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

Yuda Bi; Vince D Calhoun

q-fin.CPMar 14, 2026

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

Authors: Yuda Bi, Vince D Calhoun

Organizations: Center for Translational Research in Neuroimaging and Data Science (TReNDS), Atlanta, Georgia 30303, USA · Georgia State University, Georgia Institute of Technology, and Emory University, Atlanta, Georgia 30303, USA

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.

Figures & tables

Rival explanation	Primary discrimination	Outcome in this manuscript
Bare intrinsic persistence or autocorrelation-only inheritance	M0 versus M2, placebo surrogates, quiet-regime ACF collapse, field-stripped residual ACF, disjoint weekly reconstructions, and anchored holdouts	Strongly disfavored
Purely mechanical overlap between VIX and $\psi_{1}$	Mechanical/informational decomposition, standalone field tests, and recipe sensitivity across freeze and weighting choices	Strongly disfavored dynamically
Observed VIX as the exact hidden coordinate behind projected memory	Exact two-dimensional linear-Gaussian comparison and projected-kernel timescale check	Only partially supported
VIX-orthogonal residual as an operational predictor	Q2-versus-Q3 future-VIX test across 30-, 60-, and 90-day horizons	Not supported

Table 1: Layered attribution logic. The strongest claim is the one that survives the largest stack of discrimination tests.

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.

Model	BIC	$\Delta$ BIC vs M2
M_RS,c+VIX	-33011.9	-765.4
M_RS,c	-32274.4	-27.9
M2’	-32255.3	-8.7
M2	-32246.5	0.0
M3	-32230.0	+16.5
M0	-32137.5	+109.0

Table 2: Core one-dimensional and hybrid stochastic models on the aligned daily sample.

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

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.

Horizon	Mean Q2	Mean Q3	Mann–Whitney $p$	Rank-biserial
30 days	0.078	0.083	0.183	0.021
60 days	0.082	0.135	0.736	-0.015
90 days	0.110	0.141	0.986	-0.052

Table 3: Future VIX changes after low-VIX days split by the sign of the orthogonal residual. Positive rank-biserial values would favor the proposed Q2 signal.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

$W$	$\theta_{0}$	$\tau_{0}$	$\theta$	$\tau_{\mathrm{cond}}$	$\beta$	$\beta/\theta$	SCPA	$\Delta$ BIC
30	0.00955	104.67	0.02927	34.17	0.00982	0.336	0.674	104.0
45	0.00553	180.79	0.02093	47.78	0.00706	0.337	0.736	99.4
60	0.00335	298.10	0.01640	60.99	0.00572	0.349	0.795	109.0
90	0.00194	516.68	0.01317	75.93	0.00488	0.370	0.853	153.6
120	0.00119	838.72	0.01036	96.55	0.00390	0.376	0.885	165.4

Appendix

Table 4: Exact window-size robustness of M0 and M2, recomputing

\psi_{1}

directly from raw returns at each window. SCPA denotes

1-\tau_{\mathrm{cond}}/\tau_{\mathrm{auto}}

Field	$\theta$	$\tau_{\mathrm{cond}}$	$\beta$	$\Delta$ BIC vs M0 (decomp. sample)
Actual $\log(\mathrm{VIX})$	0.01723	58.05	0.00623	108.6
Mechanical-only proxy	0.00393	254.18	0.00026	-8.3
Informational residual	0.00850	117.67	0.00471	78.6

Appendix

Table 5: M2 fits with actual, mechanical-only, and informational-only fields on the decomposition sample.

Recipe	Mechanical fraction	Informational fraction	$\Delta$ BIC mech-only	$\Delta$ BIC info-only
Median freeze, equal weights	0.773	0.227	-8.35	78.59
Mean freeze, equal weights	0.776	0.224	-8.36	78.73
Pre-2016 median freeze, equal weights	0.779	0.221	-8.34	78.62
Median freeze, inverse-volatility weights	0.815	0.185	-8.27	77.71
Median freeze, volatility-share weights	0.721	0.279	-8.38	79.57

Appendix

Table 6: Sensitivity of the mechanical/informational decomposition to volatility-freezing and weighting choices. The static split is recipe dependent, but the standalone dynamic conclusion is invariant across all tested variants.

Construction	$n_{\mathrm{obs}}$	$\Delta$ BIC M2 vs M0	$\tau_{0}$ (trading days)	$\tau_{\mathrm{cond}}$ (trading days)	SCPA
Daily overlapping baseline	4973	109.0	298.10	60.99	0.795
Disjoint weekly $\psi_{1}$	976	151.2	8.24	5.88	0.286
Disjoint weekly mean correlation	976	140.1	8.43	6.05	0.282
Disjoint 60-day blocks, end-of-block VIX	77	1.7	89.48	64.69	0.277
Disjoint 60-day blocks, block-mean VIX	77	-3.4	89.48	75.05	0.161

Appendix

Table 7: One-dimensional non-overlapping robustness checks. The disjoint weekly reconstructions preserve strong support for M2, whereas same-horizon 60-day block reconstructions are retained only as low-power boundary checks.

Split date	$n_{\mathrm{train}}$	$n_{\mathrm{test}}$	M0 test ll/obs	M2 test ll/obs	M2 $-$ M0 test gap	M2 test/train ratio
2010-01-01	1451	3522	3.1931	3.2000	0.0070	0.964
2012-01-01	1955	3018	3.2227	3.2373	0.0146	0.994
2014-01-01	2457	2516	3.2294	3.2437	0.0143	0.999
2016-01-01	2961	2012	3.2219	3.2346	0.0128	0.995
2018-01-01	3464	1509	3.1653	3.1805	0.0152	0.972
2020-01-01	3967	1006	3.2301	3.2324	0.0023	0.995

Appendix

Table 8: Anchored chronological out-of-sample sweep. For each split date, M0 and M2 are fit on the pre-split prefix and evaluated on the full held-out suffix.

Rolling-median VIX band	Segments $\geq 120$ d	Quiet e-folding lag	Full-sample e-folding lag
$[13,21]$	5	27	69
$[14,20]$	3	27	69
$[15,19]$	0	—	69

Appendix

Table 9: Quiet-regime band sensitivity. The strict daily band

[15,18]

with a 120-trading-day minimum yields zero qualifying segments over 2004–2023. For the baseline rolling-median band

[14,20]

, the quiet e-folding estimate of 27 days carries an episode-bootstrap

95\%

confidence interval of

[18,33]

days.

Parameter or derived quantity	Estimate
$\theta$	0.02348
$\mu_{\mathrm{calm}}$	-0.6021
$\mu_{\mathrm{stress}}$	1.0000
$\beta$	0.00781
$\sigma$	0.00818
$p_{\mathrm{calm}\rightarrow\mathrm{stress}}$	0.01170

Appendix

Table 10: Parameters of the constrained regime-switching plus VIX model M_RS,c+VIX on the aligned daily sample.

Dataset	Winning structure	$\Delta$ BIC vs next-best	$\Delta$ BIC vs decoupled	Kernel timescale (days)
Daily aligned $\psi_{1}$	Feedforward	2.44	125.26	36.46 (full model only)
Naive weekly thinning	Feedforward	4.17	74.45	9.10 (full model only)
Wand-faithful weekly $\psi_{1}$	Bidirectional	0.78	154.10	33.47
Wand-faithful weekly mean correlation	Bidirectional	2.69	143.87	36.46

Appendix

Table 11: Exact two-dimensional model comparisons and projected-kernel timescales across the daily and Wand-faithful datasets. The reported

\Delta

BIC values are relative to the winning model in each panel.

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