GARCH models with function coefficients of an exogenous process.

Summary In this thesis, we study the probabilistic properties and the statistical inference of parametric models of conditional volatility, whose coefficients are functions of an observed exogenous process. A first part of the thesis is devoted to the study of the stability properties of a GARCH (1,1) model belonging to this class. Necessary and sufficient conditions for the existence of a solution, generally non-stationary, are established, as well as conditions for the existence of moments for these solutions. These conditions concern the coefficients of the GARCH model in the various regimes of the exogenous process and the stationary probabilities of these regimes. In a second part, the asymptotic properties of the quasi-maximum likelihood estimator are studied. The convergence and the asymptotic normality of this estimator are demonstrated under regularity assumptions implying the stability of the solution and the strict positivity of the parameters but not requiring the existence of moments of the observed process. The study of the asymptotic behavior of the estimator when some coefficients of the model are zero is the subject of a last part. In this case, the asymptotic distribution of the estimator is non-standard and corresponds to the projection of a Gaussian distribution on a convex cone. We also obtain the asymptotic distributions of nullity tests of some coefficients of the model as well as their local asymptotic power. The main asymptotic results are illustrated by stimulus experiments. The model appears to be particularly suitable for energy price dynamics. For gas prices, we highlight the existence of GARCH volatilities depending on several temperature-related regimes.