Estimation of functional ARMA models

Abstract

Functional auto-regressive moving average (FARMA or ARMAH) models allow for flexible and natural modelling of functional time series. While there are many results on pure autoregressive (FAR) models in Hilbert spaces, results on estimation and prediction of FARMA models are considerably more scarce. We devise a simple two-step method to estimate ARMA models in separable Hilbert spaces. Estimation is based on dimension-reduction using principal components analysis of the functional time series. We explore two different approaches to selecting principal component subspaces for regularization and establish consistency of the proposed estimators both under minimal assumptions and in a practical setting. The empirical performance of the estimation algorithm is evaluated in a simulation study, where it performs better than competing methods.