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

Wuzhou Lian; Haochun Wang; Yuefan Deng

cs.LGOct 9, 2026

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

Authors: Wuzhou Lian, Haochun Wang, Yuefan Deng

Organizations: University of Michigan · Stony Brook University

Abstract

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.

Figures & tables

Figure 1: Illustration of constant upward bias and its impact on price reconstruction. (a) A small upward shift in the predicted return rate preserves the original temporal shape but induces directional misjudgments (red areas), which are not distinguished by standard MSE. (b) During price reconstruction, the effect of this persistent return shift accumulates through compounding, leading to a progressively widening deviation between the predicted and ground-truth price trajectories.

Figure 2: Illustrative example of MSE-based test-time adaptation on AAPL. The figure shows return predictions from iTransformer with TAFAS without POGT, using both a look-back window and a forecasting horizon of 60 trading days. The vertical dashed line marks the beginning of the forecasting window. In this example, the adapted predictions remain within a narrow range of small positive values while the observed returns frequently change sign.

Figure 3: Overview of Direction-Aware Loss.

Figure 4: Overview of Price Calibration Loss.

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL
Magnificent 7	60	1.617	1.137	0.547	0.754	0.742	0.477	0.530	0.528	0.476	1.502	0.764	0.489	1.784	1.118	0.515
	90	6.164	3.634	1.482	2.062	1.993	0.943	1.152	1.152	0.933	5.265	2.293	1.139	5.761	2.864	1.379
	120	17.717	9.087	2.780	3.879	3.685	1.503	2.030	2.032	1.616	13.147	5.506	2.075	15.585	6.792	3.004
S&P 500 Excerpt	60	0.310	0.236	0.104	0.127	0.126	0.086	0.080	0.079	0.075	0.287	0.138	0.094	0.360	0.239	0.115
	90	1.084	0.758	0.286	0.307	0.293	0.164	0.153	0.152	0.137	0.794	0.319	0.197	1.187	0.717	0.319

Table 1: Forecasting performance comparison of TAFAS. Lower MSE is better. The best result within each backbone and horizon is highlighted in green.

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL
Magnificent 7	60	1.617	1.079	0.782	0.754	0.765	0.502	0.530	0.578	0.469	1.502	0.709	0.489	1.784	1.107	0.646
	90	6.164	3.459	1.847	2.062	1.826	1.013	1.152	1.199	0.888	5.265	1.968	1.106	5.761	2.454	1.299
	120	17.717	7.588	3.900	3.879	3.603	1.552	2.030	2.142	1.463	13.147	5.283	2.654	15.585	5.352	2.783
S&P 500 Excerpt	60	0.310	0.172	0.122	0.127	0.144	0.084	0.080	0.087	0.075	0.287	0.108	0.103	0.360	0.180	0.125
	90	1.084	0.434	0.320	0.307	0.322	0.164	0.153	0.164	0.135	0.794	0.211	0.200	1.187	0.430	0.333

Table 2: Forecasting performance comparison of PETSA. Lower MSE is better. The best result within each backbone and horizon is highlighted in green.

Figure 5: Representative relative-price trajectories across consecutive, non-overlapping forecast blocks. The left and right columns show TAFAS and PETSA, respectively. The dark gray line denotes the ground truth, the red line denotes the frozen base model, the blue line denotes the baseline TTA method, and the green line denotes the corresponding TTA method with DPAL. Vertical dotted lines mark the forecast origins. Each price trajectory is initialized to

1

only once at the beginning of the full sequence and is then accumulated across non-overlapping

H

-day forecast blocks.

(a) TAFAS
Median MSE change relative to the common configuration
$\bm{\alpha\backslash\beta}$	$10^{0}$	$10^{1}$	$10^{2}$	$10^{3}$	$10^{4}$
$10^{3}$	20.01%	18.43%	23.05%	22.75%	22.63%
$10^{4}$	4.58%	3.90%	7.46%	13.25%	14.73%
$10^{5}$	0.66%	0.61%	0.00%	2.46%	7.80%
$10^{6}$	0.27%	0.22%	0.12%	-0.43%	2.07%

Table 3: Robustness of DPAL to the loss coefficients

\alpha

and

\beta

under (a) TAFAS and (b) PETSA. Each cell reports the median relative change in reconstructed-price MSE, expressed as a percentage, from the common configuration

(\alpha,\beta)=(10^{5},10^{2})

used in the main experiments across all evaluated settings within the corresponding TTA framework. Positive values indicate higher MSE and worse performance relative to the common configuration, whereas negative values indicate lower MSE and better performance. The outlined cells identify the common configuration, and the red and green shading denote performance degradation and improvement, respectively.

