A simple test for white noise in functional time series

TL;DR

This paper introduces a straightforward, asymptotically normal test for functional white noise based on spectral density distance, avoiding complex variance estimation or resampling, and enabling testing for relevant deviations.

Contribution

It presents a novel, simple spectral density distance-based test for functional white noise that does not require long-run variance estimation or resampling.

Findings

Test is asymptotically normal under null and alternative hypotheses.

Does not require estimation of long-run variance or resampling.

Allows testing for relevant deviations from white noise.

Abstract

We propose a new procedure for white noise testing of a functional time series. Our approach is based on an explicit representation of the $L^2$-distance between the spectral density operator and its best ($L^2$-)approximation by a spectral density operator corresponding to a white noise process. The estimation of this distance can be easily accomplished by sums of periodogram kernels and it is shown that an appropriately standardized version of the estimator is asymptotically normal distributed under the null hypothesis (of functional white noise) and under the alternative. As a consequence we obtain a very simple test (using the quantiles of the normal distribution) for the hypothesis of a white noise functional process. In particular the test does neither require the estimation of a long run variance (including a fourth order cumulant) nor resampling procedures to calculate critical values. Moreover, in contrast to all other methods proposed in the literature our approach also allows to test for "relevant" deviations from white noise and to construct confidence intervals for a measure which measures the discrepancy of the underlying process from a functional white noise process.