papermomentummachine-learningworking-paper

Moskowitz, Sabbatucci, Tamoni & Uhl (2025) — "Nonlinear Time Series Momentum"

The theoretical-and-empirical paper that introduces a nonlinear signal-to-position mapping for Time-Series Momentum, motivated by the Ferson-Siegel Optimal Weighting from FS (2001) and validated by a neural network trained to maximize out-of-sample Sharpe ratio. Working paper, December 17, 2025.

Citation

Moskowitz, T.J., Sabbatucci, R., Tamoni, A., & Uhl, B. (2025). "Nonlinear Time Series Momentum." Working paper, December 17, 2025.

Authors

Core contribution

The standard Time-Series Momentum literature uses either a linear weighting f(s_t) = s_t or a binary weighting f(s_t) = \text{sign}(s_t). This paper shows:

  1. Theory: A mean-variance investor's optimal weight on a noisy expected-return signal is a nonlinear S-curve — linear in the middle, saturating at the extremes — exactly the Ferson-Siegel Optimal Weighting.
  2. Empirical ML: A neural network trained to maximize out-of-sample Sharpe recovers a function f that closely matches the theoretical S-curve.
  3. Performance: NLTSMOM outperforms both linear and binary TSMOM on Sharpe ratio and out-of-sample R^2 across daily/weekly/monthly frequencies and 1M/3M/12M lookbacks, in 8 equity index, 24 commodity, and 21 rates/FX futures.
  4. Tail hedging: Most of NLTSMOM's outperformance is concentrated in down markets — consistent with the FS intuition that muting extreme signals is most valuable when signals are noisiest.
  5. Robustness: The result holds when applied to long-short equity factors (Fama-French; the 153 factors in Jensen, Kelly & Pedersen, 2023), and when the function is estimated non-parametrically (Nadaraya-Watson).

Key empirical results

Strategy Daily, 12M lookback (Sharpe)
Linear 0.70***
Binary 0.83***
F&S Nonlinear 0.83***
Empirical NL (ANN) 0.84*** (statistically distinguishable at p < 0.03)

Appraisal-ratio decomposition (vs. cross-sectional momentum at the same horizon):

  • Volatility-scaling alone: ~0.27 incremental Sharpe
  • Nonlinear mean forecasting: ~0.44 incremental Sharpe

Theoretical foundation

The optimal weight, after Ferson & Siegel (2001), for a perfect expected-return signal with \mu(s_t) = s_t and unit-noise variance, normalized to \text{var}(f_{FS}(s)) = 1:

f_{FS}(s) = \lambda \, s / \sigma_\varepsilon^2, \quad \lambda \approx 0.394

with concave shape for s > 1 and convex shape for s < -1. See Ferson-Siegel Optimal Weighting for full derivation.

Implementation recipe (from the paper)

# Simplified ANN training to recover f(s_t)
# - Symmetric training data: include (-s, -r) pairs
# - Loss: negative Sharpe ratio of f(s) * r_{t+1} / sigma_{t+1}
# - Architecture: 1-2 hidden layers, 2-16 nodes, tanh activation
# - Regularization: L2 (1e-4 to 1e-1), dropout 25-50%
# - Optimizer: Adam, learning rate 1e-6 to 1e-3, ~50 epochs
# - Constraints: f(-s) = -f(s), f(0) = 0, var(f(s)) ≈ var(s)

See also