Framework	Settings	$G_{s}\leq 0.05$	Share	Median $G_{s}$
TAFAS	120	108	90.0%	0.98%
PETSA	120	94	78.3%	1.54%
Combined	240	202	84.2%	1.30%

Table 4: Comparison between the common configuration

(\alpha,\beta)=(10^{5},10^{2})

and the lowest-MSE configuration for each setting. The third column reports the number of settings with

G_{s}\leq 0.05

, corresponding to a reconstructed-price MSE no more than 5% higher than the lowest MSE.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Table A.1 : Ablation results for different combinations of the loss components under the TAFAS and PETSA frameworks. Base denotes the frozen backbone without test-time adaptation. The check-marked columns represent the seven loss configurations. In the PETSA tables, the additional column labeled original reports results using PETSA’s original adaptation objective. Each entry reports the reconstructed price MSE for a given backbone model and forecast horizon

H

, with lower values indicating better performance. The best result (lowest MSE) in each row is highlighted in green, and the second-best result (second-lowest MSE) is marked in blue. All values are rounded to five decimal places, whereas the rankings are determined using the original unrounded results.

Table 11

Table 12

Table 13

Table 14

Table 15

Table 16

Table 17

Model	$H$	Base	original	mse	✓			✓	✓		✓
				dir		✓		✓		✓	✓
				price			✓		✓	✓	✓
iTransformer	60	0.01886	0.01564		0.01422	0.01209	0.01602	0.01211	0.01361	0.01196	0.01157
	90	0.04027	0.02911		0.02804	0.02405	0.03165	0.02404	0.02737	0.02279	0.02004
	120	0.07258	0.04806		0.04782	0.04021	0.05915	0.04006	0.04824	0.03831	0.03643
DLinear	60	0.01248	0.01361		0.01309	0.01091	0.01249	0.01085	0.01268	0.01082	0.01064

Appendix

A.1.1 PETSA Macro and Market Series

Table 19

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL
Magnificent 7	60	1.150	1.139	1.126	1.114	1.113	1.110	1.135	1.134	1.125	1.145	1.129	1.128	1.139	1.127	1.120
	90	1.152	1.141	1.135	1.123	1.122	1.119	1.137	1.136	1.129	1.154	1.136	1.139	1.146	1.133	1.130
	120	1.162	1.152	1.144	1.133	1.132	1.130	1.144	1.143	1.138	1.160	1.144	1.144	1.155	1.144	1.140
S&P 500 Excerpt	60	1.513	1.508	1.488	1.476	1.476	1.472	1.502	1.501	1.489	1.507	1.494	1.486	1.500	1.494	1.482
	90	1.497	1.493	1.476	1.471	1.470	1.466	1.485	1.485	1.477	1.502	1.491	1.483	1.491	1.486	1.475

Appendix

Table A.2.3 : Forecasting performance of TAFAS measured by MSE in the standardized return space. Lower values are better. The best result within each backbone and horizon is highlighted in green.

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL	Base	TAFAS	with DPAL
Magnificent 7	60	0.761	0.755	0.750	0.742	0.742	0.741	0.754	0.753	0.748	0.758	0.751	0.750	0.756	0.750	0.747
	90	0.759	0.754	0.752	0.745	0.744	0.742	0.752	0.752	0.748	0.760	0.752	0.753	0.756	0.750	0.749
	120	0.761	0.757	0.754	0.747	0.746	0.746	0.753	0.753	0.750	0.760	0.753	0.753	0.757	0.753	0.751
S&P 500 Excerpt	60	0.822	0.820	0.811	0.806	0.806	0.805	0.817	0.817	0.812	0.820	0.814	0.811	0.817	0.814	0.809
	90	0.816	0.814	0.807	0.805	0.805	0.803	0.811	0.811	0.808	0.818	0.813	0.810	0.813	0.811	0.807

Appendix

Table A.2.4 : Forecasting performance of TAFAS measured by MAE in the standardized return space. Lower values are better. The best result within each backbone and horizon is highlighted in green.

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL
Magnificent 7	60	1.150	1.144	1.143	1.114	1.123	1.112	1.135	1.136	1.132	1.145	1.136	1.138	1.139	1.137	1.129
	90	1.152	1.144	1.138	1.123	1.123	1.121	1.137	1.137	1.135	1.154	1.142	1.141	1.146	1.137	1.135
	120	1.162	1.153	1.145	1.133	1.133	1.131	1.144	1.145	1.142	1.160	1.150	1.145	1.155	1.147	1.144
S&P 500 Excerpt	60	1.513	1.515	1.490	1.476	1.477	1.472	1.502	1.502	1.494	1.507	1.503	1.489	1.500	1.502	1.485
	90	1.497	1.492	1.478	1.471	1.472	1.466	1.485	1.485	1.481	1.502	1.493	1.488	1.491	1.486	1.477

Appendix

Table A.2.5 : Forecasting performance of PETSA measured by MSE in the standardized return space. Lower values are better. The best result within each backbone and horizon is highlighted in green.

