Volatility-Clustering Adaptation for Financial Time Series

Manh Nguyen; Minh Hoang Nguyen; Huu Hiep Nguyen; Van Dai Do; Hung Le

cs.LGSep 29, 2026

Volatility-Clustering Adaptation for Financial Time Series

Authors: Manh Nguyen, Minh Hoang Nguyen, Huu Hiep Nguyen, Van Dai Do, Hung Le

Organizations: Applied Artificial Intelligence Initiative, Deakin University, Australia

Abstract

Time-series foundation models are increasingly adapted to new domains through fine-tuning on target data, under the implicit assumption that more target data yields better forecasts. We show that this assumption can fail in financial forecasting, where individual price changes are difficult to predict, but large moves tend to cluster, creating alternating calm and turbulent periods. Using financial foundation models trained on price bars of open, high, low, close, and volume, we argue that adapting to financial domains requires training signals beyond next-token prediction. We introduce Volatility-Clustering Adaptation (VCA), which augments next-token cross-entropy with a differentiable penalty on the autocorrelation of squared returns, the standard statistical signature of volatility clustering. This additional objective provides a multi-step training signal by matching the resulting dependence structure of autoregressive rollouts to those of the realized future. Across three asset sets and two evaluation conventions, VCA improves adaptation over the pre-trained model, with the strongest gains under the primary evaluation (\textsc{fore}), driven primarily by reduced variance error. Overall, our results suggest that effective financial adaptation requires objectives that capture domain-specific temporal structure beyond token-level prediction.

Figures & tables

Figure 1: Overview of Volatility-Clustering Adaptation (VCA). ➊ Mean ACF of squared returns against lag $k$ on CSI300: the realized (ground-truth) path stays positive and well above the dotted zero-clustering null, the signature of volatility clustering. The frozen model and CE fine-tuning deviate from the ground-truth curve, while VCA matches its shape more closely. ➋ The baseline objective $\mathcal{L}_{\text{CE}}$ is single-step and teacher-forced: it scores the next token against the ground truth but gives no signal about the shape of the multi-step path the model generates at test time. ➌ VCA rolls out autoregressively over the horizon and adds $\mathcal{L}_{\text{ACF}^{2}}$ , which matches the batch-mean ACF of squared returns $\bar{\rho}_{k}$ to the realized $\bar{\rho}_{k}^{\star}$ . VCA’s total objective is $\mathcal{L}_{\text{VCA}}=\mathcal{L}_{\text{CE}}+\lambda\mathcal{L}_{\text{ACF}^{2}}$ .

Figure 2: Meaning of

\hat{\rho}_{k}>0

on a real trajectory (BTCUSDT, 15-minute bars). Blue is the log-return

r_{t}

; red is the rolling mean

|r_{t}|

(12h), a realized-volatility proxy. (a) Real order: volatility moves in sustained stretches, so a large

|r_{t}|

predicts a large

|r_{t+k}|

. (b) Same returns, time-shuffled: the returns have the same values and distribution, but once their order is shuffled, the clustering disappears, and the envelope flattens. The ordering impact is the structure that

\hat{\rho}_{k}

measures and what Eq. 5 trains the model to match.

		$H{=}8$		$H{=}16$		$H{=}32$		$H{=}48$		average over $H$
DS	Method	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	Score	WR
CSI300	Frozen	+0.291	3.4e-3	+0.317	6.6e-3	+0.333	1.4e-2	+0.333	2.3e-2	-	-	-	-
	GARCH	-	-10.5	-	-13.3	-	-7.3	-	-0.6	-	-7.9	-	-
	HAR-RV	-	-6.2	-	-5.2	-	+7.9	-	+15.8	-	+3.1	-	-
	CE	-12.6	+6.7	+7.5	-6.4	-11.8	-11.8	-8.8	-8.5	-6.4	-5.0	-5.7	2/8
	MSE	-16.8	-5.3	+5.9	-13.0	-6.5	-11.8	-1.7	-4.7	-4.8	-8.7	-6.7	1/8

Table 1: Main comparison under fore . The Frozen row (for reference) is raw RankIC and MAE; GARCH(1,1) and HAR-RV forecast only variance, so RankIC and Score are not applicable to them ( - ). Every other row is a relative delta against Frozen in percentage (higher is better), and the last group averages both deltas and the Score over horizons. Bold marks the best fine-tuned method in each column. WR denotes win rate, computed as the fraction of the eight per-horizon columns in which a method has the best result. Per-horizon values are in Appendix Tables 15 – 18 .

Figure 3: Volatility-trend channel, simulated vs. real rollouts (

H{=}32

), one panel per dataset. Simulated is

\mathbb{E}[\hat{\rho}_{1}]

from

r_{t}=\sigma_{0}e^{gt}\varepsilon_{t}

(no genuine clustering) as

g

varies; Real shows the mean

\hat{\rho}_{1}

within bins of the per-window trend rate

\hat{g}

from the predicted rollout, computed as the log-ratio of second-half to first-half squared returns. We use second-half over first-half so that

\hat{g}

has the same sign as

g

, following Proposition 1 (b).

Dataset	Slice	vol(first)	vol(last)	vol(test)	vol(val)	test/first	test/last	val/first	val/last	matched
CSI300	$10\%$	0.0298	0.0192	0.0253	0.0241	0.85 $\times$	1.32 $\times$	0.81 $\times$	1.26 $\times$	first
CSI300	$1\%$	0.0287	0.0182	0.0253	0.0241	0.88 $\times$	1.39 $\times$	0.84 $\times$	1.33 $\times$	first
CSI500	$10\%$	0.0299	0.0213	0.0282	0.0267	0.94 $\times$	1.32 $\times$	0.89 $\times$	1.25 $\times$	first
CSI500	$1\%$	0.0235	0.0196	0.0282	0.0267	1.20 $\times$	1.44 $\times$	1.13 $\times$	1.36 $\times$	first
Crypto	$10\%$	0.0105	0.0042	0.0051	0.0050	0.48 $\times$	1.21 $\times$	0.47 $\times$	1.18 $\times$	last
Crypto	$1\%$	0.0133	0.0051	0.0051	0.0050	0.38 $\times$	0.99 $\times$	0.38 $\times$	0.97 $\times$	last

Table 2: Regime-matched definition. vol(

\cdot

) is the standard deviation of log returns on close price (Appendix C ). test/first and test/last are the ratios used to pick the matched slice (bold: closer to one); val/first and val/last are the same ratios computed from the validation period.The shaded column identifies whether the first or last slice is selected.

Figure 4: Data efficiency across all three methods, Score against the frozen model (

H{=}32

, fore ). Solid lines are the regime-matched slice and dashed the mismatched one, with bands of one standard deviation over 3 seeds. The regime-matched slices are identified in Table 2 . Detailed values are in Appendix Table 11 .

Figure 5: Ablation sweeps under fore , mean over the three datasets, band is one standard deviation of the per-dataset delta.

\Delta_{\text{RankIC}}

and

\Delta_{\text{MAE}}

are the two halves Score averages; the black line at 0 is the frozen backbone. Detailed values are in Appendix Tables 12 and 13 .

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Source	Freq.	#Symbols	Splits	#Samples ( $n_{\text{tr}}/n_{\text{val}}/n_{\text{te}}$ )
Crypto top-20	Binance API	15-min	20	2021–23 / 2024H1 / 2024H2–25	63514 / 10860 / 19308
CSI300	Qlib	daily	515	2015–17 / 2018 / 2019–20	3991 / 1836 / 3557
CSI500	Qlib	daily	937	2015–17 / 2018 / 2019–20	6246 / 2991 / 5749

Appendix

Table 3: Dataset details. Equity universes are the survivorship-free point-in-time membership union sourced via Qlib, and Crypto is from the Binance API.

	Setting	Value
Backbone	Model	Kronos-base ( $102$ M), decoder-only
Backbone	Adaptation	full fine-tuning, no adapters
Optimization	Epochs	$10$
	Learning rate	$5\times 10^{-5}$
	Optimizer	$\beta=(0.9,\,0.95)$ , AdamW ( Loshchilov and Hutter, 2019 )
	Weight decay	$0.1$

Appendix

Table 4: Hyperparameter details.

