Subordinator tracking on the S&P 500

Phases 0–3, 2026-08-13. Derivations: Subordinator-tracking portfolios. Code lives in notebooks/subordinator_tracking/ and uses only the public API; normix is unchanged.

The mathematics says a unique portfolio direction is the least-noisy linear readout of the latent clock \(Y\), and that a second, non-tradable quadratic statistic can read \(Y\) even when that direction carries no signal. This page measures both claims on daily US large-cap returns. The algebra holds. The linear channel is not there. The quadratic channel is.

What was asked

Two study modes, written down before the fits.

1. Static. Fit one distribution to the full history. Report the signal-to-noise numbers \(\tilde q\) and \(\kappa\), test them against a null that kills odd moments, and check whether the model’s max-skewness portfolio is the tracker.

2. Dynamic. Update the model daily with EWMA-weighted online EM and rebalance \(w^\star_t\). The question is extraction of a latent clock from existing data, not prediction. Evaluation is in-sample by design.

Four pre-registered hypotheses.

• H1 (partial revelation). Fitted \(\kappa\ll 1\): daily equity portfolios are residual-dominated. A back-of-envelope from univariate skew and excess kurtosis in the gauge \(E[Y]=1\),

  \[ \kappa \approx \frac{\mathrm{skew}(X)^2}{3\,\mathrm{kurt}(X)}, \]

  with daily-index \(\mathrm{skew}\approx -0.5\) and excess kurtosis \(5\)–\(15\), gives \(\kappa\sim 10^{-2}\). Per-day tracker correlation would then be \(\sqrt{\kappa/(1+\kappa)}\approx 0.1\). The open question is how much the cross-section adds.

• H2 (saturation). \(\tilde q_d\) saturates as \(d\) grows because fitted \(\gamma\) aligns with the market covariance factor. Under market-only skewness the \(d\to\infty\) tracker is the index, so the ceiling is the index’s own \(\kappa\). Any excess must come from idiosyncratic skewness dispersion \(\lVert\delta\rVert^2\).

• H3 (max-skew = tracker). For fitted VG/NIG, \(t^\dagger\le 0\le 1/\tilde q\) analytically, so the model’s max-skewness portfolio is exactly \(w^\star\). For GH the inequality is checked numerically. On sample third moments, the direct skew maximiser should point near \(\hat\Sigma^{-1}\hat\gamma\) up to estimation noise.

• H4 (responsiveness). Short EWMA half-lives track volatility regimes faster but inject noise into \(\hat\gamma_t\) — direction churn, turnover, and upward-biased \(\hat\kappa_t\) — while long half-lives approach the static fit. There should be an interior optimum tied to the persistence of \(Y_t\).

Verdicts, up front. H1 accepted. H2 rejected in both of its intended readings. H3 split: true on every fitted model, false on sample third moments. H4 split: short half-lives do inflate \(\kappa_t\) and churn \(w^\star\); they do not produce a better \(\hat Y\).

Data and cohorts

Panel. data/sp500_returns.csv: daily log returns, 2015-12-15 → 2026-02-09 (\(T=2552\) days), \(d=468\) current constituents with near-complete history. The window covers the 2018 Volmageddon, the 2020-03 COVID crash, and the 2022 hiking cycle.

Hygiene. No winsorisation — tails are the signal. Twelve observations with \(|r|>0.5\) were flagged and kept. No zero-variance columns. Survivorship bias is accepted: this is a current-constituent panel of large caps, not an investable point-in-time universe. Fine for a mechanism study; not a trading backtest.

Nested universes. For each random seed we draw a permutation of the 468 tickers and take prefixes of size \(d\in\{5,10,25,50,100,200,468\}\). Nesting means the size-\(10\) set contains the size-\(5\) set, and so on, so \(\tilde q_d\) comparisons along a seed are monotone in the name list rather than in a fresh draw. Five seeds. At \(d=468\) every seed is the full panel, so the five runs coincide.

Primary working universe. \(d=50\), seed 0. Small enough that a full \(\Sigma\) is well conditioned (\(T/d\approx 51\)), large enough that the quadratic channel has something to do. Headline claims use this universe; the \(d\)-sweep is the robustness check. Fits at \(d=468\) have \(T/d\approx 5.5\) and are stress cases, not the basis of the punchline.

Market proxy. Equal-weight panel mean \(m_t=\frac1d\sum_i x_{t,i}\). A univariate NIG fit of \(m_t\) is the index baseline \(\kappa_{\mathrm{index}}\) — the \(\delta=0\) ceiling from the equicorrelation calculation.

What “cohort” means here. A cohort is one \((d,\mathrm{seed})\) universe: a fixed list of names, the full \(T=2552\) days on those names, one fitted model (and, in Phase 3, one online path). Phase 1 reports seed 0 at every \(d\). Phase 2 averages the five seeds and puts a standard error on \(\hat{\tilde q}\). Phase 3 is seed 0, \(d=50\) only, plus a secondary \(d=20\) unshrunk check.

Models and fitting

Primary: NIG, NormalInverseGaussian. No unbounded-likelihood boundary (unlike VG, whose density requires Gamma shape \(\alpha>d/2\)), closed-form InverseGaussian M-step, and regularization='a_eq_b' pins \(\mu_{\mathrm{IG}}=1\), which is exactly the identifiability gauge \(E[Y]=1\). Then \(\kappa_{\mathrm{lev}}=\tilde q\) and \(\kappa=\tilde q\,\mathrm{Var}(Y)\) directly.

