Forecast collapse of transformer-based models under squared loss in financial time series
Forecast collapse of transformer-based models under squared loss in financial time series
This paper investigates why Transformer-based models often fail in financial time-series forecasting despite their success in structured domains like electricity or traffic. The authors show that under squared loss, the Bayes-optimal predictor for a martingale price process is trivial: a flat trajectory for prices or zero for returns. When the conditional mean is degenerate, increasing model expressivity does not reduce bias but instead amplifies noise through interpolation, leading to higher prediction variance. The paper formalizes this by comparing a simple linear model with a rich class containing interpolating predictors, proving that the latter yields strictly larger expected squared error. Empirical experiments on high-frequency EUR/USD data confirm that a PatchTST model produces larger errors than a linear benchmark across most forecasting windows, supporting the theoretical claim that the collapse is inherent to the learning objective rather than a result of optimization or data limitations.
Highlights
- 1Shows that under squared loss, the Bayes-optimal predictor for financial time series with martingale dynamics is trivial: flat for prices and zero for returns.
- 2Demonstrates that increased model expressivity in such regimes does not reduce bias but amplifies noise, leading to larger prediction errors.
- 3Provides a process-level explanation for the degradation of Transformer-based forecasts on financial time series.
- 4Validates theoretical findings with empirical experiments on high-frequency EUR/USD data, showing Transformer models underperform a simple linear benchmark.
Methods
- MEmpirical risk minimization (ERM) under squared trajectory loss with uniform convergence assumptions.
- MComparison of a simple linear hypothesis class (one-parameter) with a rich class containing interpolating predictors (e.g., nearest neighbors).
- MMartingale theory and self-normalized concentration inequalities for asymptotic analysis.
- MNumerical experiments using PatchTST on high-frequency EUR/USD exchange rate data with 30-second sampling.
Results
- RUnder martingale assumptions, the Bayes-optimal predictor is constant (flat for prices, zero for returns).
- RInterpolating predictors yield expected squared error at least 2Hσ², while the linear predictor achieves Hσ² + O(Hσ²/n).
- RTransformer-based models produce systematically larger trajectory-level errors than a simple linear benchmark on EUR/USD data.
- RThe excess error is attributed to variance from noise reuse rather than bias, consistent with the theoretical mechanism.
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