Modeling and forecasting high-dimensional functional time series

Abstract

The demand to handle increasing volumes of data with complicated structures has given rise to research into modeling high-dimensional functional time series (HDFTS). Researchers face three problems in modelling HDFTS: the large number of sets of functional time series (FTS), the infinite dimensionality of functional data, and serial dependence among functions in each set of FTS. This thesis offers solutions to these problems by proposing three novel methods in modeling and forecasting HDFTS.

First, the thesis considers the problem of clustering FTS. Not all sets of FTS are homogeneous and heterogeneity within the data has a negative impact on the forecast results. The characteristics of FTS can be decomposed into two components: functional time trend and functional pattern (mode of variations of functions). The functional time trend reflects the dynamics across time, while the functional pattern captures the fluctuations within curves. To achieve more accurate forecasts, it is necessary to search for the homogeneous FTS before any modeling. This thesis develops a functional panel data model with fixed effects to characterize the two components of FTS, and proposes a novel clustering algorithm based on the model. Using Monte Carlo simulations, this thesis demonstrates the superior accuracy of the proposed clustering technique. An empirical data analysis of age-specific mortality rates in $32$ countries reveals that, with the aid of the clustering algorithm, joint modeling of mortality rates of homogeneous countries produces more accurate forecasts than several benchmark methods in forecasting age-specific mortality rates.

Second, the thesis investigates the issue of forecasting HDFTS using multivariate factor models. Despite the functional nature of the data, in practice, functional data can only be observed at discretized grid points. What makes functional data analysis different from multivariate/high-dimensional data analysis is the smoothness assumption of functional data. After smoothing, the densely observed HDFTS can be arranged into matrix-valued time series. Modeling such data in a matrix structure is crucial as the correlations among observations along the curve and the correlations among observations of different sets of FTS contain different information. This information is preserved in rows and columns of matrices in a structural way. This thesis adopts the matrix factor model to extract the temporal information contained in the HDFTS into low-dimensional matrix-valued time series, which enables forecasts. Empirical studies of both high-frequency financial data and demographic data demonstrate the superiority of the matrix factor model in forecasting the HDFTS, compared to various benchmark methods.

Third, the thesis considers modeling and forecasting HDFTS using functional factor models by extending the initial functional panel data model with additive fixed effects. By including the interaction of functional loading and the factors, the heterogeneity among different sets of FTS is incorporated and the need to search for homogeneous objects is eliminated. By allowing for different functional loadings for different cross-sections, the HDFTS can be reduced to FTS with lower dimensions. Another functional dynamic factor model is applied to the FTS with lower dimensions such that all the temporal dynamics contained in the original HDFTS are extracted to low-dimensional scalar factor matrices, which makes forecasting feasible. An empirical application to the Japanese subnational age-specific mortality rates illustrates that the proposed model produces more accurate forecasts than several existing methods. Show more