	ACF 2 gap		Vol. level (close)		Vol. level (Parkinson)
Dataset	gap	Model/Naive	$R^{2}$	Model/Naive	$R^{2}$	Model/Naive
CSI300	0.0392	1.96 $\times$	$-4.62$	5.62 $\times$	$-0.70$	1.70 $\times$
CSI500	0.0394	1.95 $\times$	$-3.17$	4.17 $\times$	$-1.60$	2.60 $\times$
Crypto	0.0417	1.93 $\times$	$+0.02$	0.98 $\times$	$+0.17$	0.83 $\times$

Appendix

Table 5: Headroom on the frozen backbone at

H{=}16

, model error against a naive baseline. ACF 2 : model gap vs. a

\hat{\rho}{=}0

(no-clustering) floor. Vol. level (close/Parkinson): model realized-variance error vs. always predicting the training-set mean. Detailed metrics are in Appendix C .

		$H{=}8$	$H{=}16$	$H{=}32$	$H{=}48$	average over $H$
DS	Method	$\Delta_{\text{QLIKE}}$	$\Delta_{\text{QLIKE}}$	$\Delta_{\text{QLIKE}}$	$\Delta_{\text{QLIKE}}$	$\Delta_{\text{QLIKE}}$	WR
CSI300	Frozen	3.13	3.36	4.65	6.04	-	-
	GARCH	+84.3	+88.5	+92.6	+94.8	+90.1	-
	HAR-RV	+84.3	+88.6	+93.2	+95.3	+90.3	-
	CE	-27.0	-49.4	-90.5	-69.9	-59.2	0/4
	MSE	-257.1	-128.4	-97.1	-14.7	-124.3	0/4

Appendix

Table 6: Secondary comparison, QLIKE, computed under the fore sampling convention. Frozen’s row is the raw QLIKE; every other row is a relative delta against Frozen,

\Delta_{\text{QLIKE}}\%

signed so positive means improvement, with the last column averaging over horizons. Bold marks the best fine-tuned method in each column, and WR is the win rate over the 4 per-horizon columns to its left.

		$H{=}8$		$H{=}16$		$H{=}32$		$H{=}48$		average over $H$
DS	Method	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	Score ${}_{\text{MAE}}$	WR
CSI300	Frozen	+0.291	3.4e-3	+0.317	6.1e-3	+0.333	1.3e-2	+0.333	2.0e-2	-	-	-	-
	GARCH	-	-11.1	-	-22.4	-	-22.1	-	-17.3	-	-18.2	-	-
	HAR-RV	-	-6.8	-	-13.7	-	-4.7	-	+1.8	-	-5.8	-	-
	CE	-12.6	+8.7	+7.5	-1.0	-11.8	-8.3	-8.8	-9.8	-6.4	-2.6	-4.5	2/8
	MSE	-16.8	-1.9	+5.9	-15.6	-6.5	-19.7	-1.7	-12.8	-4.8	-12.5	-8.7	1/8

Appendix

Table 7: Same layout as Table 1 , with

\sigma^{2}

-MAE recomputed under vol in place of fore . RankIC is unchanged from Table 1 , since it is computed once from the fore rollout and shared across conventions; only

\Delta_{\text{MAE}}

and the resulting Score differ. Frozen row is the same as in Table 1 .

$H$	Dataset	CE	MSE	VCA
$8^{\dagger}$	CSI300	.05772 $\pm$ .00042	.05949 $\pm$ .00082	.05749 $\pm$ .00028
	CSI500	.05651 $\pm$ .00037	.05821 $\pm$ .00091	.05419 $\pm$ .00070
	Crypto	.06436 $\pm$ .00007	.06773 $\pm$ .00110	.06456 $\pm$ .00006
16	CSI300	.03517 $\pm$ .00015	.03663 $\pm$ .00013	.03523 $\pm$ .00053
	CSI500	.03804 $\pm$ .00020	.04019 $\pm$ .00061	.03730 $\pm$ .00045
	Crypto	.03958 $\pm$ .00008	.04275 $\pm$ .00091	.03885 $\pm$ .00026

Appendix

Table 8: Aggregate ACF 2 gap by horizon and dataset (lower is better). Bold marks the lowest (best) arm in each row.

