On FTSE 100, SRT-GARCH gains from its variance baseline

On FTSE 100, SRT-GARCH owes its forecast advantage over constant-variance RT-GARCH to the slow-moving variance baseline; same-day feedback adds nothing detectable. The natural gas result is closer. At one step, the baseline reduces MSE by 0.82% against RT-GARCH. SRT-GARCH improves by 0.73% against S-GARCH, which leaves out feedback. For QLIKE, those gains are 2.98% and 0.74%.

Li, Chen, Xu and Jiang say their real-time mechanism "provides more pronounced gains in the natural gas application than in the FTSE application." Their natural gas section makes a stronger claim: the long-run scale and feedback both "contribute meaningfully to out-of-sample forecast accuracy." We find the gas evidence thinner. Only MSE gives a significant gain over S-GARCH. And the models use different fitted bandwidths, which means a different variance baseline accounts for some of the gap. How much does "more pronounced" amount to?

Inside the model

The authors express daily returns as a stationary short-run process multiplied by the square root of τ(t/T), a smooth deterministic function. Normalizing the short-run process to unit variance makes τ the unconditional return variance at each sample point. They estimate τ by kernel-smoothing squared returns and choose the bandwidth through cross-validation.

The short-run scale obeys h_t = ω + αu²_{t−1} + βh_{t−1} + φε²_t. Today's own shock supplies the final term, following Smetanina's real-time GARCH idea. With φ = 0, the model becomes S-GARCH, a semiparametric GARCH retaining the smooth baseline. Keeping τ constant instead produces stationary RT-GARCH. The paper's SRT-GARCH includes both features. Each benchmark removes one.

The theory is the paper's strongest contribution. Estimating τ first adds only a rank-one positive semidefinite term to the covariance available when τ is known. Standard errors then follow in closed form, without choosing a HAC lag. Simulated coverage ranges from 0.9130 to 0.9530.

For the empirical tests, the authors use 2,700 FTSE 100 days (2004 to 2014) and 3,000 natural gas futures days (May 2013 to October 2024). They score each series over its last 650 days, re-estimating on rolling windows. The Xiao-Lima test rejects covariance stationarity at 1% in both series. Its statistics are 2.8374 and 4.7216.

How does today's return reach tomorrow's forecast?

Feedback reaches tomorrow through h_T alone. Since h_t includes φε²_t, the paper describes it as "a real-time short-run scale rather than as the conventional one-step-ahead conditional variance." Under Gaussian innovations, the quantity known at the previous close is τ_t(b_{t−1} + 3φ). Today's return feeds tomorrow's forecast through b_T = ω + αu²_T + βh_T. As in any GARCH, αu²_T provides a direct channel. The φε²_T contribution passes through βh_T, with h_T recovered from the return using a quadratic-root inversion.

Feedback changes the distribution's shape as well. Under Gaussian innovations, any φ > 0 changes the conditional tails from Gaussian-type to exponential-type. The fitted φ is 0.0262 on FTSE (t = 2.7579) and 0.0345 on gas (t = 4.0588). For FTSE, the authors acknowledge that these t-statistics disregard the φ ≥ 0 boundary, excluded by Theorem 3's interior-point assumption. They nevertheless argue that the effect is substantial. Gas carries the same caveat.

FTSE 100: the baseline earns the forecast gains

SRT-GARCH records MSE of 0.9492, 0.9689, 0.9855 and 0.9866 at 1, 5, 10 and 15 steps against constant-variance RT-GARCH. The corresponding RT-GARCH figures are 0.9616, 0.9981, 1.0419 and 1.0609. Longer horizons bring a wider gap. Under MSE, significance arrives only at 10 steps (p = 0.0409) and 15 steps (p = 0.0298); the one-step MSE p-value is 0.2376. QLIKE offers stronger support for the baseline. The gain is significant at 5% from five steps onward (p = 0.0177, 0.0041, 0.0042), and at 10% for one step (p = 0.0778).

