An arXiv preprint reports that PGA-Trans-HAR ranked best on average in a test of volatility forecasts for eight international equity indices. It recorded the lowest average mean absolute error (MAE) at the one-day lead, and the lowest average MAE and mean squared error (MSE) at the 5- and 22-union-calendar-day leads. MAE and MSE were the study's two measures of forecast loss.
The advantage was clearest at five days
Against the study's HAR benchmark, PGA-Trans-HAR lowered both losses for every market at the 1- and 5-day leads. At 22 days, it lowered both for seven markets. The exception was HSI, where MSE rose 0.9% and MAE 0.3% relative to HAR. The comparison is descriptive and market-specific, so the overall ranking should not be read as a clean sweep across every horizon and market.
The study then compared forecasts pair by pair with HAC-adjusted Diebold-Mariano tests, a time-series test used to assess uncertainty around differences in forecast loss. At the 1-day lead, PGA-Trans-HAR had lower MSE in 35 of 48 comparisons and lower MAE in 42 of 48. Nine MSE and 23 MAE differentials favored it at the 5% level, and no external benchmark had a statistically lower daily loss at that level. The comparisons were multiple pairwise tests without a multiple-testing adjustment, so these counts describe the strength and spread of the evidence rather than delivering one all-purpose verdict.
At the 5-day lead, the advantage was more consistent. PGA-Trans-HAR had lower MAE in 46 of 48 pairings and lower MSE in 41 of 48. Twenty-seven MAE comparisons and seven MSE comparisons produced favorable rejections at the 5% level. One comparison still favored DCRNN-HAR for SPX MAE, with a p-value of 0.045.
A separate 90% Model Confidence Set analysis retained PGA-Trans-HAR broadly at the 5- and 22-day leads, while daily retention was more selective. A Model Confidence Set is a statistical shortlist, so retention does not establish superiority over every model that remains in it.
Designed for markets that do not share a clock
The model's design was built around markets that do not always provide observations on the same dates. Its input window ends at the forecast origin and excludes future observations, while target loss is calculated only for active target nodes. PGA-Trans-HAR combines a rolling ridge-VAR/GFEVD network, masked spatio-temporal attention, a frozen HAR anchor, market-specific gates and a bounded inverse-softplus correction. In plain terms, it mixes a structured cross-market prior with a neural correction while keeping the information set tied to the moment of the forecast.
The data were daily five-minute realized-variance measures for eight major equity indices on a 4,079-date union calendar. A date entered that calendar when at least one exchange was active; closures for individual markets remained missing. The evaluation used a chronological 60%/10% development split, then refit parameters from a fresh initialization on the combined 70% in-sample data for the selected epoch count. That setup was intended to keep the comparison out of future information while matching the asynchronous panel.
Different pieces mattered at different horizons
The component tests point to a horizon-dependent trade-off. Using only the rolling prior in the aggregation reduced daily MSE by 2.58% and MAE by 1.20%. Relative to a fixed 0.5 gate, learned market-specific gates worsened daily MSE by 0.65% and MAE by 0.43%, but reduced MSE by 1.75% and MAE by 1.29% at 5 days, and by 1.92% and 3.04% at 22 days. These are nested ablation comparisons, not a full decomposition assigning one percentage to one isolated component.
Diagnostics across 120 market-horizon-seed permutations found all learned gates below 0.5. The global data-attention weights were 0.389 at 1 day, 0.397 at 5 days and 0.414 at 22 days; corresponding prior weights were 0.611, 0.603 and 0.586. The model therefore leaned more heavily on the prior overall, even as the data-attention share rose at longer leads.
The test has clear boundaries
A robustness check using only common trading days preserved the positive comparison with HAR at all three leads. Relative to HAR, average MSE and MAE fell by 4.93% and 2.58% at 1 day, 2.05% and 1.46% at 5 days, and 2.49% and 1.67% at 22 days. MCS retention was complete at the two longer leads under both losses. Because this panel is a restricted subset, the result supports consistency within the tested data rather than broader generalization.
The paper's evidence still has clear boundaries. It reports omitted covariates, and its point losses do not characterize tail risk or predictive uncertainty. External validation on more markets and asset classes, along with portfolio exercises that include turnover and transaction costs, remains to be done. The document is an arXiv preprint, version 2 dated 5 September 2026, so the result is best read as a historical forecasting comparison, not an answer about economic value.
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
Original paper · Full text