The authors propose a functional threshold GARCH (fTGARCH) model and extend the news impact curve (NIC) and the cumulative impact response function (CIRF) within the functional heteroskedastic framework.
With improvements in data acquisition, functional versions of the heteroskedastic models have emerged to deal with the high-frequency observations. The model considered in the article is: \[ r_t(\tau) = \sigma_t(\tau)\epsilon(\tau), \] \[ \sigma^2(\tau) = \gamma(\tau)+\int_0^1 [\alpha^+(\tau,s)r^{+2}_{t-1}(s)+\alpha^-(\tau,s)r^{-2}_{t-1}(s)+\beta(\tau,s)\sigma^2_{t-1}(s)] ds, \] where \({r_t(\tau)}_{t=1}^n\) denotes a sequence of random curves over \(\tau\in[0,1]\) in the real Hilbert space \(L^2[0,1]\), the space of square integrable functions concerning the Lebesgue measure and inner product \((f,g)=\int_0^1 f(s)g(s)\,ds,\) any \(f,g \in L^2[0,1]\), \(\epsilon_t(\tau)\), \(t\in Z\), is a sequence of the i.i.d. random functions, and \(\alpha^+(\tau,s),\alpha^-(\tau,s)\) and \(\beta(\tau,s)\) are integral kernel functions which are elements of Hilbert space \(L^2[0,1]^2\).
In this model, the authors find that the fTGARCH model can describe the asymmetry of the observation data, which are revealed by the sample cross-correlation functions. The slope of the NIC changes with time for functional GARCH class models and the changes are asymmetrical for the fTGARCH model.
The persistent effects of volatility for the functional GARCH class models are explored by using the generalized CIRF. By fitting the models to the S&P500 stock market index, the authors conclude that the fTGARCH model has some flexibility and superiority in regard to volatility asymmetry.