Secondary: GH at \(d\le 50\), started from the fitted NIG’s exact GH embedding (to_generalized_hyperbolic) and then freeing \((p,a,b)\). Nested continuation, not a cold start. The same regularization='a_eq_b' flag now pins \(a=b\) on the GIG, which is not \(E[Y]=1\) unless \(p=-1/2\). Report \(\kappa_{\mathrm{lev}}\) and \(\kappa\), not raw \(\tilde q\). VG only at \(d\le 10\), with alpha_min='density' so the marginal stays bounded (\(\alpha\ge d/2+\varepsilon\)). That clamp is a different estimand from unconstrained VG; see the family comparison below.

EM. BatchEMFitter, default_init cold start, tol=1e-5. The package default \(10^{-3}\) stops after one iteration on this panel: daily-equity \(\gamma\) is small enough that \(\mathrm{rms}(\Delta\gamma)/(1+\mathrm{rms}(\gamma))\) already meets it. \(\kappa\) and \(\kappa_{\mathrm{lev}}\) are gauge-invariant, so the regularisation does not create them.

Online loop (Phase 3). Hand-rolled from the public E-step / M-step and EWMAUpdate. IncrementalEMFitter draws random batches, so it is the wrong tool for a chronological clock. Cappe2009 online EM with a constant step — the EWMA regime that tracks slowly-varying parameters rather than converging. No regularize_a_eq_b inside the loop: re-gauging every step would mix \(Y\)-gauges inside the EWMA \(\eta\) recursion. Report gauge-invariant \(\kappa_t\); \(\tilde q_t\) is not comparable across half-lives without a gauge.

Metrics glossary

Every number in the tables below is one of these.

Signal-to-noise

Symbol	Meaning
\(\tilde q=\gamma^\top\Sigma^{-1}\gamma\)	Energy of \(\gamma\) in the \(\Sigma\) metric. Scale-dependent: under \(Y\mapsto cY\) it maps to \(\tilde q/c\). Quoted in the \(E[Y]=1\) gauge that 'a_eq_b' pins for NIG.
\(\kappa_{\mathrm{lev}}=\tilde q\,E[Y]\)	Level SNR. Tracker MSE relative to \(E[Y]^2\) is \(1/\kappa_{\mathrm{lev}}\). Gauge-invariant.
\(\kappa=\tilde q\,\mathrm{Var}(Y)/E[Y]\)	Fluctuation SNR. Tracker MSE relative to \(\mathrm{Var}(Y)\) is \(1/\kappa\). Gauge-invariant. This is the headline “can a portfolio read \(Y\)?” number.
\(\sqrt{\kappa/(1+\kappa)}\)	Model correlation \(\mathrm{corr}(\hat Y,Y)\) if the mixture is well specified and \(Y\) were observed.
\(1/\kappa\)	Relative tracker MSE. At \(\kappa=0.05\) this is \(20\): the unbiased portfolio is twenty times noisier than \(Y\)’s own variance.
\(\kappa_{\mathrm{index}}\)	\(\kappa\) from a univariate NIG fit of the equal-weight mean \(m_t\). The \(\delta=0\) ceiling.

Estimators of \(Y\)

Object	What it is
\(\hat Y_t=(w^\star)^\top(x_t-\hat\mu)\)	Linear tracker. A portfolio, after subtracting location. Conditionally unbiased for \(Y\) in the model.
Linear Bayes	\(E[Y]+\frac{\kappa}{1+\kappa}(\hat Y-E[Y])\). Best affine rule. Same direction, shrunk toward the prior mean.
\(E[Y\mid x_t]\)	Posterior mean from the fitted joint. Uses \(q(x_t)\), hence both the squared tracker and \(q_\perp\). MMSE among functions of \(X\).
\(q_\perp(x_t)=q(x_t)-\tilde q\,\hat Y_t^2\)	Orthogonal Mahalanobis radius. The part of the quadratic statistic that is invisible to any portfolio.

On real data \(Y_t\) is latent, so we cannot score \(\hat Y\) against truth. We score it against volatility proxies (next block) and against the model’s own moment laws (variance of \(\hat Y\) should be \(\mathrm{Var}(Y)+E[Y]/\tilde q\); ACF of \(\hat Y\) should be \(0\) under i.i.d. mixing).

Volatility proxies

Proxy	Construction	Role
21-day RV	Centered 21-day realized variance of \(m_t\)	Slow, model-free clock. The comparison the plan treats as primary.
EWMA RV	EWMA of \(m_t^2\) at a matched half-life	Fairness check: same smoother family as the online model.
Cross-sectional dispersion	\(\frac1d\sum_i(x_{t,i}-m_t)^2/\hat\sigma_i^2\)	Model-free cousin of \(q_\perp\). A day when names disagree after scaling is a high-\(Y\) day under the mixture.

Correlations are Pearson unless stated. A number such as “corr\((\hat Y,\mathrm{RV})=0.07\)” means the linear tracker does not move with the 21-day RV series. “corr\((E[Y\mid X],\mathrm{RV})=0.58\)” means the posterior does.

