis is the fourth-order cross-moment between two variables. Standardize zx and zy, then compute E[zx²·zy²] — this is the 'raw' co-kurtosis k_{X²Y²}. Under a bivariate Gaussian null, k_{X²Y²} = 1 + 2ρ², so the 'excess' co-kurtosis is kappa_{X²Y²} = k_{X²Y²} - 1 - 2ρ² (Ang-Chen-Xing 2006). If kappa > 0, the tails cluster jointly MORE than a Gaussian with the same correlation predicts (relevant for portfolio-tail-risk); if kappa ≈ 0, the joint tail structure is Gaussian even if marginal tails are fat.

How does co-kurtosis differ from co-skewness (Stats #32)?

Co-skewness γ_{X²Y} = E[zx²·zy] is third-order, ASYMMETRIC: it measures whether Y's MEAN shifts when |X| is large. Under Gaussian null it's zero (unconditional). Co-kurtosis kappa_{X²Y²} is fourth-order, SYMMETRIC: it measures whether Y's VARIANCE is bigger than expected when |X| is large. Under Gaussian null it's zero (after removing the 2ρ² correlation floor). Yesterday's Stats #32 found aggregate co-skew ≈ zero but strong bucket-conditional skewness on GBPCAD. Today's Stats #33 finds aggregate co-kurt ≈ zero AND unstructured bucket-conditional kurtosis on GBPJPY — a genuinely different picture.

Why does 'in_line' contain the biggest outliers?

Two of the top-3 largest-magnitude 15m moves in the sample are actually in the SMALL-MISS bucket (2024-10-16 z=-0.99 at -106.1p; 2023-07-19 z=-0.73 at -101.8p), and 2 of the top-10 are in_line prints (2013-07-16 z=0.00 at -72.3p; 2024-08-14 z=-0.37 at -67.2p). Reason: 'in-line' and 'small' buckets are LARGE (n=62 and n=53) so they contain more prints; and macro-idiosyncratic days can produce big moves even when the surprise-z on THIS event is small. The tail-buckets (n=15) are small samples so their extreme is capped by sample size. Same pattern as Stats #6 (NFP × EURUSD 'in_line' contains the biggest outliers).

So which tail-modeling tools DO help on this pair?

Marginal Y kurtosis is +1.36 (z=+3.92), so any inference on Y that assumes normality (t-CI for the mean, bucket-comparison ANOVA) has coverage penalties. Use BCa bootstrap (Stats #14) or stationary bootstrap (Stats #15) for tail-robust CIs. Bucket-conditional Sortino ratio (Stats #9) captures asymmetric downside on the miss side (small_miss has left-skew -0.73). Direct co-kurtosis regressors (Ang-Chen-Xing 2006 style) DON'T add signal here — the joint tail structure is Gaussian conditional on ρ, so co-kurtosis-adjusted asset-pricing is a no-op on this event × pair.

How does this build on prior Stats installments?

Direct sibling of Stats #32 (co-skewness, 2026-09-01) — same UK Core CPI event class, different pair. Direct extension of Stats #7 (marginal skewness) and Stats #6 (fat tails / kurtosis via whisker-to-IQR ratio) — Stats #33 formalizes the higher-moment machinery on a bivariate joint distribution. Uses the same variance-decomposition ideas as Stats #23 (partial correlation) and Stats #24 (one-way ANOVA / η²), but at fourth-moment order rather than second-moment. Cross-verified via Python numpy 2.4.6 on 2026-09-02.

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Statistics2 September 2026· 6 min read

Statistics for Traders #33: Sample co-kurtosis on today's slot-1 UK Core CPI × GBPJPY sample — Y has heavy marginal excess kurtosis (+1.36, z=+3.92) but excess co-kurtosis after correlation is kappa_{X²Y²} = -0.027 (essentially zero), and the mid-buckets carry the fat tails not the tail-buckets

Q: What's the one-line summary?

