# Subordinator tracking on the S&P 500 Phases 0–3, 2026-08-13. Derivations: {doc}`subordinator_tracking`. 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**, {py:class}`~normix.distributions.normal_inverse_gaussian.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.** {py:class}`~normix.fitting.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 {py:class}`~normix.fitting.eta_rules.EWMAUpdate`. {py:class}`~normix.fitting.em.IncrementalEMFitter` draws random batches, so it is the wrong tool for a chronological clock. {ref}`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 ({ref}`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 ({doc}`../theory/online_em`). | $h$ | $\tau$ | $n_{\mathrm{eff}}$ | 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