Smooth Threshold GARCH: Quasi Maximum Likelihood Estimation and Empirical Evidence

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.