The functional AR(1) process with a unit root

TL;DR

This paper introduces the concept of strong and weak unit roots in functional AR(1) processes, demonstrating their implications for cointegration and illustrating these phenomena with demographic data analysis.

Contribution

It defines functional unit roots, explores their properties, and applies functional PCA to identify them in real demographic datasets, advancing understanding of functional time series.

Findings

Existence of functional unit roots in demographic data

Finite number of common trends in functional cointegration

Application of functional PCA to detect unit roots

Abstract

We define strong and weak unit roots for the functional AR(1) process and give some theoretical examples. It is shown that a functional form of cointegration occurs in which only a finite number of common trends exist. Using functional Principal Component Analysis we illustrate the presence of functional unit roots in two demographic data sets. We close with some remarks concerning our assumptions and the possibility of generalizing our results.