What's the one-line takeaway?

Correlation captures the linear part of the joint (X, Y) distribution but misses higher-moment structure. On today's slot-1 UK Core CPI × GBPCAD sample, ρ(X,Y) = +0.508 explains 25.8% of Y variance linearly, and the aggregate Harvey-Siddique β_SKD (co-skewness γ_{X²Y}) is only +0.090 — small. But bucket-conditional skewness reveals that big_beat has skew = +1.14 (strong right tail), big_miss = +0.58 (also right-tailed), small_beat = +0.56, small_miss = -0.73. The tail buckets have fat tails IN THE DIRECTION OF the pair's response, hidden by the aggregate averaging because between-bucket skewnesses partially cancel.

What are the definitions I need to know?

For zx = (X - μ_x)/σ_x, zy = (Y - μ_y)/σ_y (standardized): correlation ρ(X,Y) = E[zx·zy] captures the LINEAR co-move. Marginal skewness γ_1(X) = E[zx³], γ_1(Y) = E[zy³] captures the ASYMMETRY of each variable alone. Sample co-skewness γ_{X²Y} = E[zx²·zy] = E[(X-μ_x)²(Y-μ_y)] / (σ_x²·σ_y) captures how Y's mean shifts when X² is large — the directional co-move of Y with |X|. The mirror γ_{XY²} = E[zx·zy²] captures how |Y| shifts when X is large. These are the four independent 3rd-order moments of the standardized joint distribution.

Why is the aggregate co-skewness so small?

Aggregate γ_{X²Y} = +0.090 because the buckets are directionally symmetric: big-miss and big-beat produce move_pips medians that are near-symmetric around zero (-18.9p vs +20.0p), and small-miss and small-beat produce near-symmetric medians (-13.6p vs +6.0p). The sum-over-i of zx²·zy is dominated by cross-terms that partially cancel: extreme +X prints paired with +Y contribute +z², and extreme -X prints paired with -Y also contribute +z² (both raise β_SKD), but middle-magnitude prints in the OTHER direction (small-miss paired with counter-directional +Y outliers, small-beat paired with counter-directional -Y outliers) pull it toward zero. The whole-sample number IS informative — it says the co-skew signal isn't systematically directional — but it hides the WITHIN-BUCKET tail structure.

What does the big_beat skew = +1.14 actually mean?

Big_beat (n=15) has mean move_pips +27.35 and standardized third moment γ_1 = +1.14 (strongly right-tail heavy). The 3 loudest big-beat prints since 2010 are 2015-08-18 (+105.9p), 2014-07-15 (+80.8p), and 2018-09-19 (+56.4p) — all in the same direction as the bucket's central tendency, and all much larger than any counter-directional print in the bucket. Right-tail-heavy in the direction the bucket says is 'the winning direction' means the winning trades in this bucket are asymmetrically fat-tailed: the 3 loudest movers scaled +2 to +2.5x the bucket median. Practical implication for a trader: the expected value of an in-bucket bet is higher than the median suggests, but Sharpe ratio (Stats #9) is worse because those +100p outliers inflate the variance without a matching negative tail.

Why is small_miss the only bucket where the skew is significant?

Under Gaussian null, marginal-skew SE ≈ sqrt(6/n). For small_miss n=53: SE = 0.34, observed skew = -0.73, |skew|/SE = 2.17 — rejects at α=0.05 (p ≈ 0.03). For big_miss n=15: SE = 0.63, observed skew = +0.58, ratio 0.92 — fails to reject. For big_beat n=15: skew = +1.14, ratio 1.80 — borderline p ≈ 0.07. For small_beat n=53: skew = +0.56, ratio 1.66 — fails to reject at α=0.05. Only small_miss's negative skew clears the Gaussian-null test on its own sample size, but the DIRECTION of all four tail buckets is consistent with the same fat-tail-toward-bucket-direction story.

How does this compare to the marginal skewness of X and Y?

γ_1(X) = +0.258, γ_1(Y) = -0.123. Both are small in absolute value; neither individually rejects the Gaussian null at α=0.05 on n=198 (|γ_1(X)|/SE = 1.48, |γ_1(Y)|/SE = 0.71). The marginal skewness numbers say each of the two univariate distributions is CLOSE to symmetric. But the bucket-conditional numbers say the JOINT distribution has structure that the marginals mask. Correlation captures the linear part of this joint structure; higher-order cumulants (γ_{X²Y}, γ_{XY²}) capture some but not all of the asymmetric part. Bucket-conditional moments are the operational tool that catches the fat-tail-toward-bucket-direction signal that the aggregate 3rd-order cumulants dilute.

