Statistical inference in semiparametric locally stationary ARCH models
Résumé
In this paper, we develop a complete methodology for semiparametric inference in the time-varying ARCH model (tv-ARCH) introduced by Dahlhaus and Rao (2006) and studied by Fryzlewicz et al. (2008). Our first motivation is to detect and estimate non time-varying coefficients in a tv-ARCH process. Using kernel estimation, we construct $\sqrt{T}-$consistent estimates for non time-varying coefficients and, with a two-step procedure, asymptotically efficient estimates in the semiparametric sense when the noise is Gaussian. Then a statistical test for detecting a parametric component is considered, following the approach used by Zhang and Wu (2012) for time-varying regression models. Our methodology is also shown to be useful in practice for testing two well-known ARCH type models introduced in the econometric literature: the multiplicative component model of Engle and Rangel (2008) and the simple time-varying unconditional variance model of Granger and Starica (2005). An application to currency exchange rates and stock market indices illustrates the usefulness of our approach. While we find nonstationnarity in all the data sets considered, our methodology suggests three possible behaviors for the lag coefficients of the tv-ARCH model: time-varying, non time-varying or non significant.