Volatility-Correlation Tensors Have Crisis-Specific Leading Eigenvectors; Return-Correlation Tensors Do Not. A Null-Check Study on CRSP (2014–2024)

Keywords: tensor decomposition, low-rank assumption, financial correlation networks, uncertainty, systemic risk, null check, negative control, crisis eigenvector, CRSP, volatility correlations

TL;DR: Global CP/Tucker on US-equity correlation tensors explains only 16% of variance, but a per-measure null test reveals a crisis-specific leading eigenvector in volatility-type correlat

Abstract: Financial correlation tensors are routinely de- composed under the assumption that a small global rank captures their structure. We stress- test this assumption on a four-way uncertainty- network tensor U ∈ RN ×N ×T ×M built from CRSP daily returns (2014–2024, via WRDS), comparing four correlation-based measures m∈ {return, squared-return, centered-absolute-return, absolute-return}against null-distribution checks. Four findings. First, global CP at rank 16 ex- plains only 16% of variance; per-measure CP at rank 4 explains 28.8% on return correlations but ≤6.3% on the other three. Second, crisis-episode leading-eigenvector agreement (0.86–0.90 across three episodes) fails a null-distribution test on m=0 (p=0.47—the market mode) but passes on m∈{1,2,3}(p<0.001 to 0.010), strictly at the leading eigenvector (2nd-eigenvector null fails everywhere) and only at episode-sized triplets (n ≤5; the effect is absorbed at n=11). Third, crisis-conditional CP at rank 4 achieves 43.3% VE on the 29-window return-correlation subten- sor with only 916 parameters; a null test over 200 random 29-window non-crisis subsets gives null- mean VE 0.27 (4.9σbelow observed; p<0.005), so the compact fit is a genuine signal, not a sample-size artifact. Fourth, a plain EWMA of past correlation slices beats every tensor fit on one-step-ahead forecasting. The paper’s intended service is methodological: null checks on all mea- sures, ranks, and window counts—each with an explicit negative control—separate crisis-specific structure from market-mode artifacts, and this dis- cipline rescues a real finding that a single-measure analysis would have missed.

Submission Number: 49