Functional time series identification and diagnosis by means of autocorrelation analysis

Quantifying the serial correlation across lags 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. This paper proposes a lagged autocorrelation function for functional time series, which is based on the L2 norm of the lagged covariance operators of the series. Diagnostic plots utilizing large sample results for the autocorrelation function of a strong white noise sequence are proposed as a tool for selecting the order and assessing the adequacy of functional SARIMAX models. The proposed methods are studied in numerical simulations with both white noise and dependent functional processes, which show that the structure of the processes can be diagnosed using the techniques described. The applicability of the method is illustrated via applications to two real-world datasets, Eurodollar future contracts and spanish electricity price profiles.