{}^{\dagger}H{=}8

gives only

7

squared returns per window, short of the

K{=}15

lags used here (needs at least

2K

squared returns), reported for completeness.

DS	Method	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	Score	$\Delta_{\text{RankIC}}$	$\Delta_{\text{MAE}}$	Score	WR
		$H{=}16$			$H{=}32$
CSI300	Frozen	+0.0181	7.4e-3	-	+0.0042	1.6e-2	-	-
	CE	-70.7	-2.2	-36.4	+364.3	-2.2	+181.1	4/6
	MSE	+18.8	-3.9	+7.4	+181.0	-2.5	+89.2	2/6
	VCA	-248.4	-1649079.1	-824663.8	-512.9	-133064.2	-66788.5	0/6
CSI500	Frozen	+0.0537	9.9e-3	-	+0.0534	2.2e-2	-	-

Appendix

Table 9: Chronos backbone,

H{=}16,32

. Bold indicates best score across fine-tuning methods, underlined marks a cell where

\sigma^{2}

-MAE ( fore ) is orders of magnitude above Frozen;

\Delta

RankIC does not depend on that blown-up price scale. Full per-cell details in Tables 19 – 20 .

Dataset	Slice	$\hat{\rho}_{1}$ (first)	$\hat{\rho}_{1}$ (last)	$\hat{\rho}_{1}$ (test)	$\hat{\rho}_{1}$ (val)	test-closer / val-closer
CSI300	10%	+0.099	+0.102	+0.041	+0.054	first / first
CSI300	1%	+0.070	+0.058	+0.041	+0.054	last / last
CSI500	10%	+0.172	+0.127	+0.047	+0.050	last / last
CSI500	1%	+0.148	+0.074	+0.047	+0.050	last / last
Crypto	10%	+0.306	+0.134	+0.207	+0.210	last / last
Crypto	1%	+0.247	+0.179	+0.207	+0.210	last / last

Appendix

Table 10: Same regime-match exercise as Table 2 , with the raw lag-1 sample ACF of squared returns

\hat{\rho}_{1}(\cdot)

in place of

\mathrm{vol}(\cdot)

. Bold marks a cell where

\hat{\rho}_{1}

disagrees with

\mathrm{vol}(\cdot)

’s match in Table 2 .

	Dataset	Method	$1\%$ -first	$1\%$ -last	$10\%$ -first	$10\%$ -last	$100\%$
RankIC	CSI300	CE	+1.6 $\pm$ 0.2	-18.3 $\pm$ 0.4	-0.1 $\pm$ 1.1	+0.7 $\pm$ 1.1	-11.8 $\pm$ 1.1
		MSE	-16.5 $\pm$ 1.7	-32.9 $\pm$ 4.5	-22.0 $\pm$ 6.4	-34.7 $\pm$ 4.1	-6.5 $\pm$ 1.5
		VCA	-1.3 $\pm$ 1.3	-23.0 $\pm$ 4.2	-0.4 $\pm$ 12.4	-28.7 $\pm$ 9.7	+8.0 $\pm$ 1.0
	CSI500	CE	+2.0 $\pm$ 0.1	-5.9 $\pm$ 0.1	-0.1 $\pm$ 0.1	-3.6 $\pm$ 0.1	-0.6 $\pm$ 0.1
		MSE	-5.1 $\pm$ 0.6	-10.2 $\pm$ 0.1	-6.5 $\pm$ 0.8	-10.0 $\pm$ 0.2	-3.3 $\pm$ 0.8
		VCA	+1.0 $\pm$ 1.0	-0.5 $\pm$ 2.6	+0.2 $\pm$ 1.1	-1.8 $\pm$ 3.0	-1.3 $\pm$ 2.1

Appendix

Table 11: Full per-cell numbers behind Figure 4 , under both evaluation conventions. Bold marks the best method in each column within a (convention, dataset). Shaded cells indicate matched regime .

