A flexible model tests how stock volatility responds to returns

Stock-market volatility may react differently to gains and losses. Jonas Fiedler Frederiksen and Frederik Vilandt study a model designed to capture that asymmetry and test it against daily index returns.

What the study found

In their arXiv preprint, Frederiksen and Vilandt examine Smooth Threshold GARCH (STGARCH), a model that relates volatility to the previous return. It estimates a threshold at which volatility is lowest and allows different slopes on either side. The authors say this framework encompasses two existing models, GJR and A-GARCH, while accommodating patterns that each of those models rules out.

Using daily returns from five major stock indices, the authors report rejecting GJR for all five and A-GARCH for three. They estimate that volatility is lowest following returns 0.3–0.6 percentage points above their conditional mean. They also reject, for two indices, the hypothesis that returns above the threshold do not increase volatility.

Why it matters

The authors establish properties of their estimation method despite a mathematical complication: the likelihood is not twice continuously differentiable in the threshold. They report that the estimator is consistent and converges at the usual square-root-of-sample-size rate. When the slope above the threshold is zero, its limiting distribution is non-Gaussian; their results nevertheless support specified tests using standard normal critical values. This provides a way to assess competing descriptions of asymmetric volatility within one framework.

Limits and unknowns

The findings are from a preprint and the supplied abstract does not identify the five indices, the sample period, data sources, test significance levels or detailed model assumptions. Rejection of a simpler model is not, by itself, proof that STGARCH will forecast better, and the results do not establish that returns cause subsequent volatility.

Read the research

Authors: Jonas Fiedler Frederiksen, Frederik Vilandt

Published: 2026-10-29

Source basis: Based on the abstract only

Original abstract

It is well established that the volatility of daily stock returns responds asymmetrically to news, as described by the news impact curve (NIC), which relates volatility to the previous return. Empirical studies suggest that (i) volatility reacts more strongly to negative returns than to positive returns of the same magnitude, (ii) the point of asymmetry, where the NIC attains its minimum, is positive, and (iii) returns above this point often do not increase volatility at all. The popular GJR and A-GARCH models each rule out one of these features. We study the Smooth Threshold GARCH (STGARCH) model, which nests both and allows for (i)-(iii) by combining an estimated threshold, the point of asymmetry, with separate slopes on either side of it. Although the likelihood is not twice continuously differentiable in the threshold, we show that the quasi maximum likelihood estimator is consistent and converges at the usual $\sqrt{n}$-rate, whether or not (iii) holds. If (iii) does not hold, the estimator is asymptotically normal, while if it holds, the slope above the threshold is zero, a value on the boundary of the parameter space, and the limit is non-Gaussian. In that case, the $t$-statistic for (iii) converges to the positive part of a standard normal variable, and the $t$-test of the GJR model remains valid. These results allow us to test the GJR and A-GARCH models, as well as (i)-(iii), within the STGARCH using standard normal critical values. For daily returns on five major stock indices, we reject the GJR model for all five and the A-GARCH model for three, suggesting that the STGARCH improves upon both. Volatility is lowest after returns 0.3-0.6 percentage points above their conditional mean, in line with (i) and (ii). Finally, the hypothesis in (iii) is rejected for two of the five indices.

This article is an AI-generated adaptation, not a substitute for the original research. Preprints may not have been peer reviewed; AI can make mistakes.