A volatility-targeting book could use this paper's persistence estimates: 0.9637 after down days and 0.9061 after up days. The table meant to establish their forecasting advantage is much harder to defend.
The model from Guerbyenne, Hamdi and Hemis
The authors start with their earlier log GARCH-SV model. Its latent log-volatility follows an autoregression with a separate noise term, while yesterday's squared return enters through a feedback term in logs. Realized returns thus feed partly into a stochastic volatility model.
Their threshold version, T-log GARCH-SV, allows the feedback coefficient (alpha) and autoregressive coefficient (beta) to change with yesterday's return sign. Zero is a fixed threshold, not an estimated one. Regime 0 follows positive returns; Regime 1 follows negative returns.
For strict stationarity, the theory supplies the sufficient condition |α0+β0|^p0 |α1+β1|^p1 < 1. It also establishes geometric ergodicity and the existence of log-moments. The authors estimate by quasi-ML using an EM algorithm with a closed-form M-step. A particle filter and smoother supply the E-step, with 200 particles in the simulations.
The empirical sample contains 3986 daily S&P 500 log-returns, running from January 3, 1991 to October 20, 2006. Comparisons use basic SV, Breidt's threshold SV, the symmetric log GARCH-SV and an EViews EGARCH. In Breidt's model, only beta switches and there is no return feedback.
The paper claims no trading application. We would use the forecasts for sizing, dividing a volatility target by the one-step forecast to set exposure. Since the index cannot be held, implementation would use SPY and incur tracking differences and trading costs. All discussion of exposure below is our proposed application. The authors report forecasting results for the index alone.
How much does yesterday's sign matter?
The fitted coefficients suggest it matters chiefly through beta: β0 = 0.9122 after up days, compared with β1 = 0.9812 after down days.
The alphas, −0.0061 and −0.0175, are both slightly negative. Conditional on the latent state, a large lagged squared return therefore pushes next-day log-volatility a little lower. Almost all of the leverage story comes from the difference in persistence.
That difference accumulates in a sizing rule. With the regime held fixed for five sessions, a unit log-volatility shock retains 0.9637^5 ≈ 0.83 of its size following down days. Following up days, it retains 0.9061^5 ≈ 0.61. A volatility-targeted SPY position would consequently remain underweight longer after a sell-off than after an equally large rally.
There are two qualifications. Positive lagged returns account for 0.4723 of the sample, so neither regime dominates. Figure 2(a) shows no return autocorrelation, and the regime changes often. We also found no standard errors in the empirical parameter table. The difference between 0.9122 and 0.9812 therefore remains untested.
At 0.9361, composite persistence in the threshold model is lower than the symmetric model's 0.9469. The text gives 0.9367 for that same quantity.
Wider intervals win the coverage contest
Both the abstract and conclusion say T-log GARCH-SV "outperforms competing SV-type models in volatility forecasting." Table 4, which reports one-step coverage, is the only numerical evidence offered; the remaining evidence consists of volatility plots. The authors write that coverage rates "are generally close to their nominal levels across all estimated models." They then say T-log GARCH-SV "demonstrates a slight advantage." The table supports the first statement. It does little for the second.
At nominal 50%, coverage reaches 58.68% for T-log GARCH-SV, compared with 54.52% for EGARCH and 56.30% for the symmetric model. At 90%, T-log GARCH-SV records 92.19%; EGARCH records 90.29% and the symmetric model 90.71%. The threshold model is closest only at 99%, covering 98.85% against EGARCH's 98.67%.
All five models over-cover in the center, and the threshold model does so most.
Across all seven levels, T-log GARCH-SV has the highest coverage of the five, implying the widest intervals. A sizing rule consumes that width. Forecasts that run high on ordinary days leave the book below its target.
We found no reported interval widths, QLIKE or MSE loss, Kupiec or Christoffersen tests, or Diebold-Mariano comparison. The parameters appear to have been estimated over the full 1991 to 2006 sample. We found no held-out window, so each coverage result appears to be in-sample.
The simulations raise a further concern for equity data. The authors acknowledge that larger departures from Gaussianity can increase bias in some parameters, particularly under mixture noise. They also say these biases decline as the sample grows. Some do not.
For the first model under Student-t(5) observation noise, estimated δ² rises from 1.3112 at n = 500 to 1.3787 at n = 2000. Its true value is 1. In the second model under mixture noise, α1 is estimated at −0.0268 for n = 500 and −0.0265 for n = 2000, against a true −0.12. This down-regime feedback coefficient is one of the additions made by the threshold model. Quadrupling the sample barely changes its estimate.
Mixture noise also biases β1, the coefficient carrying the S&P asymmetry. Its estimates are 0.9511 at n = 500 and 0.9576 at n = 2000, versus a true 1.05. Even with correctly specified Gaussian noise, β0 is estimated at 0.8726 against 0.9 at n = 2000.
A SPY sizing test remains to be run
With the sample ending in October 2006, the whole of 2008 lies outside it. A credible test would begin expanding-window re-estimation from 2007 onward, using the forecasts to size SPY to a fixed volatility target. The baseline would be symmetric log GARCH-SV. Faster mean reversion after up days should, if anything, increase the threshold model's turnover. The comparison therefore needs to be net of costs.
Pending that test, the 0.9637 versus 0.9061 persistence split is a plausible sizing input. The coverage table supplies no evidence either way.
We would change our view if the threshold model beat the symmetric model on out-of-sample QLIKE after 2006 and delivered a net-of-cost Sharpe gain on SPY.