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-TGARCH MODEL: CAPTURING THE TAYLOR EFFECT 39
class, these theoretical expressions are very hard to obtain and are not available for all k but are indispensable for the latter approach, He and Ter¨asvirta [13] are the first to suggest the analysis of the theoretical relation corresponding to Taylor’s inicial findings, i.e.,
⇢n (1) > ⇢n (2), for all n, (1.4)
which they named Taylor property.
He and Ter¨asvirta [13] consider the conditionally Gaussian first-order ab-
solute value GARCH (AVGARCH) model and examine the Taylor property restricted to the autocorrelations of lag n = 1. They find the property to be present for some parameterizations associated with very high values of the model kurtosis. Gon¸calves, Leite and Mendes-Lopes [5] also evaluate this re- stricted version of the property, establishing its presence for some parameter- izations of the first-order threshold ARCH (TARCH) model but without pre- defining the generating process distribution. They also point to the influence the kurtosis of the generating process margins has on the parameterizations set which verify the property, since larger values seem to expand this set. Later, Haas [12] turns back to the first-order AVGARCH model, also not pre-defining the generating process distribution, and fully explores the Taylor property (for any lag n), extending the previous findings.
The Taylor property has also been investigated with similar conclusions in stochastic volatility models, by Mora-Gal´an, P´erez and Ruiz [17], Veiga [21], P´erez, Ruiz and Veiga [18] and Malmsten and Ter¨asvirta [15], and also in bilinear models, by Gon¸calves, Martins and Mendes-Lopes [7].
In this paper, we consider the first-order power threshold GARCH ( -TGARCH) model, a general model that incorporates many popular GARCH models including the ones already mentioned, and discuss its ability to capture the Taylor e↵ect. Section 2 is dedicated to the presentation of the -TGARCH model. In Section 3, setting = 1, the presence of the Taylor property (for any lag) is established for parameterizations of the 1-TGARCH model that can reproduce the leverage e↵ects. Additionally, taking Student’s t-distribution with several degrees of freedom for the variables of the generating process, the influence of its kurtosis in this property is illustrated. Finally, in Section 4, an exploratory simulation study compares -TGARCH models regarding their ability to reproduce the Taylor e↵ect, considering several values of and Gauss- ian and Laplace distributions for the generating process margins.
2. The first order power threshold GARCH model
Let X = (Xt,t 2 Z) and Z = (Zt,t 2 Z) be real stochastic processes. In the following, Xt+ = max {Xt, 0}, Xt = max { Xt, 0} and Xt 1 is the -field