Nulls and geometry

Object	Meaning
Sign-flip null	For each day \(t\), multiply the whole demeaned cross-section by an independent \(S_t=\pm 1\). Kills every odd joint moment (including all skewness) and preserves \(\Sigma\) and the mixing structure. Refitting produces the distribution of \(\hat{\tilde q}>0\) that estimation noise alone can generate. The 95% quantile of that distribution is the floor in the tables; the \(p\)-value is the fraction of null fits with \(\hat{\tilde q}\) at least as large as the real fit.
\(c=0\) synthetic floor	Phase 0 analogue: draw from the fitted NIG with \(\gamma\) set to \(0\), refit, collect \(\hat{\tilde q}\). Same idea, with known truth.
Direction cosine	\(\cos\angle(\hat w^\star, w^\star)\) in the \(\Sigma\) metric, or \(\cos\angle(w^\star_t, w^\star_{t-21})\) for stability. \(1\) is parallel, \(0\) is orthogonal, negative is flipped.
Block-bootstrap cone	Moving-block bootstrap with 21-day blocks, to respect volatility clustering. The 5/50/95 percentiles of the cosine of \(\hat w^\star\) against the full-sample \(w^\star\) are a confidence cone on the direction.
PC1 share of \(\tilde q\)	Write \(\Sigma=\sum_k\lambda_k u_k u_k^\top\) and \(\tilde q=\sum_k(u_k^\top\gamma)^2/\lambda_k\). The \(k=1\) term is the fraction of \(\tilde q\) that lives on the market eigenvector. High and stable \(\Rightarrow\) market-aligned skewness (H2’s saturation story). Falling with \(d\) \(\Rightarrow\) the extra \(\tilde q\) is in small-\(\lambda\) directions, which is also where \(\gamma\)-estimation noise lives.
\(g\mathbf{1}+\delta\) split	\(\gamma=g\mathbf{1}+\delta\) with \(\mathbf{1}^\top\delta=0\). Market loading versus idiosyncratic residual. \(\lVert\delta\rVert^2/\lVert\gamma\rVert^2\) rising with \(d\) is the idiosyncratic-dispersion branch — or noise.
\(t^\dagger\)	\(2\mathrm{Var}(Y)/E[Y]-\mu_3/\mathrm{Var}(Y)\). Model max-skewness equals \(w^\star\) iff \(t^\dagger\le 1/\tilde q\).
\(n_{\mathrm{eff}}\)	EWMA effective sample size \((2-w)/w\approx 2.9\,h\) for half-life \(h\). A full \(\Sigma\) at \(d=50\) wants \(n_{\mathrm{eff}}\gtrsim 150\); half-lives below \(\sim 63\) days are under-determined without shrinkage.
Turnover	Daily unit-gross turnover of \(w^\star_t\). How much the tracker portfolio is being rewritten.
\(\tau\)	Shrinkage weight toward a static \(\eta_0\) in the online \(\eta\) update. \(\tau=0\) is pure EWMA; \(\tau=0.1\) pins the gauge and conditions \(\Sigma_t\).

Phase 0 — synthetic validation

Truth is available. Generator: the Phase 1 NIG at \(d=50\) (\(E[Y]=1\), \(\tilde q=0.0749\), \(\mathrm{Var}(Y)=0.638\), \(\kappa=0.0478\)), with \(\gamma\mapsto c\gamma\) for \(c\in\{0,1,3,10\}\). That sweeps \(\kappa\in\{0,\,0.0478,\,0.430,\,4.78\}\). \(T=2552\), \(R=20\) i.i.d. draws. MSE and correlation laws use the true parameters (no refit). \(\hat{\tilde q}\) and direction recovery use cold-start EM.

MSE and correlation laws (true parameters)

\(c\)	\(\kappa\)	Tracker MSE rel. err.	Linear-Bayes MSE rel. err.	\(\mathrm{corr}(\hat Y,Y)\) rel. err.	Post. MSE \(/\) linear-Bayes bound	\(\mathrm{corr}(E[Y\mid X],Y)\)
0	0	—	1.1%	—	0.089	0.954
1	0.0478	0.63%	0.84%	6.5%	0.095	0.954
3	0.430	0.74%	0.10%	1.6%	0.125	0.956
10	4.78	0.19%	0.06%	0.05%	0.336	0.970

What worked. Tracker MSE \(E[Y]/\tilde q\) and linear-Bayes MSE \(\mathrm{Var}(Y)/(1+\kappa)\) hold to \(<1\%\) relative error. The correlation law \(\sqrt{\kappa/(1+\kappa)}\) is within \(2\%\) at \(\kappa\ge 0.43\).

What is already the empirics’ punchline, in simulation. Even at \(c=0\) (no drift channel) the posterior mean has \(\mathrm{corr}(E[Y\mid X],Y)=0.95\) and MSE \(0.09\times\) the linear-Bayes bound. At \(d=50\) the quadratic channel saturates. A portfolio cannot dominate \(Y\) in this regime; \(E[Y\mid X]\) still can.

A small miss. At the weakest nonzero point (\(c=1\), the generator’s own \(\kappa\)) sample \(\mathrm{corr}(\hat Y,Y)\) is \(0.228\) vs \(0.214\) (\(6.5\%\) relative). Twenty replications of \(2552\) days: this is a small systematic, not Monte Carlo noise. The \(5\%\) acceptance flag fails only there.

Estimation noise of \(\hat{\tilde q}\)

Null floor at \(c=0\), \(T=2552\), \(R=20\):

\(d\)	Mean \(\hat{\tilde q}_0\)	95% quantile	Std
10	0.0130	0.0206	0.0043
25	0.0261	0.0395	0.0091
50	0.0578	0.0775	0.0100

The floor grows with \(d\). At \(d=50\) the generator’s own \(\tilde q=0.0749\) sits on the null 95% quantile: a full-history NIG fit on this panel is not distinguishable from \(\gamma=0\).

Bias at \(T=2552\), \(d=50\):

\(c\)	True \(\tilde q\)	Mean \(\hat{\tilde q}\)	Mean \(\cos\angle\)
0	0	0.058	—
1	0.075	0.138	0.76
3	0.674	0.750	0.94
10	7.49	7.90	0.98

What failed as a point estimate. \(\hat{\tilde q}\) is upward-biased. At \(c=1\) the bias is the whole signal: you would report \(\tilde q\approx 0.14\) for a true \(0.075\). Direction recovery is still usable at \(\kappa=0.05\) given \(T=2552\) (cosine \(0.76\)) and essentially exact at \(\kappa=4.8\). Cutting \(T\) to \(500\) drops the \(c=1\) cosine to \(0.40\).