Practical trader takeaway?

(a) In the tail buckets (big_miss / big_beat), the WINNING-direction fat tails are meaningfully larger than the LOSING-direction fat tails — so a bet in the bucket direction has positively-skewed win/loss profile. (b) Aggregate co-skewness doesn't reveal this. To find it you need bucket-conditional skewness, or equivalently a Q-Q plot of within-bucket returns against a Gaussian reference. (c) Sharpe ratio understates the strategy attractiveness in these tail buckets because it penalizes the right-tail fat-tail on the same footing as a left-tail fat-tail would be penalized. Sortino (Stats #9) is a better metric when the fat tail is asymmetric IN the winning direction; the Sortino / Sharpe divergence is a diagnostic for exactly this kind of asymmetric-cumulant profile.

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Statistics1 September 2026· 7 min read

Statistics for Traders #32: Sample co-skewness on today's slot-1 UK Core CPI × GBPCAD sample — the aggregate Harvey-Siddique β_SKD is only +0.09 (small), but the bucket-conditional skewness of move_pips is +1.14 in big_beat and +0.58 in big_miss — asymmetric fat tails IN THE DIRECTION OF THE SURPRISE that correlation and aggregate co-skew both miss

Higher-moment cumulants on today’s slot-1 UK Core CPI × GBPCAD 15m sample (n=198). Correlation ρ(X,Y) = +0.508 captures the LINEAR part (r² = 25.8% of Y variance). Aggregate Harvey-Siddique β_SKD = γ_{X²Y} = +0.090 — small. But bucket-conditional skewness of Y reveals asymmetric fat tails IN THE DIRECTION OF the surprise: big_beat skew = +1.14, big_miss = +0.58 — hidden by aggregate averaging.

All numbers computed in Python via numpy 2.4.6 + scipy 1.17.1 on the same full 198-release /api/v1/news-impact/releasesresponse used to build today’s slot-1 post. Cross-verified.

Vertical bar chart of bucket-conditional Fisher-Pearson biased skewness of GBPCAD 15m move_pips across the five UK Core CPI surprise buckets. Big_miss n=15: +0.58 (coral bar, small positive right-tail). Small_miss n=53: -0.73 (down bar, left-tail heavy). In_line n=62: -0.25 (small down bar). Small_beat n=53: +0.56 (up bar, right-tail heavy). Big_beat n=15: +1.14 (largest coral bar, strongly right-tail heavy). Two horizontal reference lines mark the aggregate marginal skewness of X (surprise_z, +0.26) and Y (move_pips, -0.12), both near zero. The tail buckets big_miss and big_beat are highlighted in coral. — Bucket-conditional biased Fisher-Pearson skewness of GBPCAD 15m move_pips. Big_beat +1.14 and big_miss +0.58 have fat right tails in the winning direction; small_miss -0.73 has counter-directional left tail. Aggregate marginals near zero.

Definitions

For X = surprise_z and Y = move_pips, standardize: zx = (X - μ_x)/σ_x and zy = (Y - μ_y)/σ_y. The four independent 3rd-order joint moments are:

Moment	Formula	Interpretation
γ_1(X) [= XXX]	E[zx³]	Marginal skew of X
γ_1(Y) [= YYY]	E[zy³]	Marginal skew of Y
γ_{X²Y} [= XXY]	E[zx² · zy] = β_SKD	Y's mean shift when X² large (directional Y ~ \|X\|)
γ_{XY²} [= XYY]	E[zx · zy²]	\|Y\| shift when X large (Y variance ~ X)

Aggregate cumulants on the n=198 sample

Quantity	Value	\|value\|/SE	Read
μ_x	+0.042	—	surprise_z centered near zero (expected: consensus is unbiased)
μ_y	-2.24p	—	GBPCAD drifts slightly down through 15m post-release
σ_x	1.093	—	surprise_z scaled to standard-Normal
σ_y	30.59p	—	typical 15m GBPCAD move std ≈ 30 pips
ρ(X,Y)	+0.5075	—	moderate positive linear r; r² = 0.258
γ_1(X)	+0.258	1.48	surprise_z slightly right-skewed, fails α=0.05
γ_1(Y)	-0.123	0.71	move_pips slightly left-skewed, fails α=0.05
γ_2(X) excess	+0.536	—	surprise_z slightly heavy-tailed
γ_2(Y) excess	+1.318	—	move_pips heavily-heavy-tailed
γ_{X²Y} = β_SKD	+0.090	0.52	small — Y's mean barely shifts with X² beyond linear r
γ_{XY²}	+0.015	0.09	essentially zero — \|Y\| doesn't shift with X

SE under Gaussian null (Bowman-Shenton) for the marginal and aggregate co-skew moments: sqrt(6/n) = sqrt(6/198) = 0.174. Neither marginal skew nor either co-skew clears α = 0.05 on their own. The aggregate picture says: the joint (X, Y) distribution looks approximately elliptical, with a strong linear part and small higher-order asymmetry. The bucket-conditional analysis, below, tells a different story.

