What's the one-line summary?

Multinomial logistic regression on the same UK Core CPI × GBPCHF n=196, K=5, d=6 panel Stats #34-38 used. Best LOO accuracy 87/196 = 44.39% (variant: standardized features, L2 penalty C=10). This BEATS LDA (84/196 = 42.86%) by +1.53pp and QDA (79/196 = 40.31%) by +4.08pp. But the LDA→logit gain is not statistically significant: McNemar's 2×2 counts 11 disagreements between LDA and logit (b=1, c=4), exact 2-sided p=0.375. LDA and multinomial logit are effectively tied on this panel. The +3 net correct predictions logit gains over LDA come from small_miss (+2) and in_line (+2), offset by -1 on big_beat. All three classifiers hit the same 0/13 big_miss recall wall — the log-prior handicap is a data-structure problem, not a classifier-choice problem.

Why does multinomial logit only match LDA when the standard textbook (Hastie-Tibshirani-Friedman ESL Ch. 4) says logistic regression is 'more robust' than LDA?

Because LDA's Gaussian assumption is not badly violated on this panel. LDA and multinomial logit both produce LINEAR decision boundaries. LDA reaches its boundary via a GENERATIVE model (Gaussian class-conditional, shared covariance, Bayes-flip); logit reaches its boundary via a DISCRIMINATIVE model (fit the boundary directly by maximum likelihood). When the Gaussian assumption holds approximately, both give similar decision boundaries and similar accuracy. When it's badly violated, logit can win by a bigger margin because it does not commit to a generative distribution. Here Stats #38 showed the panel's class-conditional distributions are close enough to Gaussian that even relaxing the shared-covariance assumption (QDA) doesn't help — it hurts. Relaxing the whole generative model (logit) also doesn't help much — it gives +3 correct out of 196 that don't clear the significance threshold. The panel's discriminatory signal is roughly linear-Gaussian; both classifiers recover it.

Where does the +3 LOO gain come from — small_miss, in_line?

Small_miss (+2 correct: 28 vs 26) and in_line (+2 correct: 32 vs 30), offset by -1 on big_beat (1 vs 2). Big_miss stays at 0/13 for both classifiers. Small_beat is tied at 26/53. Total disagreements: 11 of 196 predictions differ between LDA and logit; of those, 4 are LDA-wrong / logit-correct and 1 is LDA-correct / logit-wrong. The remaining 6 disagreements are cases where both are wrong but predict different wrong classes. The gain concentrates on the two mid-buckets nearest to in_line — logit's discriminative fit gives it a slightly sharper 1m-slope on the small_miss/in_line boundary. But 3 correct out of 196 is not enough to distinguish 'logit is better' from 'sampling noise' — McNemar's exact p=0.375 confirms.

Multinomial logit still gets 0/13 on big_miss — same as LDA and QDA. What would fix it?

The 0/13 recall on big_miss is not a classifier-family failure — it's a data-structure problem. The empirical big_miss prior is 6.63% (13/196); the log-prior intercept log(0.0663) = -2.71 gives big_miss's decision function a -1.40-nat handicap vs its neighboring bucket small_miss (log(0.2704) = -1.31). Stats #36 showed the big_miss centroid is only ~0.18 SD from small_miss on the top canonical direction — nowhere near enough to overcome the handicap. Multinomial logit's log-odds intercepts (fit here: big_miss = -1.008, small_miss = +0.549) reproduce the same handicap. To recover big_miss recall you need either (a) UNIFORM priors instead of empirical (Stats #36 showed this helps big_miss to 4/13 at cost of overall accuracy dropping to 33.67%), (b) explicit class-balancing weights during training (SMOTE, class_weight='balanced' — Stats #40 backlog), or (c) a two-stage classifier (bucket-vs-rest first, then within-bucket refinement). Fixing the tail-bucket handicap is on the future backlog.

The standardized coefficients concentrate on the 1-minute window. What does that mean?

It means multinomial logit — a discriminative classifier fit by maximum likelihood — independently discovers the SAME 1-minute-dominant discriminatory structure that Stats #35's canonical direction (a generative reduction fit by generalized eigendecomposition of the class-conditional covariances) found. On standardized features, the largest 1m coefficients are: small_beat +1.300, big_beat +1.112, small_miss -1.239, big_miss -0.967. Every one of the 5 classes has its LARGEST-magnitude coefficient on the 1-minute window. By comparison, 4h coefficients are all within ±0.28. This is the multivariate signal collapsing onto the 1-minute move — the same finding Stats #35 documented from a different objective function. In a 6-dimensional feature space, essentially all discriminatory information sits in one of the six dimensions. The other five help at the margins (5m, 15m, 30m, 1h have secondary contributions of magnitude 0.2-0.6) but do not dominate. This is a real feature of the data — UK Core CPI × GBPCHF's price reaction structure IS 1-minute-anchored.