Figure 6: Same layout as Figure 4 , at

H{=}16

: every fine-tuning method stays well below the frozen model at

1\%

and

10\%

data, matched or mismatched, recovering only at

100\%

Ablation	Value	CSI300		CSI500		Crypto
		RankIC	$\sigma^{2}$ -MAE	RankIC	$\sigma^{2}$ -MAE	RankIC	$\sigma^{2}$ -MAE
Lags $K$	1	+0.3280	0.00788	+0.8615	0.00860	+0.0552	0.00028
	2	+0.0480	0.00614	+0.1242	0.00834	+0.0147	0.00030
	4	+0.0447	0.00626	+0.1437	0.00830	+0.0106	0.00289
	8	+0.0818	0.00626	+0.1582	0.00859	+0.0199	0.00030
Lookback $W$	80	+0.0902	0.00652	+0.0774	0.00824	+0.0072	0.00031

Appendix

Table 12: Per-cell values for ablation at

H{=}16

, under fore , mean over three seeds. No setting is uniformly optimal across all three datasets .

Ablation	Value	CSI300		CSI500		Crypto
		RankIC	$\sigma^{2}$ -MAE	RankIC	$\sigma^{2}$ -MAE	RankIC	$\sigma^{2}$ -MAE
Lags $K$	1	+0.3596	0.01501	+0.8522	0.01882	+0.0482	0.00071
	2	+0.3162	0.01494	+0.8280	0.01955	+0.0445	0.00066
	4	+0.2739	0.01441	+0.8473	0.01813	+0.0491	0.00065
	8	+0.2931	0.01414	+0.8683	0.02492	+0.0569	0.00069
Lookback $W$	80	+0.3105	0.01427	+0.8272	0.02659	+0.0413	0.00058

Appendix

Table 13: Same ablations as Table 12 , at

H{=}32

		$H{=}16$			$H{=}32$
%data	Dataset	CE	VCA	Ratio	CE	VCA	Ratio
100%	CSI300	2.34	3.38	1.4 $\times$	0.90	3.78	4.2 $\times$
	CSI500	4.70	5.77	1.2 $\times$	1.38	4.95	3.6 $\times$
	Crypto	5.08	6.95	1.4 $\times$	1.94	5.77	3.0 $\times$
10%	CSI300	0.49	0.85	1.7 $\times$	0.60	0.81	1.3 $\times$
	CSI500	1.18	1.29	1.1 $\times$	0.96	1.08	1.1 $\times$

Appendix

Table 14: Relative weight drift

\|\theta-\theta_{0}\|_{2}/\|\theta_{0}\|_{2}

from the frozen model, whole-model ( all ) weights, in units of

10^{-2}

, mean over three seeds. Ratio is VCA/CE. The

10\%

rows (shaded) are the regime-matched slice.

DS	Arm	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.2906	+0.0630	+0.0537	+0.0142	0.05812	0.00345	0.00342
	CE	+0.2539	+0.0044	-0.0092	-0.0100	0.05772	0.00322	0.00313
	MSE	+0.2418	-0.0096	+0.0031	-0.0025	0.05949	0.00363	0.00349
	VCA	+0.2731	+0.0331	+0.0115	-0.0067	0.05749	0.00310	0.00312
CSI500	Frozen	+0.8485	+0.0985	+0.0411	+0.0444	0.05537	0.00446	0.00448
	CE	+0.8196	-0.0787	+0.0168	+0.0185	0.05651	0.00419	0.00420

Appendix

Table 15: Raw per-cell results at

H{=}8

, mean

\pm

sd over 3 seeds. No quantity is normalized against the frozen model.

\sigma^{2}

-MAE columns are reported to 5 decimal places; all other columns to 4.

DS	Arm	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.3167	+0.0817	+0.0240	-0.0304	0.03641	0.00662	0.00612
	CE	+0.3405	+0.1118	+0.0304	+0.0183	0.03517	0.00704	0.00618
	MSE	+0.3352	+0.1045	+0.0424	+0.0463	0.03663	0.00747	0.00708
	VCA	+0.3280	+0.0990	+0.0219	+0.0030	0.03523	0.00788	0.00812
CSI500	Frozen	+0.8529	+0.1210	+0.0642	+0.0314	0.03759	0.00901	0.00840
	CE	+0.8146	-0.1335	+0.0138	+0.0373	0.03804	0.00873	0.00807

Appendix

Table 16: Raw per-cell results at

H{=}16

, mean

\pm

sd over 3 seeds. No quantity is normalized against the frozen model.