Every later \(\hat{\tilde q}\) is therefore reported against this floor (or its sign-flip sibling), not as a raw SNR.

Online EM rehearsal (H4, with known \(Y\))

Path: \(\gamma_t\) rotates by \(\pi/4\) in the \(\Sigma\) metric over \(T=2552\); InverseGaussian scale jumps \(\times 3\) at \(t=T/2\). Oracle start at the true \(t=0\) model. Filtered tracker (using \(\theta_{t-1}\) on \(x_t\)) versus true \(Y\). EWMA realized variance of the equal-weight mean is the RV baseline. True \(\kappa=0.096\) on this path.

\(h\)	\(\tau\)	\(n_{\mathrm{eff}}\)	corr\((\hat Y^{\mathrm{filt}},Y)\)	After \(h_s=21\)	corr RV	Mean \(\kappa_t\)	Mean \(\cos\angle\)	Turnover
21	0	61	0.32	0.46	0.47	1.36	0.18	0.084
21	0.1	61	0.14	0.09	0.47	0.50	0.47	0.139
63	0	182	0.26	0.30	0.48	0.64	0.31	0.054
252	0	727	0.30	0.38	0.46	0.23	0.57	0.024

What H4 got right. Short \(h\) inflates gauge-invariant \(\kappa_t\) (\(14\times\) at \(h=21\)) and raises turnover.

What H4 got wrong as a recipe. Short \(h\) does not recover the rotating direction — mean cosine is worse (\(0.18\) vs \(0.57\) at \(h=252\)). The rotation is slow (\(\pi/4\) over ten years); estimation noise in \(\hat\gamma_t\) dominates the tracking benefit. Shrinkage \(\tau=0.1\) at \(h=21\) buys cosine (\(0.47\)) and spends \(Y\)-tracking and turnover. The smoothed \(h=21\) tracker matches EWMA RV (both \(\approx 0.46\)) and does not beat it. A frozen true-model posterior on the same path still has corr \(0.96\) — again the quadratic channel, which does not need \(\gamma_t\).

In-sample \(\hat Y\) (using \(\theta_t\) on \(x_t\)) has corr \(0.63\)–\(0.89\). That is same-day overfit. The filtered numbers above are the honest ones.

Phase 1 — static S&P 500 (H1, H3)

Nested seed-0 universes. NIG at all \(d\); GH nested continuation at \(d\le 50\); VG at \(d\le 10\). Sign-flip \(B=50\) for \(d\le 50\), \(B=20\) for \(d\in\{100,200,468\}\).

H1 — no linear extraction

\(d\)	\(\hat{\tilde q}\)	Null 95%	\(p\)	\(\hat\kappa\)	\(\sqrt{\kappa/(1+\kappa)}\)	\(1/\kappa\)
5	0.0043	0.0098	0.41	0.0082	0.090	122
10	0.018	0.018	0.078	0.020	0.139	51
25	0.036	0.040	0.12	0.029	0.167	35
50	0.075	0.085	0.20	0.048	0.214	21
100	0.127	0.164	0.52	0.071	0.258	14
200	0.219	0.298	0.81	0.115	0.321	8.7
468	0.509	0.808	1.00	0.233	0.435	4.3

Read a row as: “the fit produced this \(\tilde q\); a sign-flip that kills all skewness produces a larger \(\tilde q\) this often; the implied fluctuation SNR and the correlation you would get if \(Y\) were observed and the model were true.”

Equal-weight univariate NIG: \(\kappa_{\mathrm{ew50}}=0.0093\), \(\kappa_{\mathrm{ew468}}=0.0125\). The \(d=50\) panel \(\hat\kappa=0.048\) exceeds the index but sits inside the sign-flip floor. \(\hat{\tilde q}\) grows with \(d\); the null grows faster. At \(d=468\), \(p=1\): the fit is less skewed than a typical sign-flip.

H1 holds. Point estimates sit in \(10^{-2}\) to \(10^{-1}\) and are indistinguishable from odd-moment noise. Secondary GH/VG fits do not create a linear signal either; the VG \(\kappa\) gap is a constraint, not leftover scale — next subsection.

Family comparison — why VG \(\kappa\) is smaller

The H1 table is NIG. The question is whether the secondary families disagree on \(\gamma\)-energy or on \(\mathrm{Var}(Y)\) after the scaling gauge is removed, and whether a smaller VG \(\kappa\) is an EM or simulation artifact.

The mixture has the orbit \((\gamma,\Sigma,Y)\mapsto(\gamma/c,\Sigma/c,cY)\). Under it \(\tilde q\mapsto\tilde q/c\), \(E[Y]\mapsto c\,E[Y]\), \(\mathrm{Var}(Y)\mapsto c^2\mathrm{Var}(Y)\), so raw \(\tilde q\) and raw \(E[Y]\) are not comparable across families. Two invariants factor \(\kappa\):

\[ \kappa_{\mathrm{lev}} = \tilde q\, E[Y], \qquad \mathrm{cv}^2 = \mathrm{Var}(Y)/E[Y]^2, \qquad \kappa = \kappa_{\mathrm{lev}}\,\mathrm{cv}^2. \]

\(\kappa_{\mathrm{lev}}\) is \(\tilde q\) after \(E[Y]=1\) (scale-free \(\gamma\)-energy). \(\mathrm{cv}^2\) is \(\mathrm{Var}(Y)\) in that same gauge.