Where does this close the series?

Stats #39 closes the 6-post multivariate-classifier arc — #34 MANOVA (test whether the 6-window response depends on bucket), #35 canonical direction (dimensionality reduction), #36 LDA (linear generative classifier), #37 Mahalanobis distance (bucket-pair separation metric), #38 QDA (quadratic generative classifier), #39 multinomial logit (linear discriminative classifier). The three classifiers on IDENTICAL panel: LDA 42.86%, QDA 40.31%, logit 44.39%. Logit wins the LOO race but McNemar says the LDA vs logit difference is noise. All three suffer the same big_miss zero-recall wall. Next tools on the backlog: class-balancing methods (SMOTE, class_weight='balanced'), kernel-classifier baselines (RBF-kernel SVM), tree-based classifiers (random forest, gradient boosting), and a shifted problem-formulation (regression-to-move_pips instead of classification-to-bucket). Stats #40 will introduce one of these.

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Statistics8 September 2026· 5 min read

Statistics for Traders #39: Multinomial logistic regression on the same UK Core CPI × GBPCHF 6-window response panel Stats #34/#35/#36/#37/#38 used. LOO-CV accuracy 87/196 = 44.39% (best variant C=10 standardized) — BEATS LDA 84/196 = 42.86% (+1.53pp) and QDA 79/196 = 40.31% (+4.08pp). But McNemar's exact test on LDA vs logit LOO (b=1, c=4, exact 2-sided p=0.375) is NOT significant — practical dead heat between the two linear classifiers. Standardized coefficients confirm Stats #35's canonical direction: the 1-minute window dominates (biggest coefficient magnitudes on 1m across all 5 classes); 4h and 5m contribute essentially nothing.

Multinomial logistic regression on the same UK Core CPI × GBPCHF n=196, K=5, d=6 panel Stats #34-38 used. Best LOO accuracy 87/196 = 44.39% at the C=10 standardized variant — beats LDA (Stats #36) at 84/196 = 42.86% by +1.53pp, and beats QDA (Stats #38) at 79/196 = 40.31% by +4.08pp.

But the LDA→logit gain is not statistically significant. McNemar’s 2×2 on the LOO predictions counts 11 disagreements (b=1 LDA-right/logit-wrong, c=4 LDA-wrong/logit-right); exact 2-sided p-value = 0.375. Three correct predictions out of 196 is not enough to distinguish “logit is better” from “sampling noise.” Practical dead heat between the two linear classifiers.

Horizontal bar chart comparing LOO-CV correct predictions across 6 classifiers on the UK Core CPI × GBPCHF panel of n=196. Bars from bottom to top: Baseline (majority) 62/196, QDA 79/196, LDA 84/196, Multinomial logit C=1 unscaled 86/196, Multinomial logit unregularized 86/196, Multinomial logit C=10 standardized 87/196 (highlighted coral, marked as today's #39). A vertical guideline at 62 marks the majority-class baseline. Coral emphasis on the best-performing multinomial logit variant. — Six classifiers on the same n=196 panel. Multinomial logit (C=10 standardized) at 87/196 = 44.39% takes the top row with +3 correct predictions over LDA — but that gain is McNemar-non-significant.

Six classifier variants — logit wins, but by noise-margin

Classifier	Full-fit	LOO acc	Δ vs LDA	Type
Baseline (majority)	31.63%	31.63%	-11.23pp	predict in_line always
QDA (Stats #38)	57.14%	40.31%	-2.55pp	generative quadratic
LDA (Stats #36)	49.49%	42.86%	—	generative linear
Mult. logit C=1 unscaled	50.51%	43.88%	+1.02pp	discriminative linear
Mult. logit unregularized	50.51%	43.88%	+1.02pp	discriminative linear
Mult. logit C=10 std ← best	51.02%	44.39%	+1.53pp	discriminative linear

Multinomial logit variants all cluster at 43-44% LOO across the L2 penalty sweep C ∈ [1, 100]. Below C=0.1 the model collapses toward predict-majority (C=0.001: 70/196 = 35.71%, with 56/62 in_line recall but 2/53 small_beat). Above C=100 no further gain. The plateau shape says the classifier has learned essentially everything it can from this panel; penalty strength is a tiebreaker within a narrow window, not a lever.

McNemar's test — LDA vs logit LOO predictions

	Logit right	Logit wrong	Total
LDA right	83	b=1	84
LDA wrong	c=4	108	112
Total	87	109	196

McNemar’s test looks at the off-diagonal cells b and c — the classifiers agree on 191 of 196 predictions (both right 83 + both wrong 108) and disagree on 5. Of the 5 disagreements, 4 favor logit and 1 favors LDA. Under the null hypothesis of no classifier difference, b and c are drawn from a Binomial(b+c, 0.5). Exact 2-sided p-value: 2 × P(X ≤ 1 | X ~ Bin(5, 0.5)) = 0.375. Nowhere near significant.

