What's partial correlation, in one sentence?

The partial correlation r_XY|Z is the Pearson r between X and Y after subtracting off (partialing out) the linear influence of a third variable Z on both — algebraically r_XY|Z = (r_XY − r_XZ·r_YZ) / sqrt((1−r_XZ²)(1−r_YZ²)), and equivalently the ordinary Pearson r between the residuals of X and Y after regressing each on Z.

Walk me through today's worked example.

Sample: CAD CPI m/m × USDCAD 15m, 195 non-contaminated releases (today's slot 1). Encode the 5 buckets as an ordinal: big_miss=−2, small_miss=−1, in_line=0, small_beat=+1, big_beat=+2. Compute three Pearson r's: r(surprise_z, move_pips) = −0.4256 (overall); r(surprise_z, bucket_ord) = +0.9488 (bucket is essentially a coarse-quantized surprise_z, so this is near ±1 by construction); r(move_pips, bucket_ord) = −0.4327 (the move responds to bucket, sign flipped by USDCAD convention). Plug into the partial-correlation formula: r_partial = (−0.4256 − 0.9488·(−0.4327)) / sqrt((1−0.9488²)·(1−0.4327²)) = (−0.4256 + 0.4106) / sqrt(0.0999·0.8127) = −0.0151 / 0.2848 = −0.0531. Fisher 95% CI (dof = n−1−3 = 191): tanh(arctanh(−0.0531) ± 1.96/sqrt(191)) = [−0.192, +0.089] — crosses zero. Controlling for bucket collapses the correlation to statistical noise.

How does this compare to Stats #11's within-bucket correlations?

Stats #11 (2026-08-11) reported within-bucket r values on the CAD CPI × CADJPY sample (mirror pair with sign-flipped convention). The current CAD CPI × USDCAD sample gives: big_miss n=13 r=−0.18, small_miss n=46 r=+0.10, in_line n=84 r=−0.35, small_beat n=32 r=+0.02, big_beat n=20 r=+0.25. Only in_line has a within-bucket r above 0.2 in magnitude — because in_line has the widest z-range (roughly −0.5 to +0.5, covering the full ambiguous zone) so it has room for within-bucket surprise-direction to carry information. Every other bucket's within-bucket r has a wide Fisher CI that crosses zero. Pooled across all 5 buckets the within-bucket r is −0.06 — essentially the same as the partial correlation with categorical bucket (−0.06 by construction: the categorical partial is algebraically identical to the pooled within-bucket r).

Why is the between-bucket r so extreme (−0.97) when the partial is essentially zero?

Because bucket assignment is defined by surprise_z (buckets are quantiles of z), the sample bucket variable captures ~91% of the variance in z (the between-bucket SS(z) is 235.26 out of total 258.83 — 90.9%). It captures only 19.5% of the variance in move_pips (Between SS(m) 33,031 out of total 169,211). The 19.5% is entirely explained by the between-bucket differences in mean move — that's the −0.97 between-bucket r. The remaining 80.5% of move-variance is within-bucket noise that surprise_z does not explain. So: total r = −0.43 = (bucket-explained part, big) + (within-bucket-noise part, essentially zero); the partial correlation is measuring exactly that second part.

What's the practical takeaway for trading?

The bucket boundaries ARE the granularity of the surprise → move signal on this event. Once you know a release is 'big_miss', additional information about HOW big-miss it is (surprise_z = −1.6 vs −3.0) adds essentially no forecast power for the 15m move on USDCAD. You should size your trades based on the bucket you're in, not on the specific z-value inside the bucket. This confirms the working assumption behind the tool's bucket-first display: p25/median/p75 columns by bucket are the actionable summary; the individual z's are only interesting for outlier flags and context contamination checks (like today's slot-1 four big-beat up prints, all with specific contextual explanations).

Does the ordinal-bucket-partial and the categorical-bucket-partial ever disagree materially?

