White noise testing for functional time series. Application to model identification and diagnosis

White noise characterization is a crucial step in the identification and diagnosis of a model for scalar time series, where the autocorrelation and partial autocorrelation functions of the time series are the most common tools used for this purpose. An autocorrelation function for functional time series is proposed, based on the L2 norm of the lagged covariance operators of the time series. The distribution of this sequence of statistics has been established under the assumption of functional white noise, hence providing a method to test the adequacy of functional time series models by checking if the residuals of a
fitted model do not exhibit serial autocorrelation. This method is validated by numerical simulations of both white noise and dependent functional processes, where the structure of the process is identified by its autocovariance norms and a linear model is fitted and diagnosed using the techniques described in this
paper. The applicability of the method is illustrated via an application to two real-world datasets, including spanish electricity prices profiles.