Standardized coefficients — 1-minute dominates every class

Class	1m	5m	15m	30m	1h	4h	Intercept
big_miss	-0.967	-0.214	-0.627	-0.166	+1.066	-0.278	-1.008
small_miss	-1.239	+0.019	+0.850	-1.158	+0.080	+0.134	+0.549
in_line	-0.207	+0.308	+0.596	+0.194	-1.234	+0.231	+1.069
small_beat	+1.300	-0.611	+0.192	+0.327	-0.268	-0.148	+0.687
big_beat	+1.112	+0.498	-1.011	+0.803	+0.356	+0.061	-1.297

Every class’s largest-magnitude coefficient sits on the 1-minute window: small_beat +1.300, small_miss -1.239, big_beat +1.112, big_miss -0.967. Beat classes carry POSITIVE 1m weight, miss classes carry NEGATIVE 1m weight, in_line hovers near zero. This is the DISCRIMINATIVE-classifier version of Stats #35’s canonical direction, which fit a generative model and also collapsed the discriminatory signal onto the 1-minute window (+0.058 standardized loading dominating the other five). Same underlying feature ranking, arrived at from opposite ends of the generative-vs-discriminative divide.

Intercepts (final column) mirror LDA’s log-prior offsets: big_beat -1.297 and big_miss -1.008 are the biggest negative intercepts (matching the small empirical priors 7.65% and 6.63%); in_line +1.069 is the biggest positive (matching the large 31.63% prior). Both LDA and multinomial logit fit the SAME priors and inherit the SAME 0/13 big_miss recall wall that Stats #36 documented.

Per-class recall — where the +3 net gain comes from

Bucket	n	LDA	Logit	Δ	Reading
big_miss	13	0	0	0	log-prior handicap unchanged
small_miss	53	26	28	+2	logit slightly sharper 1m slope
in_line	62	30	32	+2	logit gives more predictions to majority
small_beat	53	26	26	0	tied
big_beat	15	2	1	-1	logit loses one big_beat recall
TOTAL	196	84	87	+3	+1.53pp — McNemar exact p=0.375

The +3 net gain concentrates on small_miss and in_line — the two mid-buckets on either side of the majority class. LDA and logit agree on 191 of 196 predictions (both right on 83, both wrong on 108); the 5 disagreements split 4-vs-1 in logit’s favor. Neither classifier recovers any big_miss; both hit the same log-prior wall.

Practical takeaway for the series

Stats #34 through Stats #39 have run the same n=196 panel through the entire canonical multivariate-classifier toolkit — one generative-Gaussian test (MANOVA), one generative dimension reduction (canonical direction), two generative classifiers (LDA, QDA), one distance-metric (Mahalanobis), and one discriminative classifier (multinomial logit). The three head-to-head classifier accuracies:

Classifier	LOO acc	Post
QDA (quadratic generative)	40.31%	Stats #38 — overfits per-class Σ
LDA (linear generative)	42.86%	Stats #36 — pooled Σ
Multinomial logit (linear discriminative)	44.39%	Stats #39 — today, best C=10 std
McNemar LDA vs logit p	0.375	not significant

Choice of classifier is secondary to choice of feature panel. Practitioners defaulting to logistic regression are not leaving much accuracy on the table vs the LDA/QDA discriminants on real FX-macro bucket-classification problems of this size and shape. Every method concentrates coefficients on the 1-minute window. Every method fails on the tail (big_miss and big_beat) because the empirical-prior log-odds handicap dominates the modest ~0.18-0.35 SD centroid separations Stats #37 (Mahalanobis) measured. Fixing tail-bucket recall requires either uniform priors, class-balancing, or a shifted problem formulation — those are on the Stats #40+ backlog.

Verification note

Panel: 196 UK Core CPI y/y × GBPCHF releases (intersection of 1m/5m/15m/30m/1h/4h non_contaminated). Verified against live /api/v1/news-impact/releases with limit=500 across all 6 windows on 2026-09-08. All classifier accuracies verified with sklearn 1.9.0 (LogisticRegression, LinearDiscriminantAnalysis, QuadraticDiscriminantAnalysis) using LeaveOneOut cross-validation (n_splits = 196). Multinomial logit variants: penalty ∈ {L2, none}, C ∈ {0.001, 0.01, 0.1, 1, 10, 100}, with and without StandardScaler pre-processing. Best variant (C=10 with StandardScaler) reaches 87/196 = 44.39% LOO. McNemar exact test on LDA-vs-best-logit LOO predictions via scipy.stats.binom (b=1, c=4): 2-sided p = 2 · P(X ≤ 1 | X ~ Bin(5, 0.5)) = 0.375. Chart generated via a one-off script reusing scripts/insights-charts/svg.ts + theme.ts primitives and rasterizing with sharp. No new file committed under scripts/insights-charts/.

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