Recommended gauge for cross-family display: pin \(E[Y]=1\). 'a_eq_b' does that only for NIG (\(\mu_{\mathrm{IG}}=1\)). On GH it sets \(a=b\), so \(E[Y]=K_{p+1}(a)/K_p(a)\), which is \(0.10\) at \(d=50\) (\(p\approx-2.47\)). On VG (\(b=0\)) the flag is a no-op. \(|\Sigma|=1\) is the classical GH convention and also fails to align \(E[Y]\). The representative that puts every family on the same \(Y\)-scale is a post-fit rescale \(s=1/E[Y]\) along the existing orbit (NormalMixture._rescale; notebook helper pin_mean_y):

Family	\(Y\mapsto sY\) rule	After \(E[Y]=1\)
NIG	\(\mu_{\mathrm{IG}}\mapsto s\mu_{\mathrm{IG}}\), \(\lambda\mapsto s\lambda\)	same as 'a_eq_b'
GH	\(a\mapsto a/s\), \(b\mapsto sb\), \(p\) fixed	\(a\neq b\) in general
VG	\(\beta\mapsto\beta/s\), \(\alpha\) fixed	\(\beta=\alpha\)

The marginal law of \(X\) is unchanged. \(\kappa\) and \(\kappa_{\mathrm{lev}}\) are invariant; raw \(\tilde q\) and \(\mathrm{Var}(Y)\) become the two numbers to compare.

Seed-0 secondary fits. Columns \(\tilde q\) and \(E[Y]\) are as stored (NIG already at \(E[Y]=1\)); \(\kappa_{\mathrm{lev}}\) and \(\mathrm{cv}^2\) are the \(E[Y]=1\) numbers.

\(d\)	Family	\(\tilde q\)	\(E[Y]\)	\(\kappa_{\mathrm{lev}}\)	\(\mathrm{cv}^2\)	\(\kappa\)
5	NIG	0.0043	1	0.0043	1.91	0.0082
5	GH	0.014	0.25	0.0034	5.24	0.018
5	VG	0.0085	0.99	0.0084	0.385	0.0032
10	NIG	0.018	1	0.018	1.11	0.020
10	GH	0.105	0.14	0.014	3.39	0.049
10	VG	0.056	0.85	0.048	0.196	0.0094
50	NIG	0.075	1	0.075	0.638	0.048
50	GH	0.688	0.10	0.069	1.39	0.097

Not a leftover scale. After \(E[Y]=1\), GH and NIG \(\gamma\)-energy agree at every overlapping \(d\) (\(0.0034\) vs \(0.0043\), \(0.014\) vs \(0.018\), \(0.069\) vs \(0.075\)). Directions agree too: \(\cos\angle(\gamma_{\mathrm{NIG}},\gamma_{\mathrm{GH}})\ge 0.999\) in the NIG \(\Sigma\)-metric; \(\cos\angle(\gamma_{\mathrm{NIG}},\gamma_{\mathrm{VG}})\ge 0.988\). The families are reading the same skewness axis.

Not a simulation issue. Phase 0 draws were NIG-only. These numbers are full-history EM on the same real panel. Mean log-likelihood at \(d=10\): GH \(27.06\), NIG \(27.05\), VG \(26.73\). VG is a worse fit, not a different random draw.

GH \(\kappa\) is not higher because \(E[Y]=0.10\). \(\kappa\) is gauge-invariant. The \(0.10\) vs \(1\) contrast is the \(a=b\) gauge. GH \(\kappa\) is higher because GIG puts more relative variance on \(Y\) (\(\mathrm{cv}^2=1.39\) vs NIG \(0.64\) at \(d=50\)). \(\kappa_{\mathrm{lev}}\) agrees; extra \((p,a,b)\) flexibility does not create a linear signal.

VG \(\kappa\) is smaller because the density clamp binds. This is not the moment floor that keeps \(E[Y]\) finite. For Gamma mixing, \(E[Y]=\alpha/\beta\) is finite for every \(\alpha,\beta>0\). The clamp restricts the marginal density of \(X\). Near \(x=\mu\),

\[ f(x)\;\propto\; q(x)^{\alpha-d/2}, \qquad q(x)=(x-\mu)^\top\Sigma^{-1}(x-\mu). \]

When \(\alpha\le d/2\) this diverges at the mode — an unbounded likelihood, the same degeneracy as a Gaussian mixture with a vanishing component (Day1969). alpha_min='density' is \(\alpha\ge d/2+\varepsilon\) (\(\varepsilon=0.1\)), so the density stays bounded. It is an estimand restriction, not an arithmetic floor.

Three other VG guards are easy to confuse with it:

Guard	Where	What it does	What it does not do
ALPHA_MOMENT_MARGIN	prior \(E[1/Y]=\beta/(\alpha-1)\)	keeps that inverse moment finite when \(\alpha\le 1\)	does not touch \(E[Y]=\alpha/\beta\); does not bound \(f(x)\)
B_POST_FLOOR	E-step \(b_{\mathrm{post}}\)	keeps \(E[1/Y\mid x]\) from overflowing	leaves the likelihood unbounded; EM can still park at a spike
alpha_min='inverse_moment'	M-step \(\alpha\ge d/2+1+\varepsilon\)	density and \(E[1/Y\mid x]\) classically bounded	tighter than 'density'; not used in this study
alpha_min='density'	M-step \(\alpha\ge d/2+\varepsilon\)	\(f(x)\) bounded at \(\mu\)	does not pin \(E[Y]\)

The finite-mean constraint \(E[Y]=\beta/(\alpha-1)\) (\(\alpha>1\)) belongs to InverseGamma / NInvG, not to VG.

