Modeling and forecasting functional time series

Modeling and forecasting functional time series in the past decade have attracted increasing attention among actuarial and demographic researchers, as well as finance and health care practitioners. This thesis, considers three questions that researchers often encounter when applying univariate and multivariate functional time series, and provide novel solutions for making improved estimation and forecasting. First, the thesis considers adequate feature extraction of functional data. Most functional time series methods depend on dimension reduction techniques to collapse infinite-dimensional functional objects to finite features to facilitate estimation and forecasting. The global features concerning the dominant modes of variation over the entire function domain, and the local features of function variations over particular short intervals within the function domain are both important in functional time series analysis. Existing functional time series methods extensively rely on functional principal component analysis (FPCA) for dimension reduction. Although a key feature extraction tool, FPCA focuses only on capturing the dominant global features of functional data, neglecting highly localized features. To overcome this issue, the thesis introduces a feature extraction method that initially extracts global features of functional data via FPCA, and then extracts local features by block thresholding of wavelet (BTW) coefficients. Using Monte Carlo simulations, along with an empirical application on near-infrared (NIR) spectroscopy data of wood panels, the thesis illustrates that the proposed FPCA-BTW method produces more accurate forecasts than FPCA and sparse FPCA methods.Finally, the thesis develops asymptotic properties of FPCA-BTW estimators, discovering the interaction between convergence rates of global and local features. Second, the thesis considers modeling and forecasting mortality rates for multiple subnational populations. It is important to account for dynamically changing correlations between subnational populations to achieve improved forecast accuracy. Moreover, independent forecasts at the subnational levels should add up to the forecasts at the national level. The thesis proposes a grouped multivariate functional time series method that collectively models subnational mortality rates and makes forecasts to achieve improved modeling and forecasting accuracy. To ensure coherence, forecasts are arranged into hierarchical structures disaggregated by sex and geographic factor before applying reconciliation approaches. A case study using the regional age-specific mortality rates in Japan during the period 1975--2016 demonstrates that joint consideration of populations with similar mortality patterns can improve point and interval forecast accuracy. Third, the thesis considers the application of functional time series methods in modeling financial time series. It is often difficult for conventional time series methods to make accurate forecasts for high-frequency financial data because of the curse of dimensionality as well as the highly volatile observations. The thesis considers 15-second high-frequency intraday Chicago Board Options Exchange Volatility Index (VIX) tick values as realizations of a collection of curves observed sequentially over time, and utilizes functional data analysis techniques to produce one-day-ahead forecasts of these curves. To accommodate abrupt increases of VIX associated with sudden market movements, various robust estimation techniques are considered. With the help of dynamic updating techniques, the obtained point and interval forecasts are shown to provide improved accuracy over several conventional time series models