On today's slot-1 UK Core CPI × GBPJPY 15m sample (n=198): correlation ρ = +0.4794 explains 22.99% of Y variance linearly. Y's marginal excess kurtosis is +1.36 (z=+3.92, strongly non-Gaussian, fat tails). BUT excess co-kurtosis kappa_{X²Y²} = -0.027 (essentially zero) — the tails don't cluster jointly beyond what ρ already predicts. And the fat tails live in the MID-buckets (in_line +1.14, small_miss +1.09), not the tail-buckets — same pattern as Stats #6 on NFP × EURUSD.

Fourth cumulant follow-up to yesterday’s Stats #32 co-skewness. Same event class (UK Core CPI), different pair (GBPJPY not GBPCAD), same sample size (n=198). Marginal excess kurtosis γ_2(Y) = +1.36 (z=+3.92, strongly non-Gaussian). But excess co-kurtosis kappa_{X²Y²} = -0.027 — essentially zero. The joint tails are Gaussian conditional on ρ.

This is the natural next-order extension of Stats #32. Where Stats #32 found aggregate co-skew ≈ 0 with strong bucket-conditional directional skewness, Stats #33 finds aggregate co-kurt ≈ 0 AND unstructured bucket-conditional kurtosis— the mid- buckets (in_line, small_miss) carry the fat tails, not the tail- buckets, so "the tails live where you expect them" is false on this event × pair. Same "outliers land where you don’t expect" pattern as Stats #6 (NFP × EURUSD).

Five vertical bars, one per surprise bucket (big_miss, small_miss, in_line, small_beat, big_beat) plotting bucket-conditional excess kurtosis of GBPJPY 15m move_pips. Bars from left: big_miss n=15 kurt = -0.35 (below zero baseline, thin tails); small_miss n=53 = +1.09 (above baseline, fat); in_line n=62 = +1.14 (highest, fat); small_beat n=53 = +0.026 (near zero); big_beat n=15 = +1.18 (fat but small-n). Horizontal dashed line at Gaussian null (excess kurt = 0). Right panel: three statistic tiles showing marginal excess kurt Y = +1.36 (z=+3.92 highly significant), raw co-kurt k_{X²Y²} = 1.43 vs Gaussian null prediction 1.46, and excess co-kurt kappa = -0.03 (essentially zero). — Bucket-conditional excess kurtosis of GBPJPY 15m moves within each UK Core CPI surprise bucket. The MID-buckets (small_miss, in_line) carry the fat tails; tail-buckets sit near Gaussian (big_miss even below zero). Aggregate excess co-kurtosis kappa = -0.03 essentially zero.

Definitions

Let X = surprise_z, Y = move_pips (15m). Standardize:

zx = (X - μ_x) / σ_x, zy = (Y - μ_y) / σ_y

Marginal excess kurtosis: γ_2(X) = E[zx⁴] - 3. Gaussian null = 0; positive = fat-tailed; negative = thin-tailed (platykurtic).

Raw co-kurtosis: k_{X²Y²} = E[zx² · zy²]. Under Gaussian null with correlation ρ this quantity is exactly 1 + 2ρ² (Isserlis 1918 theorem for Gaussian moments), so it is NOT zero — it has a floor set by the linear correlation.

Excess co-kurtosis (Ang-Chen-Xing 2006):

kappa_{X²Y²} = E[zx²·zy²] - 1 - 2·ρ²

This IS zero under Gaussian null. If positive, tails cluster JOINTLY more than a Gaussian with the same ρ predicts (a portfolio-tail-risk warning). If ≈ 0, joint tails are Gaussian conditional on ρ even when marginal tails are fat.

Computed values on today’s slot-1 sample

Quantity	Value	SE (Gauss null)	z	Reject α=0.05?
ρ(X,Y)	+0.4794	0.0708	+6.77	yes (strongly)
γ_2(X) excess kurt	+0.5362	0.348	+1.54	no
γ_2(Y) excess kurt	+1.3645	0.348	+3.92	yes (strongly)
k_{X²Y²} raw	+1.4325	—	—	—
1 + 2ρ² (Gauss null)	+1.4597	—	—	—
kappa_{X²Y²} EXCESS	-0.0272	~0.35	-0.08	no
gamma_{X³Y}	+1.3387	—	—	—
gamma_{XY³}	+1.9025	—	—	—

The takeaway rows are highlighted. Y’s marginal excess kurtosis is +1.36 (z = +3.92, p < 0.001) — Y is strongly fat-tailed, unsurprising for a pip-move distribution. But the raw co-kurtosis +1.43 almost perfectly matches the Gaussian-null prediction 1 + 2ρ² = +1.46. The excess co-kurtosis kappa = -0.027 is essentially zero — the joint tails ARE Gaussian conditional on ρ.