They differ if the bucket-move relationship is non-linear (e.g., non-monotonic 'response is U-shaped in surprise-z'). Ordinal partial fits a single linear coefficient on the bucket ordinal, so it under-adjusts if the response saturates or reverses at the tails. Categorical partial fits a separate mean per bucket (4 dummies for 5 buckets), so it perfectly absorbs any bucket-level pattern including saturation. On today's sample the two agree to 1 decimal (ordinal −0.053, categorical −0.064) because the underlying response IS approximately monotonic (5-bucket pct-up walks 92.3/73.9/52.4/37.5/20.0 with only a 0.15p median inversion between small_miss and big_miss — well within CI). The categorical version is always the more conservative choice for a partial-correlation robustness check.

Are there caveats around the Fisher CI for the partial correlation?

Same caveats as ordinary Fisher (Stats #22): the SE = 1/sqrt(n−k−3) approximation, where k is the number of controlled-for variables (k=1 for ordinal bucket, k=4 for full categorical), assumes multivariate Normal residuals — mild fat tails cause 1-3% coverage error at n~100. Also: partial correlation reads causal control language into a purely statistical operation. r_XY|Z = 0 doesn't mean 'X has no causal effect on Y' — it means the LINEAR part of X's effect that isn't already carried by Z is zero. Any nonlinear or interaction effect is still there; Z might not even be causally relevant. On today's sample the interpretation is sound because bucket assignment is a deterministic function of surprise_z, so partialing out bucket is partialing out the coarse-quantized version of X itself — which reveals that the fine-grained within-bucket surprise-magnitude adds no linear predictive power.

Can I reproduce this in the tool?

Not directly — Vantage's News Impact Explorer doesn't expose a within-bucket-r column. But you can recompute: pull /api/v1/news-impact/releases?event=FF:CAD_CPI_MOM&instrument=USDCAD&window=15m and compute Pearson r(surprise_z, move_pips) overall (should give ~−0.43) and per-bucket (should give the five within-bucket r values above). All numbers verified against a Python numpy corrcoef cross-check to 4 decimal places.

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Statistics23 August 2026· 6 min read

Statistics for Traders #23: Partial correlation on today's slot-1 CAD CPI × USDCAD sample — total r is −0.43, but controlling for the 5-bucket assignment collapses it to −0.05 (95% CI crosses zero)

The Pearson r between surprise_z and 15-minute move_pips across 195 CAD CPI m/m × USDCAD releases is −0.4256— the correlation-language version of today’s slot-1 5-bucket monotonic walk. Partial correlation formally decomposes it. Controlling for the 5-bucket surprise assignment, the partial r drops to −0.05 (ordinal) or −0.06 (full categorical) and both Fisher 95% CIs cross zero. The between-bucket r (weighted by n) is −0.9691: nearly the entire structure lives in the differences between bucket means.

This is the Statistics installment that closes a queue item flagged since Stats #11 introduced Pearson r 12 days ago, and formalises the “buckets carry the signal” pattern via the partial-correlation identity r_XY|Z ≈ 0.

Horizontal-bar chart with Fisher 95% confidence-interval whiskers. Top rows: 'Overall (n=195)' Pearson r = -0.426 with tight CI [-0.534, -0.303] drawn in solid dark, clearly negative. 'Partial | bucket ordinal' r = -0.053 with CI [-0.193, +0.089] drawn in coral, crossing the r=0 coral reference line. 'Partial | bucket categorical' r = -0.064 with CI [-0.204, +0.078] also in coral, also crossing zero. Bottom rows: five within-bucket r values in muted grey. big_miss (n=13) r=-0.178 with wide CI crossing zero; small_miss (n=46) r=+0.096 crossing zero; in_line (n=84) r=-0.350 with tighter CI slightly to the negative side; small_beat (n=32) r=+0.024 crossing zero; big_beat (n=20) r=+0.247 crossing zero. Only in_line has a within-bucket r above 0.2 in magnitude and CI that doesn't fully cross zero. Coral vertical line at r=0 makes the crossings visible at a glance. — Total r vs partial r (bucket controlled) vs 5 within-bucket r’s. Coral bars cross zero; the between-bucket structure explains virtually all of the total r.

