Choosing this paper's best-likelihood model would steer a risk desk away from its best tail-risk forecasts. Zhao, Wang, Tran and Gerlach find that their least-trained variant delivers the lowest joint VaR-ES loss in 6 of 8 equity indices over January 2020 to May 2026. Yet every full-period density confidence set excludes that variant. This split is the paper's most useful finding for a risk desk. The authors flag it themselves.
A learned score does both jobs
The model builds on a score-driven (GAS) recursion for log-variance: f(t+1) = ω + (α/2){z·NN(z) − 1} + β·f(t). Here NN(z) is the innovation density's negative score, or minus the slope of its log. With Gaussian innovations, the score equals z and the variance update grows quadratically with the shock. With Student-t innovations, the score tends to zero and the update stays bounded.
The replacement is a 1-8-8-1 tanh network with 97 parameters. The authors integrate its learned score to obtain a density over a [−40, 40] grid, keeping the score constant outside it. They rescale that density to have mean zero and unit variance. The same function therefore determines the predictive density used for VaR and ES, along with tomorrow's variance response to today's return. In the tails, the learned score levels off at nonzero constants. That makes the variance update grow approximately linearly with shock size and makes the far tails exponential by construction.
Estimation follows two routes. GAS-NN-2, the two-step version, first uses score matching to fit the score to returns divided by their unconditional standard deviation. With that density frozen, maximum likelihood then fits ω, α and β. GAS-NN-1, the one-step version, takes this solution as its starting point and makes 30 joint likelihood updates to all parameters. Score matching lasts 30 or 200 epochs. The choice of 30 comes from the best two-step checkpoint on the 2016 to 2019 S&P 500 validation sample, which contains 1,006 days.
Forecasts are one day ahead. Daily re-estimation on a rolling window produces 1,562 to 1,641 forecasts per index, including 119 to 126 during January to June 2020.
Which fit wins in the lower tail?
The density results come first. For full-period predictive log-likelihood (PNLL, where lower is better), the 75% model confidence set drops GAS-NN-2 (30) in all 8 markets. It retains GAS-NN-2 (200) and GAS-NN-1 (200-30) in 6 markets apiece. Student-t GAS survives in 3.
At 2.5%, joint VaR-ES loss largely reverses the neural ranking. GAS-NN-2 (30) records 2.1782 for the S&P 500, against 2.2132 for Student-t and 2.2374 for Gaussian. During early 2020, GAS-NN-2 (30) has the lowest joint loss across all 8 markets. Its scores range from 2.7203 for N225 to 3.4616 for GDAXI.
Why does the weakest neural density fit produce the best tail forecasts? On the FTSE in-sample results, GAS-NN-2 (30) has the worst neural likelihood: 1.2999, compared with 1.2858 for the best. The authors describe a sharply peaked density whose outer tails decay more slowly than those of the other neural fits. We read that shape as pushing the 2.5% quantile and ES farther out. It sacrifices average likelihood and earns its keep in the lower tail, especially in 2020. If that reading holds, early stopping is responsible.
The conclusion makes the case for the two-step model: "The two-step specification remains useful when its fixed working density produces favourable risk forecasts, even when another model provides a better average density fit."
A trader must choose before seeing the losses. We did not find an ex-ante rule for selecting among the four neural variants for a given loss. Density NLL also determined the 30-epoch checkpoint, making its joint-loss victories more striking. The tables compare four neural columns with two benchmarks; changing the loss function changes the winner. Every window uses a single random seed, 1234. Seed dependence should be strongest for a network halted at 30 epochs after Xavier initialization. We did not find a seed-sensitivity check.
The COVID sweep shares one shock
Both sweep claims draw on 119 to 126 days per index: GAS-NN-1 (30-30) has the lowest COVID PNLL in all 8 markets, and GAS-NN-2 (30) has the lowest COVID VaR-ES loss in all 8. Those eight indices went through the same global selloff. The draws are correlated.
At the permissive 75% confidence level, Gaussian GAS remains in 7 of 8 COVID joint-loss sets. It also achieves the lowest COVID quantile score for GDAXI (0.2373) and the Australian All Ordinaries (AORD, 0.1869). The authors acknowledge the confidence-set limitation in their conclusion: "The model confidence sets retain several competitors, particularly for tail-risk losses, so these numerical rankings do not establish uniform statistical superiority."
They then say first-stage training and joint refinement "have different implications for density and tail-risk forecasts." That observation is accurate. It leaves a trader without a way to choose a variant before losses arrive. Across 119 to 126 correlated days, noise among four neural variants could also produce different winners for density and tail loss.
The abstract describes the models as competitive with Gaussian and Student-t GAS, highlighting particularly strong performance during the COVID-19 period. Competitive holds up. "Particularly strong" in 2020 depends on 119 to 126 days, while Gaussian GAS survives in 7 of 8 COVID joint-loss sets.
The larger omission is in the benchmarks. According to the abstract, the learned density accommodates asymmetry, including negative skewness. Sample skewness over 2004 to 2026 is −0.4785 for the S&P 500 and −0.7841 for AORD. Both benchmark scores are even functions of z, as the paper states. Yet the forecast tables include no skewed parametric competitor. They also leave out the kernel model of Blasques, Ji and Lucas and the spline model of van Brummelen, Gorgi and Koopman, although both are cited as precedents. We suspect that asymmetry accounts for some of the 2020 advantage and that a skewed-t would capture it too. We have no test establishing that suspicion.
Sizing SPY and QQQ from daily ES forecasts
Because we cannot trade the eight indices, our own exercise substitutes SPY and QQQ. This is not a replication of the paper; its international results do not transfer. The paper itself reports no strategy returns. It evaluates PNLL, the 2.5% quantile score and joint VaR-ES loss.
We size each ETF daily using each model's 2.5% ES forecast, with identical exposure caps across models. We pre-commit to GAS-NN-2 (30) and GAS-NN-1 (30-30), using Student-t as the baseline. For the S&P 500 over the full period, the joint-loss advantage over Student-t is 0.035, roughly 1.6% of the loss. To clear costs, that advantage must translate into sufficiently large differences in positions.
Results are still running. We will rerun GAS-NN-2 (30) against Student-t on SPY across several seeds, net of costs. The paper uses one seed, 1234.