The Bounce Has No Direction: Sign, Magnitude, and the Microstructure of Equity Return Predictability — Fourier-Residue Identities, Fejér Sums, and Evidence from US Equity and Cross-Asset Markets, 1993–2026

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Directly actionable for quantitative trading. The FRI decomposition reveals that SPY's famous lag-1 autocorrelation (-0.081, 7.4 SEs) is entirely magnitude-driven (sign channel p=0.11), invalidating naive contrarian directional strategies while supporting volatility-sizing strategies. The lag-3 directional reversal (p=0.02) identifies a partial-adjustment channel invisible to standard ACF tests. Cross-asset results show mean reversion is confined to exchange-traded composites (equity ETFs, Treasuries) and absent in OTC credit, commodities, FX, and crypto—critical for cross-asset strategy design. The subsample persistence diagnostic (R_N≈1 for equities) confirms out-of-sample reliability. The heteroskedasticity-robust z* avoids the 10-12% over-rejection of the homoskedastic Lo-MacKinlay z under GARCH, preventing spurious mean-reversion signals. The paper provides a complete microstructure diagnosis that maps directly to tradeable strategy construction.

The mathematical framework (Fejér identity, FRI decomposition, subsample diagnostic) is elegant but requires careful implementation: k-ary symbol encoding of returns, Fourier character computations over Z/kZ, Fejér-kernel weighted autocorrelation sums, heteroskedasticity weights δ(j), Bartlett standard errors, and binomial z-statistics for the sign channel. The 27 unit tests and Monte Carlo validation add implementation overhead. However, all formulas are closed-form, no iterative optimization or model fitting is required, and the core computations (autocorrelations, variance ratios, sign/magnitude bucketing) are straightforward in Python/R. The main complexity lies in correct finite-sample edge corrections and the heteroskedasticity-robust variance estimator.

All data sourced from Yahoo Finance via yfinance Python package with auto_adjust=True. 27 unit tests and Monte Carlo studies (1000 replications) validate all estimators. Fejér identity verified numerically to <10^-3 on all series. Heteroskedasticity-robust z* calibration confirmed asymptotically N(0,1). However, no GitHub repository is explicitly linked, and the exact Python code is not provided in the extract. The methodology is fully specified with closed-form expressions, making independent replication feasible.