Functional Autoregression for Sparsely Sampled Data

TL;DR

This paper introduces a hierarchical Gaussian process model tailored for forecasting sparse and irregular functional time series data, offering improved accuracy and computational efficiency over existing methods.

Contribution

It develops a nonparametric dynamic functional factor model with Gaussian process innovations, optimized for sparse sampling and measurement error, with proven optimality properties.

Findings

Substantial forecasting improvements over competing methods.

Effective recovery of autoregressive surfaces in sparse data.

High performance in real yield curve forecasting.

Abstract

We develop a hierarchical Gaussian process model for forecasting and inference of functional time series data. Unlike existing methods, our approach is especially suited for sparsely or irregularly sampled curves and for curves sampled with non-negligible measurement error. The latent process is dynamically modeled as a functional autoregression (FAR) with Gaussian process innovations. We propose a fully nonparametric dynamic functional factor model for the dynamic innovation process, with broader applicability and improved computational efficiency over standard Gaussian process models. We prove finite-sample forecasting and interpolation optimality properties of the proposed model, which remain valid with the Gaussian assumption relaxed. An efficient Gibbs sampling algorithm is developed for estimation, inference, and forecasting, with extensions for FAR(p) models with model averaging over the lag p. Extensive simulations demonstrate substantial improvements in forecasting performance and recovery of the autoregressive surface over competing methods, especially under sparse designs. We apply the proposed methods to forecast nominal and real yield curves using daily U.S. data. Real yields are observed more sparsely than nominal yields, yet the proposed methods are highly competitive in both settings.