Statistical inference of non-linear GARCH models.

Summary In this thesis, we study the estimation and hypothesis testing problems of two large classes of nonlinear GARCH models. First, we consider several methods for estimating a class of GARCH models with a power threshold. Under very weak conditions, we study the asymptotic properties of these estimators in the following two situations. First, we assume the power is known. We establish the properties of the quasi-maximum likelihood estimator (QMV). We also consider two sequences of ordinary least squares estimators, in the pure ARCH case of the model and we show that, for some values of the power, these estimators can be more efficient than the QMV estimator. In a second step we consider the case where the power is unknown, and is jointly estimated with the other parameters. The asymptotic properties of the QMV are established under the assumption that the noise has a density. Moreover, we study a class of non-Gaussian quasi-Maximum Likelihood estimators in the concrete situation where the error density is misspecified. We show that this class of estimators can provide efficient alternatives to the standard QMV estimator, in particular, when the errors have thick distribution tails. Skewness tests are proposed. In the second part of this thesis, we introduce a general class of weak GARCH processes containing a large family of models with conditional heteroscedasticity. We propose a representation consisting of two ARMA equations: the first one deals with the observed process, and the second one with some function of the linear innovation of the observed process. Under ergodicity and mixing assumptions, and certain moment conditions on the observed process, we establish the convergence and asymptotic normality of the two-stage least squares estimator. We also consider the estimation of the asymptotic covariance matrix of this estimator. Most of these asymptotic results are illustrated by simulation experiments and are applied to financial series.