On A Family Of Contrasts For Parametric Inference In Degenerate Arch Models

In this paper, we introduce a family of contrasts for parametric inference in ARCH models the volatility of which exhibits some degeneracy. We focus specifically on ARCH processes with a linear volatility (called LARCH processes), for which the Gaussian quasi-likelihood estimator may be inconsistent. Our approach generalizes that of Beran and Schützner (2009) and gives an interesting alternative to the WLSE used by Francq and Zakoïan (2010) for an autoregressive process with LARCH errors. The family of contrasts is indexed by a single parameter that controls the smoothness of an approximated quasi-likelihood function. Under mild conditions, the resulting estimators are shown to be strongly consistent and asymptotically normal. The optimal asymptotic variance is also given. For LARCH processes, an atypical result is obtained: under assumptions, we show that the limiting distribution of the estimators can be arbitrarily close to a Gaussian distribution supported on a line. Extensions to multivariate processes are also discussed.