Gamma mixing has \(\mathrm{cv}^2=1/\alpha\). The density clamp therefore caps \(\mathrm{cv}^2\le 2/(d+0.2)\). Fitted \(\alpha\) sits exactly on that floor: \(2.6\) at \(d=5\), \(5.1\) at \(d=10\). Hence \(\mathrm{cv}^2=0.385\) and \(0.196\), against NIG \(1.91\) and \(1.11\). VG then inflates \(\kappa_{\mathrm{lev}}\) (\(0.048\) vs NIG \(0.018\) at \(d=10\)) to recover some skewness; \(\kappa=\kappa_{\mathrm{lev}}\,\mathrm{cv}^2\) still comes out about half of NIG. Matching NIG’s \(\mathrm{cv}^2\approx 1.11\) at \(d=10\) would need \(\alpha\approx 0.9\), below \(d/2=5\).

Cold-start VG with alpha_min=None on the same universes drops below the floor and the \(\gamma\)-energy lines up with NIG:

\(d\)	\(\alpha\)	vs \(d/2\)	\(\kappa_{\mathrm{lev}}\)	\(\mathrm{cv}^2\)	\(\kappa\)
5	1.16	\(<2.5\)	0.0046 (NIG \(0.0043\))	0.86	0.0040
10	1.56	\(<5\)	0.022 (NIG \(0.018\))	0.64	0.014

After \(E[Y]=1\), unconstrained VG \(\gamma\)-energy matches NIG and \(\mathrm{Var}(Y)\) is in the same order. The residual \(\mathrm{cv}^2\) gap (Gamma vs InverseGaussian tails) is a family difference, not a scale bug. Those fits have an unbounded density at \(x=\mu\), which is why the study clamped \(\alpha\). The reported \(\kappa_{\mathrm{VG}}=0.009\) at \(d=10\) is the clamped estimand. Compare it to NIG at the same \(d\) (\(\kappa=0.020\)), not to the \(d=50\) NIG headline.

None of the three families clears the sign-flip floor. Extra subordinator flexibility still does not create a linear signal.

Tracker versus quadratic channel (\(d=50\))

	Sample	Model / simulation
\(\mathrm{Var}(\hat Y)\)	14.67	\(\mathrm{Var}(Y)+E[Y]/\tilde q=13.99\)
\(\mathrm{skew}(\hat Y)\)	0.79	0.51
ACF\(_1\), ACF\(_{21}\) of \(\hat Y\)	0.012, 0.005	0
\(\mathrm{corr}(\hat Y,\,E[Y\mid X])\)	0.203	0.205
\(\mathrm{corr}(E[Y\mid X],\,q_\perp)\)	0.989	0.998
\(\mathrm{corr}(E[Y\mid X],\,\text{x-sec. disp.})\)	0.933	—
\(\mathrm{corr}(E[Y\mid X],\,\text{21-day RV})\)	0.580	—
\(\mathrm{corr}(\hat Y,\,\text{21-day RV})\)	0.071	—

What worked (model side). The linear tracker is consistent with the i.i.d. mixture: variance, ACF near zero, and the split between \(\hat Y\) and \(E[Y\mid X]\) match a draw from the fitted model.

What failed (as a clock). \(\hat Y\) is uncorrelated with realized vol. The posterior mean is essentially \(q_\perp\) (corr \(0.99\)) and is the volatility clock (corr \(0.58\) with 21-day RV, \(0.93\) with cross-sectional dispersion).

The i.i.d. misspecification the plan expected to see in \(\hat Y\) (clustered tracker residuals) is not there. It lives in \(E[Y\mid X]\), which inherits the persistence of realized vol. Phase 3 therefore scores the quadratic channel against RV, and does not expect a clustered \(\hat Y_t\).

H3 — max-skewness

\(t^\dagger<0\) for every NIG and GH fit; \(t^\dagger\approx 0\) for VG (Gamma equality). Model max-skewness is \(w^\star\) in all cases, including the GIG check on these four GH fits.

Sample third moments do not recover it. Direct skew maximisation, 20 L-BFGS starts:

\(d\)	In-sample max skew	Tracker sample skew	\(\cos\angle\)	OOS max	OOS tracker
10	3.24	0.59	0.13	\(-0.023\)	0.24
50	8.31	0.79	\(-0.20\)	0.20	\(-0.047\)

Equal-weight, min-variance, and PC1 all have \(w^\top\gamma<0\) and negative sample skew (index leverage). The tracker is the only listed portfolio with positive sample skew, and it is not the sample maximiser. In-sample skew \(8.3\) at \(d=50\) collapses out of sample. Third moments overfit.

H3 split. True on the model. False as a sample-moment recipe.

Anatomy of \(w^\star\) at \(d=50\)

Unit-gross long/short, 24 long / 26 short, long share \(0.45\), Herfindahl \(0.032\). Correlation of weights with market beta is \(0.065\) — not a market tilt. \(\mathrm{corr}(\hat Y, m_t)=-0.37\). Location \((w^\star)^\top\mu=-0.987\) against sample \(E[P^\star]=0.013\), so \(E[\hat Y]=1=E[Y]\): the location term eats the \(Y\)-premium and the raw tracker return is near zero. That is the no-near-arbitrage sign from Proposition 2, in a regime where the proposition’s tail bound is vacuous (\(\tilde q\) is small: \(P(\hat Y\le -1)=0.28\) vs bound \(0.58\)).

