DPAL: A Test-Time Adaptation Loss for Financial Time Series Forecasting

The gist DPAL introduces a Direction- and Price-Aware Adaptation Loss for financial time series forecasting, which combines pointwise return fitting with directional and price-calibration terms to improve test-time adaptation performance across various financial datasets and models.

Motivation

Existing test-time adaptation (TTA) objectives often focus primarily on prediction errors at individual time steps, which is insufficient for financial time series forecasting because they do not fully address the needs of the domain Mean squared error (MSE) measures numerical deviation but does not treat a sign reversal as a distinct type of forecasting error In addition to pointwise return fitting, return direction and the price trajectories reconstructed from sequences of returns are also important This motivates an adaptation objective that jointly incorporates pointwise return fitting, penalties for predictions whose signs differ from the observed returns, and calibration of cumulative price trajectories

DPAL Formulation

The Direction- and Price-Aware Adaptation Loss (DPAL) is formulated as LDPAL = LMSE + αLdir + βLprice The pointwise mean squared error is computed in the normalized return space, but the directional and price-aware terms are computed from returns in the original scale The directional loss Ldir is constructed by taking the negative product of the predicted and observed returns, followed by a ReLU activation function to penalize opposing signs The price calibration loss Lprice penalizes discrepancies between trajectories reconstructed from predicted and observed returns using a Huber loss on the relative price ratio qb,t,c The combined objective uses common weighting configuration α = 105 and β = 102 throughout the main experiments

Experimental Setup

The study evaluates DPAL within existing TSF-TTA frameworks without modifying the source forecaster or the adapter architecture They apply DPAL to both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025), augmenting their original objectives The evaluation uses five source forecasters: PatchTST, iTransformer, DLinear, OLS, and FreTS Models are evaluated at forecasting horizons of 60, 90, and 120 trading days across eight financial datasets

Key Findings

Across a diverse collection of financial datasets and five source forecasters, DPAL reduces reconstructed-price MSE in the large majority of the evaluated settings The results show lower reconstructed-price MSE in most configurations for both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025) Rolling price trajectories further show that the corresponding DPAL variants often stay closer to the ground-truth price paths with less cumulative drift over successive forecast blocks The statistical analysis confirms that DPAL produces a geometric mean reconstructed price MSE ratio below one relative to the original adaptation objective for both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025)

Robustness and Ablation

The evaluation of DPAL to alternative coefficient values shows that small values of α, particularly α = 103, result in substantially higher relative MSE under both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025) The ablation on the price calibration length k shows that the reference setting produces the largest number of lowest MSE results under both adaptation frameworks When k is extended to the complete forecast horizon, the number of lowest MSE results decreases to 23 under TAFAS (Kim et al., 2025) and 24 under PETSA (Medeiros et al., 2025) This common pattern indicates that applying the price calibration constraint to every forecast step is generally less reliable than restricting it to an early portion of the predicted trajectory Overall, these results indicate that DPAL can improve test-time adaptation for financial time series forecasting under the evaluated settings The results show that DPAL achieves lower standardized return errors than the corresponding original adaptation objective in 470 of the 480 comparisons

Conclusion

DPAL is an adaptation objective for financial time series forecasting that combines pointwise return fitting with directional and price-calibration terms, and it can improve test-time adaptation for financial time series forecasting under the evaluated settings The rolling price trajectories further show that the corresponding DPAL variants often stay closer to the ground-truth price paths with less cumulative drift over successive forecast blocks The results indicate that DPAL can improve test-time adaptation for financial time series forecasting under the evaluated settings.

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Conclusion. This work introduces Direction- and Price-Aware Adaptation Loss (DPAL), an adaptation objective for financial time series forecasting that combines pointwise return fitting with directional and price-calibration terms DPAL is used as the adaptation objective in TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025) without modifying the source forecaster or the adapter architecture Across the evaluated settings, using DPAL reduces reconstructed-price MSE in most configurations for both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al., 2025) The rolling price trajectories further show that the corresponding DPAL variants often stay closer to the ground-truth price paths with less cumulative drift over successive forecast blocks Overall, these results indicate that DPAL can improve test-time adaptation for financial time series forecasting under the evaluated settings.

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Limitations. Although the evaluation covers multiple financial datasets, it remains limited to the markets and time periods represented in the collected data In addition, the current formulation of DPAL uses relatively simple directional and price-calibration terms Future work could extend the evaluation to broader market regimes and longer time periods, and explore more flexible directional and price-calibration terms DPAL could also be studied in higherfrequency financial forecasting settings, where return dynamics and adaptation timing may differ from those considered here

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Table A.4: Ablation results for different price calibration lengths under TAFAS and PETSA. The lowest MSE count is the number of times that a setting achieves the lowest reconstructed price MSE among the 120 combinations of datasets, forecasting backbones, and horizons Geometric mean changes are calculated relative to the reference setting.

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B Additional Dataset Details

B.1 Data Source and Dataset Construction

All raw series were obtained from Yahoo Finance through the yfinance package For each dataset, all constituent tickers were downloaded jointly using yf.download with auto adjust=True The requested download interval was specified as [2012-05-18, 2026-01-01), with the starting date included and the ending date excluded After return construction and missing value removal, seven datasets begin on May 21, 2012, whereas the Russell 2000 Excerpt begins on August 20, 2013 because complete observations for all of its constituent series were not available before that date

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B.3 Chronological Data Splits

Each dataset was arranged chronologically and divided into training, validation, and test portions using an approximate ratio of 7:1:2 The first Ntrain observations formed the training portion, the following Nval observations formed the validation portion, and the final Ntest observations formed the test portion

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Table B.1: The eight datasets.

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B.2 Return Construction and Trading Date Alignment

Let Pt,c denote the adjusted closing value of series c on trading day t The daily simple return was calculated using pct change: rt,c = Pt,c − Pt−1,c / Pt−1,c = 1 Returns were stored as decimal values rather than percentages

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B.4 Ablation on the Price Calibration Length

To examine the effect of the truncation length k in the price calibration loss, we conduct ablation experiments under both TAFAS (Kim et al., 2025) and PETSA (Medeiros et al.

1. This improvement involves introducing Direction- and Price-Aware Adaptation Loss (DPAL) into existing Test-Time Adaptation (TSF-TTA) frameworks like TAFAS and PETSA by augmenting their original objectives with directional and price constraints, specifically by implementing the loss component: Ldir = 1/BHC X B b=1 X H h=1 X C c=1 ReLU(−rˆb,h,c · rb,h,c). This allows the adapted forecaster to penalize predictions whose signs oppose observed returns while preserving pointwise return fitting.

2. The DPAL loss includes a price calibration term that addresses cumulative errors by defining a loss Lprice = 1/BkC X B b=1 X k t=1 X C c=1 Huber(qb,t,c, 1), where the relative price ratio qb,t,c is defined as qb,t,c = pˆb,t,c / pb,t-k. This prevents trajectory drift by penalizing multiplicative discrepancies in the relative price path reconstructed by compounding returns over the forecast horizon.

3. The combined adaptation objective is formulated as: LDPAL = LMSE + αLdir + βLprice, where coefficients are set to α = 105 and β = 102 as a common weighting configuration throughout the main experiments. This balanced loss function allows the system to jointly optimize pointwise return fitting with constraints on directional accuracy and long-term price trajectory calibration.

4. The system can now perform test-time adaptation without modifying the source forecaster or adapter architecture, as DPAL is applied within existing TSF-TTA frameworks without modifying the source forecaster or the adapter architecture. This facilitates rapid deployment and fine-tuning of complex models in dynamic financial environments.

5. The improved system can reduce reconstructed-price MSE in most evaluated settings across diverse financial datasets, as evidenced by improvements reported in Table 1 and Table 2 when using DPAL compared to baseline TAFAS and PETSA methods.

6. By utilizing the scale-consistent evaluation method, the improved system can assess robustness by comparing Original TTA and TTA + DPAL under aligned standardization schemes, where Is = (1 − ρs) × 100%, indicating a measure of improvement that is not solely dependent on history-conditioned rescaling.

7. The system can be optimized for stable performance across different loss coefficient choices, as the analysis shows that the common configuration (α=105, β=102) is within 5% of the lowest MSE in 84.2% of cases, suggesting robustness against hyperparameter tuning in this crucial adaptation objective.

Test-time adaptation (TTA) enables time series forecasters to update after deployment as new observations become available. However, adaptation objectives that focus primarily on prediction errors at individual time steps do not fully address the needs of financial time series forecasting. In addition to pointwise return fitting, return direction and the price trajectories reconstructed from sequences of returns are also important. To better account for these properties, we introduce Direction- and Price-Aware Adaptation Loss (DPAL), which combines pointwise return fitting with directional and price-calibration terms. We apply DPAL within existing TSF-TTA frameworks without modifying the source forecaster or the adapter architecture. Across a diverse collection of financial datasets, DPAL reduces reconstructed-price mean squared error in most evaluated settings.