Every DM-HLN p-value against S-GARCH lies between 0.3930 and 0.8599. This comparison isolates feedback cleanly: both models retain the baseline and use a bandwidth of 0.0981. S-GARCH even delivers the lower one-step QLIKE, 0.3335 versus 0.3342. The authors themselves report the lack of significance.

A φ t-statistic near 2.76 buys no detectable improvement in FTSE forecasts.

Long-horizon gains depend on the projection of τ. The paper extrapolates using max{(k+1)τ̂s − kτ̂{s−1}, c_τ}, calling it "an empirical forecasting convention rather than a consequence of the population moment recursion." The slope comes from the final point of the window. In Remark 2, the authors identify an O(h) bias in that boundary band unless τ is flat at the endpoint. They show that parameter estimates are unaffected by this bias. The forecast slope, however, draws on precisely those points; that implication for forecasting is our reading. We did not find whether each rolling window gets a newly selected bandwidth.

Natural gas gains depend on the loss function

At one step, SRT-GARCH's MSE is 85.6325, compared with 86.3373 for RT-GARCH and 86.2642 for S-GARCH. The improvement over S-GARCH is 0.73%, with p = 0.0234. But their fitted bandwidths differ: SRT-GARCH uses 0.0843 and S-GARCH uses 0.0734. The comparison therefore combines feedback with a change in τ̂. For QLIKE, the same comparison gives 2.4721 against 2.4906, with p = 0.1368.

Against RT-GARCH, the significance switches loss functions. QLIKE supports the baseline's gain (2.4721 against 2.5480, p = 0.0296). MSE does not (p = 0.1258).

Each component thus clears significance for one loss function only. SRT-GARCH nevertheless posts the lowest point loss at every horizon under both MSE and QLIKE. MSE rewards accuracy on the big days. QLIKE is scale-free, penalizes underprediction more than overprediction, and is less dominated by the largest days. Together, the FTSE and natural gas forecast tables contain 32 DM-HLN tests (16 each). We saw no multiplicity adjustment. Several gas results highlighted by the paper clear only the 10% threshold.

For gas VaR, SRT-GARCH ranks last at 1% by 0.0004, then first at 5% and 10%. Its 1% quantile loss is 0.0670, behind S-GARCH at 0.0666 and RT-GARCH at 0.0667. The 1% tail is where exponential tails should help. Yet the losses occupy a narrow range, 0.0666 to 0.0670, and SRT-GARCH has the highest 1% coverage p-value: 0.9091, versus 0.3776 for RT-GARCH and 0.7690 for S-GARCH. SRT-GARCH leads at 5% (0.2138) and 10% (0.3446). At those levels, its Christoffersen p-values of 0.2329 and 0.1824 are the lowest or tied lowest among the three models. RT-GARCH gives 0.3357 and 0.8364; S-GARCH gives 0.5577 and 0.1824. None of the three levels rejects. For FTSE, SRT-GARCH achieves the lowest quantile loss at all three levels (0.0224, 0.0834, 0.1376).

Sizing remains untested

The paper supplies no position rule, Sharpe ratio, turnover or drawdown. That scope is fair for an econometrics paper. A desk still needs the 0.73% MSE improvement to survive the translation from variance forecast to position size. Generating those forecasts requires refitting three models on 2,350 days at each of 650 dates, kernel estimation included.

The simulations are also some distance from the fitted empirical regime. Gas persistence, α + β, is estimated at 0.9592, whereas the simulations use 0.70 to 0.80. Even in that range, the third design gives a β bias of −0.0708 at T = 1,000. It falls to −0.0431 at T = 2,000.

We could not rerun the gas application. We have no futures price data, and we did not find a description of the contract or roll rule used to construct the paper's return series. All figures above come from the authors' published results. We have not replicated them.

Use SRT-GARCH for its changing-variance baseline. Keep φ optional until a natural gas test shows the MSE advantage over S-GARCH persisting beyond one 650-day window and delivering lower realized error against a volatility target.