A preprint study reports that a volatility-forecasting model using carefully limited information from other equity markets lowered both reported error measures relative to a one-market-at-a-time HAR benchmark across all eight markets at one-day and five-day horizons. At the 22-day horizon, it lowered both measures in seven of the eight markets. Across the full panel, it had the lowest average MSE and MAE at five and 22 days. The pattern supports a qualified conclusion: cross-market information can add predictive value when its timing and use are tightly constrained, but the model was not a universal winner.
The data covered daily realized-volatility observations for eight major global equity indices from October 24, 2006, through June 28, 2022. The researchers placed them on a union calendar of 4,079 dates, including any date when at least one exchange was active. Forecasts were made directly 1, 5 and 22 union-calendar days ahead, with results assessed on active target days and on the subset of common trading days.
The challenge was timing
With some markets inactive on a union-calendar date, the model used an asymmetric mask. An inactive market was blocked as a temporal and spatial source, but it remained a destination that could receive information from active markets. In practical terms, a market that was closed could still have its forecast informed by markets that were open, while it could not contribute an observation on that date.
The proposed system, called PGA-Trans-HAR, combined an origin-admissible, regularized cross-market prior with data-driven spatial self-attention, a fixed market-specific allocation gate, the asymmetric mask and a frozen direct-horizon HAR anchor. The prior supplied a structured cross-market signal, while the attention and gates controlled how information was allocated across markets.
Timing controls also governed the prior itself. It used 22 input positions, a graph history capped at 252 dates and reuse for at most 20 consecutive exclusive forecast-origin endpoints. Refreshes were indexed by forecast origin and applied in the same way to the one-, five- and 22-day horizons. The design was meant to keep the information available to the model aligned with the point at which each forecast was made.
The evaluation followed a time-ordered split. The first 60% of the data was used for training and the next 10% for validation; the model was then refit from scratch on the first 70% and evaluated once on the final 30%. Predictions from five runs, using seeds 1 through 5, were averaged before losses and test calculations were made.
The gains were uneven
At the shortest horizon, the picture depended on the error measure. PGA-Trans-HAR had the lowest cross-market average MAE, 0.180997, while HAR-KS had the lowest average MSE, 0.089199. PGA-Trans-HAR's daily MSE was 0.090442. The daily ranking therefore changed with the way forecast error was scored.
The advantage was clearer farther out. At five union-calendar days, PGA-Trans-HAR recorded the lowest cross-market averages for both MSE, 0.150782, and MAE, 0.230454. At 22 days, it again had the lowest averages, with MSE of 0.252361 and MAE of 0.289352. These are panel averages, so they summarize the eight markets rather than guarantee the same ranking for each one.
Market-level results show why the qualification matters. Relative to HAR, the largest daily MSE reduction was 7.3% for N225. At five days, OMXSPI's MAE reduction was just 0.1%. At 22 days, HSI moved the other way, with MSE 0.9% higher and MAE 0.3% higher than HAR. The results point to incremental predictive value, not uniform dominance.
A qualified advantage
Formal comparisons of forecast losses, using a HAC-adjusted Diebold-Mariano test, generally pointed in the same direction. PGA-Trans-HAR was favored in 35 of 48 MSE comparisons and 42 of 48 MAE comparisons at one day. The figures rose to 41 and 46 of 48 at five days, then stood at 39 and 40 of 48 at 22 days. At the 5% level, favorable rejections were fewer: 9 MSE and 23 MAE comparisons at one day, 7 and 27 at five days, and 5 and 16 at 22 days.
A separate 90% Model Confidence Set analysis gave a more mixed daily picture. It retained PGA-Trans-HAR in five of eight markets under daily MSE but only one of eight under daily MAE. At five days, it was retained in all eight markets for MSE and seven for MAE. At 22 days, it was retained in all eight for both measures. Yet the weekly-MSE set retained all 11 candidate models in every market, with a mean set size of 11.0, so the procedure did not identify a lone winner there.
The panel and period set clear boundaries on the claim. The test covered eight indices from October 2006 through June 2022 and used direct forecasts at only three horizons. Its rankings also depended on the loss measure: the daily MSE leader was HAR-KS, while PGA-Trans-HAR led daily MAE and both longer-horizon averages. The result is therefore evidence about this panel, period and set of point-loss comparisons, not a blanket result for every market or forecasting task.
Taken together, the study answers a narrow question: whether origin-aligned, constrained information from other markets adds forecast accuracy beyond established volatility models. On the reported evidence, the answer is broadly yes, especially at five and 22 days, but it depends on market, horizon and how error is measured. For readers building international volatility forecasts, the practical message is limited but clear: timing rules and the treatment of inactive markets can be central parts of the model.
Paper data and sources
Original title: Forecasting Global Volatility Across Asynchronous Markets: Incremental Accuracy from Constrained Cross-Market Attention
Authors: Xinlin Zhao, Haotian Qiao, Ziyao Lin
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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