# Testing for stationarity of functional time series in the frequency domain

## TL;DR

This paper introduces a new frequency domain-based stationarity test for functional time series, utilizing spectral density operators and principal components analysis, with proven asymptotic properties and demonstrated effectiveness through simulations and temperature data analysis.

## Contribution

It proposes a novel stationarity test for functional time series that accounts for frequency-dependent dynamics using spectral methods and principal components analysis.

## Key findings

- Test performs well in finite samples
- Method effectively detects stationarity and non-stationarity
- Application to temperature data demonstrates practical utility

## Abstract

Interest in functional time series has spiked in the recent past with papers covering both methodology and applications being published at a much increased pace. This article contributes to the research in this area by proposing a new stationarity test for functional time series based on frequency domain methods. The proposed test statistics is based on joint dimension reduction via functional principal components analysis across the spectral density operators at all Fourier frequencies, explicitly allowing for frequency-dependent levels of truncation to adapt to the dynamics of the underlying functional time series. The properties of the test are derived both under the null hypothesis of stationary functional time series and under the smooth alternative of locally stationary functional time series. The methodology is theoretically justified through asymptotic results. Evidence from simulation studies and an application to annual temperature curves suggests that the test works well in finite samples.