Second-Order Properties of the Convolved Subsampling Method for Time Series

Block resampling methods provide useful nonparametric inference with time series by applying data blocks to capture dependence and develop distributional approximations for statistics. Among such methods, the block bootstrap (BB) and subsampling (SS) represent distinct approaches, where SS has advantages over bootstrap in more general applicability and computation, though bootstrap, when viable, can be more accurate. Recently, convolved subsampling (CS) has emerged as a hybrid method between SS and BB, though its accuracy has remained an open question, and its performance depends intricately on the choice of two tuning parameters (i.e., block length and convolution level). This paper establishes the formal accuracy properties of CS for a broad class of smooth-model statistics and time processes, showing that CS can be second-order correct like the BB, offering improvements over SS or normal approximations. However, the success of CS can depend heavily on the convolution level chosen and on how the CS approximation is centered or de-biased. Indeed, without such consideration, CS can even become invalid. In developing accuracy properties, this work also provides important guideposts for the convolution level needed to implement the CS. Numerical evidence suggests the method achieves good coverage accuracy and compares favorably with other block resampling approaches.