A Spectral Generalisation of the Variance Ratio: Eigenstructure of Long-Horizon Portfolio Covariance and a Multi-Memory Factor Model of U.S. Equity Returns

Battle Count: 79

Highly relevant to quantitative trading. The paper directly addresses long-horizon equity return dynamics, factor momentum persistence at sub-leading eigenmodes, volatility regime transitions, and cross-sectional structure of return/volatility memory. The five-factor decomposition provides a structured framework for understanding how different eigenmodes carry momentum, mean-reversion, and volatility persistence signals. The late-1980s regime transition finding has direct implications for backtesting period selection and strategy robustness. The channel-distinct beta structure finding suggests that momentum strategies and volatility-targeting strategies load on different cross-sections. However, the paper is primarily diagnostic/descriptive rather than prescriptive for trading strategy construction.

High implementation complexity. Requires: (1) eigendecomposition of cross-sectional covariance matrices at multiple horizons, (2) computation of per-eigenmode variance ratios and eigenvector overlap matrices for both linear and volatility channels, (3) closed-form evaluation of AR(p) variance ratios, fBm scaling, ARFIMA autocovariances via spectral density/FFT, and MSM cascade variance ratios, (4) joint least-squares optimization with 9 global parameters plus 5N per-mode weights using L-BFGS-B with 9 starting points and 3 warm-restart passes, (5) 1000-replicate moving-block bootstrap, (6) rolling-window analysis with 15 windows x 1000 replicates, (7) permutation tests with 10,000 permutations. The MSM cascade evaluation and the multi-start optimization are particularly computationally intensive.

The paper provides a detailed reproducibility manifest (Appendix H) specifying data sources (French data library), bootstrap procedure (moving-block, block size 2520, 1000 replicates), optimisation configuration (L-BFGS-B, 9 starting points, 3 warm-restart passes), and loss-conditional reporting. Code is stated to be available from the author and will be deposited at a public repository (Zenodo or analogous) at submission. All parameters, horizons, and methodology are fully specified. Minor deduction because code is not yet publicly deposited at time of paper.