Block-bootstrap cosine 5/50/95 \(=0.58/0.73/0.82\): a moderately tight cone around a direction the sign-flip calls odd-moment noise. CVaR at \(5\%\) on the \(\hat Y\) scale is \(7.50\) long / \(9.14\) short (std \(\approx 3.8\)). Mild asymmetry, Gaussian residual, not a GIG floor.

Split \(\gamma=g\mathbf{1}+\delta\): \(g=-8.5\cdot 10^{-4}\), \(\lVert\delta\rVert^2=2.1\cdot 10^{-5}\) vs \(\lVert\gamma\rVert^2=5.7\cdot 10^{-5}\). PC1 carries \(5.6\%\) of \(\tilde q\). Not market-aligned skewness — consistent with isotropic estimation noise.

Phase 2 — dimension sweep (H2)

Five nested seeds. \(d=468\) is the full panel, so seeds coincide.

\(d\)	Mean \(\hat{\tilde q}\)	SE	Sign-flip 95%	Mean \(\hat\kappa\)	Mean \(\kappa_{\mathrm{index}}\)	PC1 share	Small-\(\lambda\) 10%
5	0.012	0.006	0.010	0.014	0.009	0.39	0.024
10	0.020	0.004	0.018	0.019	0.011	0.27	0.024
25	0.035	0.007	0.040	0.025	0.013	0.14	0.075
50	0.060	0.015	0.085	0.039	0.012	0.086	0.085
100	0.107	0.014	0.164	0.061	0.013	0.045	0.080
200	0.217	0.011	0.298	0.111	0.013	0.019	0.095
468	0.509	0	0.808	0.233	0.012	0.007	0.106

\(\hat{\tilde q}_d\) grows, roughly linearly in \(d\), and does not saturate. That is the shape of the idiosyncratic-\(\delta\) branch in the theory note — except every point sits at or below the sign-flip floor. (The \(d=5,10\) means slightly exceed the seed-0 null; they do not exceed a seed-matched null, and PC1 shares at \(d=5\) range \(0.001\)–\(0.76\) across seeds.) The index ceiling \(\kappa_{\mathrm{index}}\approx 0.01\) is flat in \(d\).

Attribution is the opposite of market-aligned skewness: PC1’s share of \(\tilde q\) falls from \(0.39\) to \(0.007\). The equicorrelation ceiling \(g^2/(\bar\sigma^2\rho)\) is \(0.004\)–\(0.011\), the same order as \(\kappa_{\mathrm{index}}\). \(\lVert\delta\rVert^2/\lVert\gamma\rVert^2\) rises \(0.18\to 0.41\): the extra \(\tilde q\) lives in \(\delta\), which is where \(\gamma\)-estimation noise lives. The small-\(\lambda\) \(10\%\) column is the share of \(\tilde q\) on the weakest tenth of the spectrum; it rises, slowly, as one would expect if the “signal” is noise in ill-conditioned directions.

H2 is false in both intended readings. There is no saturating market-skewness signal, and the growing \(\tilde q_d\) is not recoverable idiosyncratic skewness. It is the \(d\)-dependent null. Cross-section does not buy a linear clock.

Phase 3 — online EM (H4)

Warm start: NIG on 2016–2017 (504 days), \(d=50\) seed 0. Warm-window \(\tilde q=0.368\), \(\kappa=0.182\) — already higher than the full-sample \(0.075/0.048\). Online period 2017-12-14 → 2026-02-09 (\(T=2048\)). No re-gauging. Half-life \(h\) with \(w=1-2^{-1/h}\); \(1/t\) is the consistency check (Online EM Algorithm).

\(h\)	\(\tau\)	\(n_{\mathrm{eff}}\)	Mean \(\kappa_t\)	Mean \(E[Y]_t\)	Turnover	\(\cos_{21}\)	corr\((\hat Y,\mathrm{RV})\)	corr\((q_\perp,\mathrm{RV})\)	corr\((q_\perp,\mathrm{disp})\)
21	0	61	1.06	4772	0.084	0.59	\(-0.14\)	0.14	0.61
63	0	182	0.41	27	0.055	0.79	\(-0.04\)	0.01	0.44
126	0	364	0.19	3.3	0.039	0.88	\(-0.05\)	0.37	0.80
252	0	727	0.11	1.9	0.027	0.94	\(-0.03\)	0.55	0.90
504	0	1454	0.084	1.6	0.017	0.98	\(-0.01\)	0.60	0.93
21	0.1	61	0.52	1.3	0.123	0.47	\(-0.22\)	0.54	0.89
\(1/t\)	0	2552	0.079	1.5	0.010	0.99	\(0.00\)	0.61	0.94

The \(1/t\) schedule ends at \(\kappa=0.051\) vs static \(0.048\) — consistency holds. A secondary \(d=20\) unshrunk run: \(h=21\) \(q_\perp\) corr with RV \(=-0.05\); \(h=252\) gives \(0.50\). Same pattern, smaller panel.

What H4 got right. Short \(h\) inflates gauge-invariant \(\kappa_t\) (\(22\times\) at \(h=21\)) and explodes the gauge itself (\(\overline{E[Y]}_t=4772\)). Direction cosine over 21 days drops to \(0.59\). The linear tracker stays uncorrelated (or anti-correlated) with RV at every \(h\).

What H4 got wrong as a clock recipe. Shrinkage \(\tau=0.1\) at \(h=21\) pins the gauge (\(\overline{E[Y]}=1.3\)) and rescues \(q_\perp\) (corr \(0.54\)) at the cost of turnover. It does not rescue \(\hat Y\). Long \(h\) or \(1/t\): \(\kappa_t\) near the static value, stable \(w^\star\), and \(q_\perp\) matches the static posterior as a vol proxy (corr with 21-day RV \(0.55\)–\(0.61\); with cross-sectional dispersion \(0.90\)–\(0.94\)). Smoothing \(q_\perp\) at half-life \(5\) lifts RV-corr to \(\sim 0.71\)–\(0.75\).

