This thesis is composed in two parts. In the first chapter, we develop the theory of a novel fast bootstrap for dependent data. We show the asymptotic refinements of the new procedure, proving that it is higher-order correct under mild assumptions on the time series, the estimating functions, and the smoothing kernel. We illustrate the applicability and the advantages of our procedure for M-estimation, generalized method of moments, and generalized empirical likelihood estimation.

In a Monte Carlo study, we consider an autoregressive conditional duration model and we compare our method with other extant, routinely-applied first- and higher-order correct methods. The results provide numerical evidence that the novel bootstrap yields higher-order accurate confidence intervals, while remaining computationally lighter than its higher-order correct competitors. A real-data example on dynamics of trading volume of US stocks illustrates the empirical relevance of our method, when the underlying distribution of the test statistic is skewed or fat-tailed.

In the second chapter, we present a robust estimator for time series processes based on a aggregation of Whittle maximum likelihood estimators. We show that this estimator is robust against corruption in the spectrum, which can arise when time series exhibit complex cycles. It allows to estimate parameters of a contaminated time series without pre-processing the data for cycles or seasonalities, making inference safer and more accurate. We control the convergence of our estimator via a concentration bound in two components - the first component is exponential and the second one follows the dependence in the periodogram ordinates. A Monte-Carlo experiment confirms the robustness and efficiency of our estimator, compared to the main extant alternatives. We apply our method to infer the sky position of the first event detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO).