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Conditioning on a Volatility Proxy Compresses the Apparent Timescale of Collective Market Correlation

Yuda Bi, Vince D Calhoun

Abstract

We address the attribution problem for apparent slow collective dynamics: is the observed persistence intrinsic, or inherited from a persistent driver? For the leading eigenvalue fraction $ψ_1=λ_{\max}/N$ of S\&P 500 60-day rolling correlation matrices ($237$ stocks, 2004--2023), a VIX-coupled Ornstein--Uhlenbeck model reduces the effective relaxation time from $298$ to $61$ trading days and improves the fit over bare mean reversion by $Δ$BIC$=109$. On the decomposition sample, an informational residual of $\log(\mathrm{VIX})$ alone retains most of that gain ($Δ$BIC$=78.6$), whereas a mechanical VIX proxy alone does not improve the fit. Autocorrelation-matched placebo fields fail ($Δ$BIC$_{\max}=2.7$), disjoint weekly reconstructions still favor the field-coupled model ($Δ$BIC$=140$--$151$), and six anchored chronological holdouts preserve the out-of-sample advantage. Quiet-regime and field-stripped residual autocorrelation controls show the same collapse of persistence. Stronger hidden-variable extensions remain only partially supported. Within the tested stochastic class, conditioning on the observed VIX proxy absorbs most of the apparent slow dynamics.

Conditioning on a Volatility Proxy Compresses the Apparent Timescale of Collective Market Correlation

Abstract

We address the attribution problem for apparent slow collective dynamics: is the observed persistence intrinsic, or inherited from a persistent driver? For the leading eigenvalue fraction of S\&P 500 60-day rolling correlation matrices ( stocks, 2004--2023), a VIX-coupled Ornstein--Uhlenbeck model reduces the effective relaxation time from to trading days and improves the fit over bare mean reversion by BIC. On the decomposition sample, an informational residual of alone retains most of that gain (BIC), whereas a mechanical VIX proxy alone does not improve the fit. Autocorrelation-matched placebo fields fail (BIC), disjoint weekly reconstructions still favor the field-coupled model (BIC--), and six anchored chronological holdouts preserve the out-of-sample advantage. Quiet-regime and field-stripped residual autocorrelation controls show the same collapse of persistence. Stronger hidden-variable extensions remain only partially supported. Within the tested stochastic class, conditioning on the observed VIX proxy absorbs most of the apparent slow dynamics.
Paper Structure (36 sections, 42 equations, 4 figures, 11 tables)

This paper contains 36 sections, 42 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Slow collective observables pose an attribution problem
  3. Financial correlation dynamics provide a stringent test case
  4. The literature gap is not whether memory exists, but what causes it
  5. This paper provides a layered attribution test
  6. Methods
  7. Data, alignment, and construction of the collective observable
  8. Experimental design and evidential logic
  9. One-dimensional stochastic hierarchy and exact daily likelihood
  10. Placebo gate and the formal attribution null
  11. Directional diagnostics and mechanical versus informational overlap
  12. Model-free persistence controls, non-overlapping reconstructions, and out-of-sample evaluation
  13. Window-size robustness and resolution dependence
  14. Exact two-dimensional extension and Wand-faithful weekly reconstruction
  15. Orthogonal residual state variable
  16. ...and 21 more sections

Figures (4)

  • Figure 1: Collective correlation tracks a moving volatility field. (a) The long-run $\psi_1$ series co-moves strongly with VIX over 2004--2023, with the largest excursions concentrated in major crisis episodes. (b) The contemporaneous association is strong but nondegenerate, with $r=0.724$ against $\log(\mathrm{VIX})$ and $r=0.681$ against level VIX. (c) Equal-count VIX bins show that the fitted conditional equilibrium $\mu_{\mathrm{eff}}(\mathrm{VIX})=\mu+(\beta/\theta)\log(\mathrm{VIX})$ tracks the empirical shift of the $\psi_1$ distribution across volatility conditions. Together these panels motivate the attribution problem addressed below.
  • Figure 2: The main attribution result. (a) Same-sample model comparison across the core, hybrid, and auxiliary-field variants centers the main interpretive story on the VIX-coupled OU model: bare OU and quartic alternatives are strongly disfavored, while rare-event hybrids are better read as an added extreme-state channel than as a replacement for the continuous field picture. (b) Under the baseline $W=60$ construction, conditioning on VIX reduces the relaxation time from $298$ to $61$ trading days. (c) One hundred AR-matched placebo fields fail to approach the real-VIX gain (maximum placebo $\Delta$BIC $=2.7$ versus $109$ for real VIX). (d) Level-based Granger tests show dominant VIX$\rightarrow\psi_1$ directionality ($F=15.1$ versus $1.6$ in the reverse direction). These panels together establish the conditional field-proxy attribution that anchors the rest of the manuscript.
  • Figure 3: Independent robustness checks. (a) Persistence collapses both in quiet regimes and after subtracting the fitted VIX-dependent equilibrium: the full-sample e-folding lag is $69$ trading days, the field-stripped residual falls to $42$ days, and the quiet-regime estimate falls to $27$ days. (b) Among auxiliary fields, VIX remains uniquely strong, while MOVE is secondary and TED does not improve on the bare benchmark. (c) On the decomposition sample, the informational residual of $\log(\mathrm{VIX})$ alone retains most of the model-selection gain ($\Delta$BIC $=78.6$), whereas the mechanical proxy alone fails to improve the fit ($\Delta$BIC $=-8.3$); the same sample still yields a $77.3\%/22.7\%$ mechanical/informational variance split. (d) Exact recomputation across $W=30$--$120$ shows that absolute timescales depend on rolling-window resolution, but the field susceptibility $\beta/\theta$ stays in the narrow range $0.336$--$0.376$ and the attribution fraction remains majority-inherited at every window. Together these panels show that the main claim is not an artifact of placebo persistence, auxiliary fields, mechanical overlap, or the baseline window choice.
  • Figure 4: Extensions beyond the main one-dimensional result. (a) On daily aligned data the 2D comparison weakly favors feedforward over bidirectional drift by $\Delta$BIC $=2.44$, whereas the Wand-faithful weekly reconstructions weakly favor bidirectionality ($\Delta$BIC $=-0.78$ for weekly $\psi_1$ and $-2.69$ for weekly mean correlation; negative values favor bidirectional coupling). The projected-kernel timescales remain in the $9$--$36$ day range and therefore do not recover the $15$-day Wand benchmark in any clean way. (b) The orthogonal residual phase space in $(\log\mathrm{VIX},\epsilon_\perp)$ is structured and visually nontrivial. (c) The proposed Q2-versus-Q3 prediction test nevertheless remains null across all tested horizons ($p=0.183$, $0.736$, and $0.986$ at 30, 60, and 90 trading days). The figure therefore frames these sections as constrained extensions rather than as new headline results.