Determining the order of the functional autoregressive model


Correspondence to: Matthew Reimherr, Department of Statistics, University of Chicago, 5734 S. University Avenue, Ryerson N375, Chicago, IL 60637, USA.


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 Lpm–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.