Bucket-conditional skewness of Y = move_pips

Bucket	n	mean	sd	skew	\|skew\|/SE	Read
big_miss	15	-20.05p	25.78	+0.58	0.92	right-tail heavy — 2 anomalous +up prints
small_miss	53	-21.72p	30.16	-0.73	2.17	LEFT-tail heavy — big-down outliers, only sig bucket
in_line	62	+1.09p	23.54	-0.25	0.80	mildly left-tail
small_beat	53	+10.02p	24.31	+0.56	1.66	right-tail heavy in beat direction
big_beat	15	+27.35p	31.36	+1.14	1.80	STRONGLY right-tail — 2015 +106p, 2014 +81p

Bucket-conditional |skew|/SE is computed against sqrt(6/n_bucket) — a Gaussian-null test on the bucket-specific sample size. small_miss (n=53) is the only bucket that individually clears α=0.05 (|skew|/SE = 2.17, p ≈ 0.03). But the DIRECTIONAL pattern is consistent across the tail buckets: big_beat and small_beat both have positive skew (right-tail-heavy in the winning beat direction), and big_miss has small positive skew driven by the 2 counter-directional outliers 2013-08-13 (+22p) and 2014-12-16 (+38p).

Why aggregate co-skew misses the bucket asymmetry

β_SKD = γ_{X²Y} = +0.090is small because the bucket-tail asymmetries partially cancel when pooled. Big_miss’s +0.58 (right tail in the WRONG direction — counter-directional +up outliers on a down-bucket) and big_beat’s +1.14 (right tail in the RIGHT direction — bigger-than-expected +up outliers on an up-bucket) contribute to β_SKD with opposite signs relative to the bucket-mean direction. The sum across buckets ends up near zero.

This is the classic Simpson-style cancellation for higher-order cumulants: an aggregate co-skew of near-zero does not mean the joint distribution is symmetric. It means the asymmetries are conditionally on X and self-cancelling on aggregation. To catch the signal you have to condition on X — via buckets, quantiles, or equivalently a nonparametric regression Y ~ X and look at the residual conditional distribution.

Practical trader read

(a) Fat tails in the winning direction on tail buckets. big_beat (+1.14) and small_beat (+0.56) both have right-tail-heavy move_pips distributions on the up (winning) side. Meaning: the distribution of returns from a "bet with the big-beat direction" strategy is positively skewed — the median (+20p) understates the mean (+27p), and the top-decile prints (+80p / +106p) drag the mean further right.

(b) Sharpe ratio understates the attractiveness. Sharpe penalizes variance regardless of sign. On a positively-skewed strategy, the right-tail variance is desirable (it’s where the upside lives), but Sharpe treats it identically to left-tail variance. Sortino (Stats #9) is a better metric when the fat tail is asymmetric IN the winning direction — the Sortino / Sharpe divergence itself is a diagnostic for exactly this kind of asymmetric-cumulant profile.

(c) Higher-order cumulants complement the linear r, they don’t replace it. r = +0.508 does most of the work (25.8% of Y variance). γ_{X²Y}= +0.09 adds only a small refinement. But conditional-on-X moments (bucket-skew here, or bucket-Sharpe, or bucket-percentile) can change the practical trade decision meaningfully. That’s the argument for looking at the tool’s per-bucket columns rather than just the correlation number.

Verification note

All numbers computed in Python via numpy 2.4.6 + scipy 1.17.1 on 2026-09-01 on the same 198-release /api/v1/news-impact/releases?event=FF:GBP_CORE_CPI_YOY&instrument=GBPCAD&window=15m&population=non_contaminated&limit=500 response feeding today’s slot-1 post. Chart via one-off script reusing scripts/insights-charts/svg.ts + theme.ts primitives + sharp; not committed under scripts/. Direct follow-up to Stats #7 (marginal skewness, 2026-08-07) — 25-day spacing. Queue rotates to suppressor-variable partial-vs-semi-partial (queued since Stats #25), higher-order VAR-b prewhitening (queued since Stats #20), co-kurtosis (natural sibling of today’s post).

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