Fairness versus RV. EWMA of \(m_t^2\) at \(h=21\) has corr \(0.66\) with 21-day RV (same window family). Unshrunk \(h=21\) \(q_\perp\) loses that comparison (\(0.14\)). The model’s quadratic channel is competitive only once \(n_{\mathrm{eff}}\) is hundreds of days — once the online model is close to the static fit.

Open problem (a) of the theory note, on this panel: the tracker plus online EM is not a real-time activity index comparable to realized variance. \(q_\perp\) is, and it does not need \(\gamma_t\).

Worked / did not work

Claim	Result
Tracker MSE \(=E[Y]/\tilde q\) in simulation	Holds to \(<1\%\).
Linear-Bayes MSE \(=\mathrm{Var}(Y)/(1+\kappa)\)	Holds to \(<1\%\).
\(\mathrm{corr}(\hat Y,Y)=\sqrt{\kappa/(1+\kappa)}\)	Holds at \(\kappa\ge 0.43\); \(6.5\%\) high at the panel’s own \(\kappa\approx 0.05\).
Posterior mean tracks \(Y\) at \(d=50\) even if \(\gamma=0\)	Holds (corr \(0.95\) in simulation).
\(\hat{\tilde q}\) is a usable point SNR	Fails. Upward-biased; at the panel’s \(\kappa\) the bias is the whole signal. Always report a matched null.
Fitted \(\kappa\ll 1\) on daily equities (H1)	Holds. Indistinguishable from sign-flip.
Cross-section lifts \(\kappa\) above the index (H2, first reading)	Point estimates do; the null does too, faster.
\(\tilde q_d\) saturates on the market factor (H2, second reading)	Fails. Grows linearly; PC1 share falls.
Growing \(\tilde q_d\) is recoverable idiosyncratic skewness	Fails. It is the \(d\)-dependent null.
Model max-skewness \(=w^\star\) (H3, model)	Holds on every VG/NIG/GH fit, including the GIG check.
Sample skew maximiser recovers \(w^\star\) (H3, sample)	Fails. Cosine \(0.13\) / \(-0.20\); OOS collapse.
Tracker has positive sample skew; index portfolios do not	Holds, and is the only listed portfolio that does.
Short online half-life inflates \(\kappa_t\) and churns \(w^\star\) (H4, noise)	Holds, including gauge explosion at \(h=21\), \(\tau=0\).
Short half-life extracts a better linear clock (H4, recipe)	Fails. \(\hat Y\) uncorrelated with RV at every \(h\).
\(q_\perp\) / \(E[Y\mid X]\) tracks realized vol	Holds, once \(n_{\mathrm{eff}}\) is hundreds of days (corr \(0.55\)–\(0.61\) with 21-day RV; \(0.90\)–\(0.94\) with dispersion).
Online EM + tracker is a real-time activity index	Fails on this panel.
Proposition 2 tail bound is informative at fitted \(\tilde q\)	Fails (vacuous). Location sign is consistent with no-near-arbitrage.
Extra GIG parameters (GH vs NIG) create a linear signal	Fails. \(\kappa_{\mathrm{lev}}\) agrees; GH \(\kappa\) is higher via \(\mathrm{cv}^2\), not via the \(a=b\) gauge \(e=0.10\).
VG \(\kappa\) smaller than NIG is leftover scale or a simulation bug	Fails as a diagnosis. \(\kappa_{\mathrm{lev}}\) and \(\gamma\)-direction agree once the gauge is removed; the gap is the binding alpha_min='density' clamp (\(\alpha=d/2+\varepsilon\)). Unconstrained VG recovers NIG \(\gamma\)-energy and a closer \(\kappa\), at the cost of an unbounded density.

None of this is a failure of the mathematics. Fitted daily-equity \(\gamma\) is small, the sign-flip says it is consistent with zero, and Proposition 1 then says no linear payoff can reveal \(Y\). That is partial revelation in the \(\kappa=O(10^{-2})\) regime the no-near-arbitrage heuristic predicted, taken all the way to “no linear revelation.”

Mencía & Sentana’s skewness fund is, on this panel, a noisy long–short with no clock content. The EM E-step’s \(E[Y\mid X]\) remains a usable latent-vol diagnostic, as the finance tutorials already treat it.

Limitations

Current-constituent panel (survivorship); large caps only; a single common subordinator; i.i.d. GH (the ACF of \(\hat Y\) being \(\approx 0\) is consistent with that, the ACF of \(E[Y\mid X]\) is not). Full-\(\Sigma\) at \(d=468\) has \(T/d\approx 5.5\). Sign-flip nulls at \(d\ge 100\) use \(B=20\). GH was a nested continuation, not a cold start, and was not run at \(d>50\). VG was \(d\le 10\) only, and the reported \(\kappa\) uses alpha_min='density' (the unconstrained MLE wants \(\alpha<d/2\)). VIX and high-frequency RV were not used; panel-based proxies suffice for the mechanism question. Multi-subordinator models and nonlinear payoffs (open problems (b), (c) in Subordinator-tracking portfolios) were out of scope.

What this is not asking normix to do

Report \(\tilde q\) and \(\kappa\) on fitted models only with a matched null floor. A tracker-versus-Bayes notebook is still worth teaching the channel split; the equity punchline is \(q_\perp\), not \(w^\star\). No package change. The linear tracker is a clean theoretical object that this panel does not support as a data-analysis tool.