The partial-correlation formula, plugged

The identity for partial correlation of X and Y controlling for Z is

r_XY|Z = (r_XY − r_XZ · r_YZ) / sqrt((1 − r_XZ²)(1 − r_YZ²))

With X = surprise_z, Y = 15m move_pips, Z = bucket ordinal (big_miss = −2, small_miss = −1, in_line = 0, small_beat = +1, big_beat = +2):

r(surprise_z, move_pips)	−0.4256
r(surprise_z, bucket_ord)	+0.9488
r(move_pips, bucket_ord)	−0.4327
r_XZ · r_YZ	−0.4106
r_XY − r_XZ·r_YZ (numerator)	−0.0151
sqrt((1−0.9488²)(1−0.4327²)) (denom)	+0.2848
r_partial (ordinal bucket)	−0.0531

Fisher 95% CI with dof = n − 1 − 3 = 191 (subtracting 1 for the controlled variable and 3 for Fisher’s empirical n−3 correction from Stats #22): [−0.192, +0.089] — crosses zero. Controlling for bucket collapses the correlation to statistical noise.

The categorical (4-dummy) version is even cleaner

The ordinal encoding fits a single linear coefficient on the bucket variable. That’s fine when the response is monotonic in bucket (as today’s is). To fully absorb any pattern including tail saturation or inversions, use a full 4-dummy categorical encoding: subtract off each bucket’s own mean of z and mean of move, then compute Pearson r on the residuals.

Result: r_partial (categorical) = −0.0643. Fisher CI with dof = n − 4 − 3 = 188: [−0.204, +0.078] — also crosses zero. The categorical version is algebraically identical to the sample-size-weighted pooled within-bucket r (both are the mean of the demeaned-in-bucket product divided by demeaned-in-bucket SDs). It comes out slightly more negative than the ordinal because the categorical absorbs the small median inversion between small_miss and big_miss that the ordinal-linear treatment mis-models.

Within-bucket r values

Bucket	n	z-range in bucket	r	Reading
big_miss	13	−3.0 to −1.5	−0.178	small negative, CI wide (crosses zero)
small_miss	46	−1.5 to −0.5	+0.096	essentially zero
in_line	84	−0.5 to +0.5	−0.350	the ONE bucket with meaningful within-bucket r
small_beat	32	+0.5 to +1.5	+0.024	essentially zero
big_beat	20	+1.5 to +4.2	+0.247	positive (mild noise), CI wide

Only in_line has a within-bucket r above 0.2 in magnitude — and it’s the only bucket with enough z-range (−0.5 to +0.5) for within-bucket surprise-direction to carry information. The four tail buckets all quantise a narrower z-range and their within-bucket r’s are indistinguishable from zero at 95%.

SS budget — where the variance lives

The ANOVA-style variance decomposition (between-bucket + within-bucket = total sum of squares) for both variables:

Variable	Total SS	Between-bucket SS	Within-bucket SS	% between
surprise_z	258.83	235.26	23.56	90.9%
move_pips	169,211	33,031	136,180	19.5%

Reading: buckets soak up 91% of the surprise-z variance (mechanical — bucket boundaries are quantiles of z). Buckets soak up 19.5% of the move-pips variance. That 19.5% is what the overall r = −0.43 is picking up: nearly all bucket-explained. The remaining 80.5% of move-pips variance is within-bucket noise that surprise_z does not linearly explain.

Same-day cross-links

Today’s slot 1 ( CAD CPI m/m × USDCAD) shows the 5-bucket monotonic-pct-up walk: 92.3 / 73.9 / 52.4 / 37.5 / 20.0 percent up-rate. That’s the between-bucket story — the −0.9691 weighted between-bucket r translated into pct-up columns. Today’s Stats installment quantifies the complement: within a bucket, surprise magnitude adds essentially nothing.

Cross-links: Stats #11 ( 2026-08-11 correlation intro, same event on CADJPY mirror pair — established the within-bucket pattern qualitatively); Stats #22 ( 2026-08-22 Fisher r-to-z, provides the CI machinery this post reuses); Stats #10 (autocorrelation caveat — the between-release sample is IID enough at monthly spacing).

Verification note

All r values computed in a scratch TypeScript script against the 195-release sample from /api/v1/news-impact/releases on 2026-08-23. Cross- verified against Python numpy.corrcoef to 4 decimal places on the total, ordinal-partial, and categorical-partial r values, plus all five within-bucket r values. Chart generated via a one-off script reusing scripts/insights-charts/svg.ts and theme.ts primitives; not committed under scripts/ since the horizontal-bar-with-whiskers layout is single-use for this post.

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