For the S&P 500, next-day variance reaches its minimum after a return roughly half a percentage point above its conditional mean. At a zero residual, the variance contribution is about 0.036 squared percentage points. At the minimum, it is zero.
Frederiksen and Vilandt's Smooth Threshold GARCH (STGARCH) allows that minimum of the news impact curve to shift away from zero. The alteration to the familiar model is small. Yesterday's shock enters variance through its squared distance from an estimated threshold γ. Above γ the slope is α1; below it the slope is α2, alongside the usual ω and βσ². With γ = 0, the specification becomes GJR. With α1 = α2, it becomes Engle's A-GARCH. Both models are nested restrictions that can be tested with standard t-tests.
The paper devotes more weight to theory. Its Gaussian quasi maximum likelihood estimator (QMLE) is consistent at the usual √n rate for all five variance parameters, including the threshold. For interior parameters, it is asymptotically normal. The result survives the jump in the second derivative in γ wherever γ equals a past return, when α1 ≠ α2. At α1 = 0, moves above the threshold add nothing and the true parameter lies on the boundary. The α1 t-statistic then converges to max{ξ,0}; the one-sided critical value remains 1.64.
Stationarity requires the same condition as GJR, with no dependence on γ. The authors write, "The introduction of the threshold parameter thus costs nothing in terms of the stationarity region." For the empirical work they fit an AR(1) mean and STGARCH variance to daily percent log returns. The sample covers five indices from 10 April 2007 to 14 August 2026 (S&P 500 n = 4865). They also fit three restricted mean variants.
Where is the S&P 500 minimum?
For the unrestricted S&P 500, the estimates are δ = 0.0287, ρ = −0.0573, ω = 0.0032, α1 = 0.0829, α2 = 0.1440, γ = 0.5028 (s.e. 0.1278, τ = 3.93), β = 0.8408. The threshold refers to the residual ε, measured in percentage points above the AR(1) prediction. After yesterday's +1% return, today's predicted mean is 0.0287 − 0.0573 ≈ −0.03. A 1% return today therefore produces a residual of about 1.03.
The location and the curvatures do different jobs. γ fixes the bottom of the curve; α1 and α2 govern how quickly variance rises on each side. A zero residual is 0.5028 below the threshold, on the steeper branch. Its contribution to next-day variance is 0.1440 × 0.5028², or about 0.036 squared percentage points. GJR assigns no contribution to a zero shock.
Across indices and mean specifications, γ̂ ranges from 0.32 to 0.62 and the τ statistics range from 2.57 to 4.47. The Swiss MI with δ = 0 supplies the weakest result: τ = 2.57 and p = 0.010. The authors describe the finding as "Markets are thus calmest when returns are about half a percent larger than their conditional mean."
A shifted minimum carries the S&P 500 result
GJR is rejected throughout. A-GARCH fares differently across markets. The equal-slopes restriction is rejected for the FTSE 100 (κ 4.59 to 4.63), EURO STOXX 50 (3.20 to 3.32) and Nikkei 225 (2.21 to 2.36). It survives for the S&P 500 (1.13 to 1.53) and Swiss MI (0.72 to 0.99).
In the unrestricted S&P 500 model, the point estimate of α2 − α1 is 0.061, with κ = 1.13. The US index's in-sample support for STGARCH thus comes from moving the minimum. The equal-slopes test cannot reject at κ 1.13 to 1.53. For the FTSE 100 and EURO STOXX 50, the fitted shape differs: α̂1 is at most 0.018, leaving the right branch nearly flat and the left branch steep.
These tests concern fitted parameters. The authors use the rejections to conclude that STGARCH improves upon GJR and A-GARCH. We did not find a comparison of likelihoods or information criteria against GJR or A-GARCH, or an out-of-sample forecast evaluation.
A significant γ leaves that question open.
Choices the implementer still has to make
The variance backcast uses an exponentially weighted average of the first 75 squared observations, with decay 0.94, held fixed in θ. For simulations, the authors use L-BFGS-B and the analytical score. They do not identify the optimisation algorithm used in the application. Inference relies on a sandwich covariance; the application's Hessian comes from numerical differentiation of the analytical score. The simulation bounds are stated explicitly: ω in [0.001, 20], α1 in [0, 2], α2 in [0.001, 2], γ in [−5, 5], β in [0.001, 0.999].