Bucket-conditional excess kurtosis: fat tails live in the middle

Bucket	n	Excess kurt	SE	z	Reading
big_miss	15	-0.346	1.265	-0.27	thin-tailed (small n)
small_miss	53	+1.089	0.673	+1.62	fat, borderline sig
in_line	62	+1.140	0.622	+1.83	fat, borderline sig
small_beat	53	+0.026	0.673	+0.04	near-Gaussian
big_beat	15	+1.177	1.265	+0.93	fat but n=15, not sig

Fat tails are in the mid-buckets: in_line at +1.14 and small_miss at +1.09 (both borderline-sig by Bowman-Shenton at α = 0.05). Tail-buckets big_miss (-0.35, thin tails) and big_beat (+1.18, fat but n=15) don’t drive the aggregate Y kurtosis. The pattern is the same "outliers land where you don’t expect" message as Stats #6 on NFP × EURUSD.

Top-10 largest-magnitude prints

Sorted by |15m move_pips|. Highlights the surprise: the 3 biggest moves are all in NON-tail buckets. Only 1 of the top-10 lives in a big-tail bucket (2015-08-18 big_beat +100.0p).

Release date	Bucket	surprise_z	15m move
2024-10-16	small_miss	-0.99	-106.1p
2023-07-19	small_miss	-0.73	-101.8p
2015-08-18	big_beat	+3.16	+100.0p
2018-04-18	small_miss	-1.31	-94.6p
2014-10-14	big_miss	-1.57	-78.5p
2023-12-20	small_miss	-1.42	-75.9p
2017-09-12	small_beat	+1.09	+75.0p
2013-07-16	in_line	0.00	-72.3p
2024-05-22	small_beat	+0.91	+67.7p
2024-08-14	in_line	-0.37	-67.2p

7 of the top-10 are in MID-buckets (small_miss, in_line, small_beat). The 3 tail-bucket prints in the top-100: 2015-08-18 big_beat +100p, 2014-10-14 big_miss -78.5p, and 2018-09-19 big_beat +46p. The tail buckets are underpowered at n=15; the mid-buckets contain the exceptional moves.

Contrast with Stats #32 co-skewness

Yesterday’s Stats #32 sample co-skewness on the GBPCAD sibling sample found aggregate co-skew γ_{X²Y}= +0.09 (small) but strong bucket-conditional directional skew: big_beat +1.14, big_miss +0.58, small_beat +0.56 — right-tails IN THE DIRECTION OF THE SURPRISE on tail buckets. That’s tradeable: strategy-return distributions are positively skewed on the winning side.

Today’s Stats #33 on the GBPJPY sample gives a genuinely different picture. Aggregate excess co-kurt kappa ≈ 0 AND bucket-conditional kurtosis is unstructured — big_miss even has NEGATIVE excess kurt (thin tails) at -0.35. Practical read: co-skewness Stats #32 added information; co-kurtosis Stats #33 does NOT. On this event × pair, the joint tail structure is Gaussian conditional on ρ, and portfolio-tail-risk sizing that would tilt for co-kurtosis is a no-op.

Verification note

All numbers computed 2026-09-02 in Python via numpy 2.4.6 and scipy 1.17.1 on the full /api/v1/news-impact/releases response for event=FF:GBP_CORE_CPI_YOY, instrument=GBPJPY, window=15m, population=non_contaminated, limit=500. Sample size n = 198 non-contaminated releases from 2010-01-19 through 2026-06-17. Cross-verified: raw co-kurtosis 1.4325 vs Gaussian null prediction 1 + 2ρ² = 1.4597; the difference -0.0272 is well within Monte-Carlo simulation error at n=198 under a bivariate Gaussian with ρ = 0.4794. Chart via one-off script reusing scripts/insights-charts/svg.ts + theme.ts primitives + sharp; not committed under scripts/.

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