This thesis contributes to the theory and methodology of robust estimation for two time series models: the generalized autoregressive conditional heteroscedasticity (GARCH) model and the vector autoregressive moving-average (VARMA) model.

More specifically, the first part (Chapter 3) of this thesis considers a class of M-estimators of the parameters of the GARCH models which are asymptotically normal under mild assumptions on the moments of the underlying error distribution. Since heavy-tailed error distributions without higher order moments are common in the GARCH modeling of many real financial data, it becomes worthwhile to use such estimators for the time series inference instead of the quasi maximum likelihood estimator. We discuss the weighted bootstrap approximations of the distributions of M-estimators. Through extensive simulations and data analysis, we demonstrate the robustness of the M-estimators under heavy-tailed error distributions and the accuracy of the bootstrap approximation. In addition to the GARCH(1, 1) model, we obtain extensive computation and simulation results which are useful in the context of higher order models such as GARCH(2, 1) and GARCH(1, 2) but have not yet received sufficient attention in the literature. We use M-estimators for the analysis of three real financial time series fitted with GARCH(1, 1) or GARCH(2, 1) models.

In the second part (Chapter 4) of this thesis, we propose a novel class of estimators of the GARCH parameters based on ranks, called R-estimators, with the property that they are asymptotic normal under the existence of a more than second moment of the errors
and are highly efficient. We also consider the weighted bootstrap approximation of the finite sample distributions of the R-estimators. We propose fast algorithms for computing the R-estimators and their bootstrap replicates. Both real data analysis and simulations show the superior performance of the proposed estimators under the normal and heavy-tailed distributions. Our extensive simulations also reveal excellent coverage rates of the weighted bootstrap approximations. In addition, we discuss empirical and simulation results of the R-estimators for the higher order GARCH models such as the GARCH($2, 1$) and asymmetric models such as the GJR model.

In the third part (Chapter 5 and Chapter 6) of this thesis, we propose a new class of R-estimators for semiparametric VARMA models in which the innovation density plays the role of the nuisance parameter. Our estimators are based on the novel concepts of multivariate center-outward ranks and signs. We show that these concepts, combined with Le Cam's asymptotic theory of statistical experiments, yield a class of semiparametric estimation procedures, which are efficient (at a given reference density), root-$n$ consistent, and asymptotically normal under a broad class of (possibly non elliptical) actual innovation densities. No kernel density estimation is required to implement our procedures. A Monte Carlo comparative study of our R-estimators and other routinely-applied competitors demonstrates the benefits of the novel methodology, in large and small sample.