Inference on common trends in functional time series

TL;DR

This paper develops statistical methods for analyzing common trends, unit roots, and cointegration in high-dimensional functional time series within a Hilbert space framework, applicable to various complex data types.

Contribution

It introduces new inference techniques for the number of common stochastic trends and hypotheses testing in infinite-dimensional time series models, valid even with unknown subspace dimensions.

Findings

Methods are asymptotically valid for high-dimensional and functional data.

Applicable to cointegrated vector and functional time series.

Includes empirical examples demonstrating practical utility.

Abstract

We study statistical inference on unit roots and cointegration for time series in a Hilbert space. We develop statistical inference on the number of common stochastic trends embedded in the time series, i.e., the dimension of the nonstationary subspace. We also consider tests of hypotheses on the nonstationary and stationary subspaces themselves. The Hilbert space can be of an arbitrarily large dimension, and our methods remain asymptotically valid even when the time series of interest takes values in a subspace of possibly unknown dimension. This has wide applicability in practice; for example, to cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor models that include a finite number of nonstationary factors, to cointegrated curve-valued (or function-valued) time series, and to nonstationary dynamic functional factor models. To illustrate our methods, we include two empirical examples.