Bootstrap for continuous-time autoregressive moving average processes

Abstract

The main focus of this thesis is to develop bootstrap approaches for the class of continuous-time autoregressive moving average processes. The first part focuses on the classical setup for time series in discrete time, especially the class of linear processes is considered in detail. Central limit results for integrated periodograms and ratio statistics are shown to change essentially when the estimates are based on low-frequency observations instead of full-time observations. The results correspond to those for continuous-time parameters based on discrete observations. This motivates the further parts of this thesis. The second part proposes bootstrap possibilities for continuous-time autoregressive processes. Samples of such processes have representations as autoregressions but with uncorrelated innovation sequence only. To circumvent the interdependencies in the innovations, the underlying continuous-time model is used. This allows for a parametric representation with truely independent innovations which is favorable for the bootstrap, however, there is a certain price to pay. Nevertheless, on this basis a valid residual-based bootstrap approach is developed. The third part considers the bootstrap for continuous-time autoregressive moving average processes. As for pure continuous-time autoregressions, the samples fulfill an autoregessive representation with uncorrelated innovations only. Due to the moving average part, residual-based proposals are not adaptable here. However, the block bootstrap is valid, since it works under very mild assumptions. The additional information on the autoregressive part of the process motivates a two-step bootstrap approach. First, an autoregressive model is fitted. In a second step - to address the remaining dependencies - the block bootstrap is applied. This approach is shown to be valid. Its applicability is tailor-made but not limited to continuous-time autoregressive moving averages. Indeed, it is as widely applicable as standard block bootstraps and thus suitably generalizes the moving block bootstrap. The approach further robustifies the residual bootstrap.