Time series of functional data with application to yield curves

We develop time series analysis of functional data observed discretely, treating the whole curve as a random realization from a distribution on functions that evolve over time. The method consists of principal components analysis of functional data and subsequently modeling the principal component scores as vector autoregressive moving averag (VARMA) process. We justify the method by showing that an underlying ARMAH structure of the curves leads to a VARMA structure on the principal component scores. We derive asymptotic properties of the estimators, fits, and forecast. For term structures of interest rates, these provide a unified framework for studying the time and maturity components of interest rates under one setup with few parametric assumptions. We apply the method to the yield curves of USA and India. We compare our forecasts to the parametric model that is based on Nelson-Siegel curves. In another application, we study the dependence of long term interest rate on the short term interest rate using functional regression.