Datasets	$H$	iTransformer			DLinear			FreTS			PatchTST			OLS
Datasets	$H$	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL	Base	PETSA	with DPAL
Magnificent 7	60	0.761	0.759	0.757	0.742	0.745	0.742	0.754	0.753	0.751	0.758	0.754	0.755	0.756	0.754	0.751
	90	0.759	0.756	0.752	0.745	0.744	0.743	0.752	0.752	0.751	0.760	0.755	0.754	0.756	0.751	0.752
	120	0.761	0.757	0.754	0.747	0.747	0.746	0.753	0.754	0.752	0.760	0.755	0.753	0.757	0.754	0.754
S&P 500 Excerpt	60	0.822	0.822	0.813	0.806	0.806	0.805	0.817	0.817	0.814	0.820	0.817	0.812	0.817	0.817	0.811
	90	0.816	0.814	0.808	0.805	0.805	0.803	0.811	0.811	0.810	0.818	0.814	0.813	0.813	0.811	0.808

Appendix

Table A.2.6 : Forecasting performance of PETSA measured by MAE in the standardized return space. Lower values are better. The best result within each backbone and horizon is highlighted in green.

Framework	Settings improved by DPAL	Median improvement	Geometric-mean improvement
TAFAS	102/120	17.24%	18.24%
PETSA	104/120	20.85%	19.66%

Appendix

Table A.2.7 : Scale-consistent reconstructed-price evaluation using training-set inverse standardization. Improvements are relative to Original TTA; positive values indicate lower MSE with DPAL.

Dataset	TAFAS	PETSA
Magnificent 7	-0.659118	-0.521236
S&P 500 Excerpt	-0.571172	-0.322853
Russell 2000 Excerpt	-0.585539	-0.524676
Technology Sector	-0.617923	-0.629437
Consumer Sector	-0.367755	-0.402655
Energy Sector	-0.270304	-0.190842

Appendix

Table A.3.1 : Dataset-level mean log MSE ratios (

K_{d}

). Negative values indicate lower reconstructed price MSE for DPAL than for the original adaptation objective.

			TAFAS		PETSA
Configuration			Best	Second-best	Best	Second-best
Base			1	3	2	3
Original			–	–	0	1
mse	dir	price
✓			0	0	0	0
	✓		5	0	12	30

Appendix

Table A.3.2 : Counts of best and second-best reconstructed price MSE results across the experimental settings under TAFAS and PETSA. Original denotes results using PETSA’s original adaptation objective; the TAFAS baseline is represented by the MSE-only configuration.

	$k$			Lowest MSE Count		Geometric Mean Change
Setting	$H=60$	$H=90$	$H=120$	TAFAS	PETSA	TAFAS	PETSA
Reference	15	30	30	51	56	—–	—–
Intermediate	30	45	60	46	40	+0.48%	+0.86%
Full horizon	60	90	120	23	24	+4.95%	+7.82%

Appendix

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.

Dataset	Number of series	Tickers
Magnificent 7	7	AAPL, MSFT, GOOG, AMZN, NVDA, META, TSLA
S&P 500 Excerpt	20	CDNS, SNPS, KLAC, LRCX, MCHP, MPWR, MRVL, ON, TER, SWKS, STX, WDC, FSLR, VRSN, MSI, ROP, FICO, PTC, ZBRA, APH
Russell 2000 Excerpt	20	MHO, WWD, FIX, AAON, ENSG, CASY, KBR, TEX, UGI, WBS, AWI, SPSC, CNO, MTG, PII, SAIA, LECO, THG, MUSA, FIBK
Technology Sector	20	AMD, AVGO, QCOM, INTC, ASML, MU, TXN, AMAT, CRM, ADBE, TSM, ORCL, ADSK, FTNT, AKAM, NTAP, INTU, NFLX, CSCO, IBM
Consumer Sector	20	CHD, KO, PEP, WMT, COST, TGT, CL, KMB, GIS, HSY, MKC, EL, KR, HD, LOW, TJX, NKE, MCD, SBUX, YUM
Energy Sector	20	XOM, CVX, COP, OXY, EOG, DVN, APA, EQT, RRC, MUR, SM, NOG, SLB, HAL, NOV, VLO, MPC, PSX, KMI, WMB

Appendix

Table B.1 : The eight datasets.

Dataset	Training	Validation	Test
Magnificent 7	2012-05-21–2021-11-24 (2,396)	2021-11-26–2023-04-11 (344)	2023-04-12–2025-12-31 (684)
S&P 500 Excerpt	2012-05-21–2021-11-24 (2,396)	2021-11-26–2023-04-11 (344)	2023-04-12–2025-12-31 (684)
Russell 2000 Excerpt	2013-08-20–2022-04-11 (2,177)	2022-04-12–2023-07-11 (312)	2023-07-12–2025-12-31 (622)
Technology Sector	2012-05-21–2021-11-24 (2,396)	2021-11-26–2023-04-11 (344)	2023-04-12–2025-12-31 (684)
Consumer Sector	2012-05-21–2021-11-24 (2,396)	2021-11-26–2023-04-11 (344)	2023-04-12–2025-12-31 (684)
Energy Sector	2012-05-21–2021-11-24 (2,396)	2021-11-26–2023-04-11 (344)	2023-04-12–2025-12-31 (684)

Appendix

Table B.2: Chronological data splits. The number in parentheses is the number of return observations in the corresponding portion.

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