\sigma^{2}

-MAE columns are reported to 5 decimal places; all other columns to 4.

DS	Arm	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.3328	+0.1078	+0.0164	-0.0309	0.01848	0.01430	0.01260
	CE	+0.2936	+0.0496	+0.0380	+0.0517	0.01771	0.01600	0.01360
	MSE	+0.3111	+0.0759	+0.0163	+0.0262	0.01850	0.01600	0.01510
	VCA	+0.3596	+0.1472	+0.0595	+0.0507	0.01749	0.01500	0.01290
CSI500	Frozen	+0.8638	+0.1946	+0.0933	+0.0321	0.01815	0.01960	0.01710
	CE	+0.8583	+0.1606	+0.0029	-0.0089	0.01773	0.02080	0.01770

Appendix

Table 17: Raw per-cell results at

H{=}32

, mean

\pm

sd over 3 seeds. No quantity is normalized against the frozen model.

\sigma^{2}

-MAE columns are reported to 5 decimal places; all other columns to 4.

DS	Arm	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.3329	+0.1110	-0.0155	-0.0548	0.01618	0.02290	0.01970
	CE	+0.3035	+0.0656	+0.0150	+0.0129	0.01466	0.02490	0.02160
	MSE	+0.3274	+0.1065	+0.0211	+0.0328	0.01489	0.02400	0.02220
	VCA	+0.3122	+0.0842	+0.0246	+0.0182	0.01417	0.02130	0.02020
CSI500	Frozen	+0.8800	+0.2900	+0.1163	+0.0429	0.01583	0.03110	0.02680
	CE	+0.8068	-0.0934	+0.0007	-0.0420	0.01478	0.03150	0.02740

Appendix

Table 18: Raw per-cell results at

H{=}48

, mean

\pm

sd over 3 seeds. No quantity is normalized against the frozen model.

\sigma^{2}

-MAE columns are reported to 5 decimal places; all other columns to 4.

DS	Method	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.0181	+0.0189	+0.0221	+0.0159	0.03561	0.00745	0.00650
	CE	+0.0053	+0.0055	+0.0498	+0.0397	0.03556	0.00761	0.00670
	MSE	+0.0215	+0.0220	+0.0654	+0.0610	0.03551	0.00774	0.00703
	VCA	-0.0269	-0.0265	-0.0059	+0.0092	0.03369	123.0	189.0
CSI500	Frozen	+0.0537	+0.0589	+0.0451	+0.0206	0.03777	0.00994	0.00868
	CE	+0.0351	+0.0425	+0.0740	+0.0601	0.03736	0.01000	0.00883

Appendix

Table 19: Raw per-cell results at

H{=}16

(Chronos).

\sigma^{2}

-MAE columns are reported to 5 decimal places, 1 for the Chronos-VCA rows; all other columns to 4.

DS	Method	price RankIC	price IC	ret. RankIC	ret. IC	ACF 2 gap	$\sigma^{2}$ -MAE fore	$\sigma^{2}$ -MAE vol
CSI300	Frozen	+0.0042	+0.0040	+0.0223	+0.0200	0.01808	0.01630	0.01410
	CE	+0.0195	+0.0192	+0.0873	+0.0933	0.01779	0.01660	0.01500
	MSE	+0.0118	+0.0124	+0.0721	+0.0493	0.01763	0.01670	0.01530
	VCA	-0.0173	-0.0197	+0.0058	-0.0035	0.01719	21.7	51.7
CSI500	Frozen	+0.0534	+0.0458	+0.0990	+0.0622	0.01798	0.02180	0.01900
	CE	-0.0142	-0.0127	+0.0856	+0.0922	0.01768	0.02190	0.01970

Appendix

Table 20: Raw per-cell results at

H{=}32

(Chronos).

\sigma^{2}

-MAE columns are reported to 5 decimal places, 1 for the Chronos-VCA rows; all other columns to 4.

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