Determining the order of the functional autoregressive model

We propose a multistage testing procedure to determine the order p of a functional autoregressive process, FAR ( p ). At its core is the representation of the FAR( p ) process as a fully functional linear model with dependent regressors. Estimating the kernel function in this linear model allows us to construct a test statistic which has, approximately, a chi–square distribution with the number of degrees of freedom determined by the number of functional principal components used to represent the data. The asymptotic justification relies on the concept of L p – m –approximability which quantifies the temporal dependence of functional time series. The procedure enjoys very good finite sample properties, as confirmed by a simulation study and applications to functional time series derived from credit card transactions and Eurodollar futures data.