Three decisions remain with the implementer, and a proof is absent. We did not find starting values or a differentiation step size. The application repeats only that α1 has a zero lower bound, "as in Section 5". We therefore chose to carry over the remaining simulation bounds. The authors also explicitly omit the proof of consistency for the sandwich estimator, describing it as "standard, but lengthy".
They flag the larger theoretical gap themselves. Their results cover the zero-mean model, whereas the headline specification uses an AR(1) mean. They conjecture that the results extend to it and describe that extension as "far from trivial." The reported zero-mean rows fall within the theory and hold up empirically. For the S&P 500 with δ = ρ = 0, γ̂ = 0.51 and τ = 4.06; the α1 statistic is 1.69.
The S&P 500's t of 1.67 meets a conservative boundary test
For the S&P 500, t lies between 1.67 and 1.76, rejecting α1 = 0 against the critical value of 1.64. The Nikkei 225 also rejects, at 3.19 to 3.25. Simulations show slow convergence of the boundary test. When α1 is truly zero and n = 5000, rejection occurs 3.0% of the time with normal innovations and 1.7% with scaled t5. Even at n = 200000, the rates reach only 4.6% and 2.7%. At n = 5000, the estimate lands exactly on zero in 57.4% of t5 replications, compared with the limiting 50%.
This under-rejection protects the S&P result from size inflation. A marginal rejection remains a rejection. The concern falls on the three non-rejections, whose t statistics are at most 1.07. A flat right branch is plausible for the FTSE 100 and EURO STOXX 50, with α̂1 at most 0.018. The Swiss α̂1 ranges from 0.050 to 0.061, comparable to the Nikkei estimate but noisy. A test that under-rejects at α1 = 0 will miss small positive α1 as well, so those non-rejections cannot establish a flat branch.
The GJR test behaves more reassuringly at the boundary. In the boundary simulations, η, the correlation between the γ and α1 estimators, is near 0.17 and 0.12. The GJR test's asymptotic sizes are 4.8% and 4.9%. At n = 5000, simulated rejection rates are 4.8% (normal) and 6.3% (t5). Fat tails cause more trouble for the interior α1 test: skewness is −0.69 and rejection reaches 8.2% at n = 5000. That rate declines to 6.2% at n = 20000 and 5.3% at 50000.
Our SPY, QQQ and DIA adaptation
We could not trade the paper's indices. Instead, we adapted the model to SPY, QQQ and DIA, using three equal sleeves with independent financing.
At every month-end, each sleeve refits AR(1)-STGARCH through bounded L-BFGS-B QMLE on a trailing five-year window. We use the paper's backcast and the simulation bounds given above. Exposure is min(1, 10% / forecast annualised volatility), with execution at the next open. We charge commissions of $0.004 a share, with a $1 minimum and a cap of 1% of trade value.
Our run returned 40.20% in total over 2020-01-01 to 2024-07-01. Sharpe was 0.91, Sortino was 1.21, and annualised volatility was 9.85%. The worst drawdown reached -14.42%, giving a Calmar of 0.54. Realised volatility of 9.85% falls just below the 10% target, consistent with a cap that often binds.
These returns come from US-only ETFs and say nothing about the paper's five-index estimates. The paper reports no trading performance of any kind, leaving no author-reported figure against which to compare ours. Our figures represent one automated pass and should be read only as an ETF sizing experiment. We also built GJR and A-GARCH sleeves with identical caps and targets. The reported figures belong solely to the STGARCH book, and we do not report the GJR sleeve here.
Two choices we made probably account for much of what the returns show. Five years contains roughly 1260 sessions, about a quarter of the paper's 4865. Scaling the S&P standard error of 0.1278 by √(4865/1260) yields about 0.25 on γ. An estimate near 0.5 would consequently be only about two standard errors from zero. Monthly refits will move the threshold around.
The 100% cap also binds whenever forecast volatility falls below 10%. When all three forecasts are under 10%, the models take identical positions. Any sizing differences must therefore arise on high-volatility days. One pass across four and a half years is a single draw, telling us more about our window length than about the threshold.
The S&P 500's in-sample evidence rests on the 0.5028 displacement of the minimum. To make that displacement worth trading, we would want an out-of-sample forecast-loss comparison with GJR on the same index. The paper does not perform that comparison. Our sizing run cannot substitute for it.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.