The SDE's 0.1306 SPX QLIKE lead over GARCH barely depends on realized-measure feedback: removing that term costs 0.0069. Yao, Chi and Gao give feedback much of the credit. We suspect more of the edge comes from resetting latent variance to yesterday's realized variance each night. The paper never isolates that reset, so the suspicion remains untested.
Where the forecast starts
The structured SDE runs two coupled states in continuous time. Latent log-variance, x, reverts toward θ and receives an extra drift, γz, from the second state. That state, z, tracks the log realized measure. It reverts toward ξ + ϕx and has its own noise. The γz drift is the "realized-measure feedback", inspired by the γz term in Realized GARCH's log-variance recursion. The authors say their model is neither a continuous-time extension nor a diffusion limit of Realized GARCH. Return shocks connect to both states through ρYx and ρYz, correlations constrained to be non-positive in the leverage direction.
Each close brings a fresh starting point: both states become the log of that day's 5-minute RV. From there, the model runs 256 Euler paths at 20 steps per day. It integrates e^x over the next day on each path, then averages those values to forecast next-day integrated variance. Parameters are fitted by Gaussian quasi-likelihood on the simulated joint distribution of (return, log RV), with gradients through the paths. The authors describe this as "a simulation-based predictive criterion rather than the exact observed-data likelihood of the latent continuous-time system."
WRDS supplies RV for 17 equity indices from 2004 to 2022. Each series is split in half; a fixed-length window is refitted every 21 trading days and scored on the second half. The SDE has the lowest out-of-sample loss on 15 of 17 indices under QLIKE, 12 under MSE and 17 under MAE. For a trader, a better next-day variance forecast matters when it changes the vol-target weight.
How much does feedback buy?
The feedback ablation sets γ = 0 and refits the other parameters. The full model posts lower QLIKE on 14 of 17 indices, with 11 differences significant at 10%. Its MAE is lower on 17, with 13 significant. MSE falls on 13, though only 2 differences are significant.
The loss gaps tell a different part of the story. SPX QLIKE is 0.0778 without feedback and 0.0709 with it. For comparison, GARCH scores 0.2015, Realized GARCH 0.1131, HEAVY 0.0743 and HAR 0.0783. On N225, the two SDE scores are 0.4526 and 0.4498; HAR gets 0.7400 and Realized GARCH 0.9133. RUT MAE moves from 0.4300 to 0.4272, against 1.0188 for the best benchmark.
The γ = 0 variant still clears the best benchmark by a wide margin on N225 and RUT: 0.4526 vs HAR 0.7400, and MAE 0.4300 vs Realized GARCH 1.0188. SPX is an exception. There, its 0.0778 trails HEAVY's 0.0743.
The starting value remains untested.
The γ = 0 run retains the nightly reset x0 = z0 = log RV, along with everything else except feedback. No alternative starting value is tested in the paper. The abstract says feedback adds value "within the proposed framework," and the DM counts support that narrower claim. The contributions section says the gains "are primarily associated with realized-measure feedback." In the Discussion, the authors call the γ = 0 deterioration "substantial". Their examples include QLIKE gaps of 0.0069 on SPX and 0.0028 on N225. Beside the benchmark losses, the no-feedback model remains well ahead on N225 and RUT, though it trails HEAVY on SPX. It trails on STOXX50E and NSEI too, at 0.8606 and 0.4903 against HEAVY's 0.7462 and HAR's 0.4067.
A few ablation gaps fit within reported seed variation: KSE 0.0005, OMXC20 0.0001 and MXX 0.0012, against an average seed range of 0.0012. That range is the per-index max-min QLIKE across seeds 7, 42 and 2024, averaged over 17 indices. The paper does not identify which specification was used for the sensitivity run.
The correlation channel contributes less. With both ρ terms set to zero, the full model leads on QLIKE on 10 of 17 indices; 3 differences are significant, and none is significant under MSE. The authors call these "more modest complementary gains."
The benchmark table's limits
Best-model counts do not establish a trading result. Under MAE, every benchmark, including Realized GARCH, HEAVY and HAR, is retained 0 times out of 17 in the 90% model confidence set. Under MSE, benchmarks remain in that set 12 or 13 times each. Most MSE wins therefore lack statistical separation from the field.
There are losses for the SDE as well. Its out-of-sample STOXX50E QLIKE is 0.8867 against HEAVY's 0.7462; the γ = 0 variant scores a better 0.8606. On NSEI, HAR wins QLIKE, 0.4067 against 0.4990. Realized GARCH also beats the full model on SPX variance-scale MSE, 2.2426 against 2.6538.
Two comparison choices matter. HAR is the usual hard RV baseline, but the authors "estimate the HAR specification using a Gaussian quasi-likelihood for daily returns to improve comparability," and we did not find an OLS fit on RV in the table. The paper lists five models in the MCS candidate set, omitting HAR, while the table gives HAR MCS counts.
The scoring target matters too. Forecasts are judged against raw 5-minute RV, the level used to start the proposed model each day. GARCH and GARCH-Itô instead update on squared daily close-to-close returns, including the overnight move. GARCH-Itô uses realized-variance information during estimation as well. The proposed model is itself trained jointly on close-to-close returns, which tempers this concern. We did not find an overnight-adjusted RV check, leaving open the possibility that part of the GARCH margin reflects a level effect.
From index variance to an ETF position
We cannot trade the 17 indices, so our backtest uses SPY, QQQ and IWM. The feedback mechanism can operate on ETF price paths. None of the paper's index-level accuracy results carries over automatically to those funds, and we did not find a portfolio, VaR or allocation evaluation in the paper.
Our daily vol-target rule sets weight to target vol divided by the square root of forecast variance. We compare the full model with the γ = 0 variant and yesterday's RV. Here the overnight mismatch becomes sharper: the ETF position stays on through the close, while the paper scores forecasts against intraday 5-minute RV.
Each ETF forecast requires 256 paths × 20 steps per day. A warm-started 300-epoch Adam refit follows every 21 days. The method needs clean 5-minute RV by each close because every forecast starts from it. Each refit keeps the lowest in-sample training loss, without a validation split. We also did not find the stability condition γϕ < κx enforced during estimation; each refit should be checked against it.
That ETF backtest is running now. If γ = 0 sizes our ETF positions as well as the full model, feedback adds nothing for that use. If yesterday's RV matches both, the SDE adds nothing for that use either.