title: Autocovariance-based statistical inference for high-dimensional function/scalar time series
Creators: name: Chen, Cheng
Publisher: name: London School of Economics and Political Science (LSE)
Year: 2021

description: High-dimensional time series analysis is an important area in modern statistics. Data arises in many fields, including finance, economics, environmental and medical studies, among others. Faced with corrupted data where measurement errors require particular attention, previous work devoted to clean data calls for further exploration. Motivated by the fact that the autocovariance of observed time series automatically filters out the noise term, this thesis investigates the autocovariance-based estimation and inferential studies based on high-dimensional functional and scalar time series models. The subsequent chapters are organised as follows. The first chapter studies the functional linear regression when observed functions are error contaminated. The second chapter extends the topic to high-dimensional functional linear models. The third chapter investigates not only the estimation but also the inference on scalar time series in high dimensions. In the first chapter, we briefly introduce the motivation, main idea, contributions and limitations of this thesis. In the second chapter, we consider functional linear regression with serially dependent observations of the functional predictor, where the contamination of the predictor by the white noise is genuinely functional with a fully nonparametric covariance structure. It is commonly assumed that samples of the functional predictor are independent realisations of an underlying stochastic process and are observed over a grid of points contaminated by i.i.d. measurement errors. In practice, however, the dynamical dependence across different curves may exist, and the parametric assumption on the error covariance structure could be unrealistic. Therefore, we propose a novel autocovariance-based generalised method-of-moments estimate of the slope function. We also develop a nonparametric smoothing approach to handle the scenario of partially observed functional predictors. The asymptotic properties of the resulting estimators under different scenarios are established. Finally, we demonstrate that our proposed method significantly outperforms competing methods through an extensive set of simulations and an analysis of a public financial dataset. In the third chapter, we model observed functional time series, which are subject to errors in the sense that each functional datum arises as the sum of two uncorrelated components, one dynamic and one white noise. We propose an autocovariance-based three-step procedure by first performing autocovariance-based dimension reduction and then formulating a novel autocovariance-based block regularised minimum distance (RMD) estimation framework to produce block sparse estimates, from which we can finally recover functional sparse estimates. We investigate non-asymptotic properties of relevant estimated terms under such an autocovariance-based dimension reduction framework. To provide theoretical guarantees for the second step, we present a convergence analysis of the block RMD estimator. Finally, we illustrate the proposed autocovariance-based learning framework using applications of three sparse high-dimensional functional time series models. With derived theoretical results, we study the convergence properties of the associated estimators. Using simulated and real datasets, we demonstrate that our proposed estimators significantly outperform the competitors. In the fourth chapter, we study the high-dimensional linear regression with scalar serially dependent predictors that are error contaminated. To mitigate the influence of measurement errors, we propose an autocovariancebased de-bias regularised generalised method of moments (DRGMM) framework to obtain a high-quality estimator for regression coefficients. Moreover, we conduct an inferential study on the estimators within this framework. Theoretical results on estimation consistency and inference accuracy are provided. Finally, the finite sample performance of the proposed inference procedure is examined through simulation studies.