Received: 12 May 2017 Revised: 11 January 2019 Accepted: 12 February 2019 DOI: 10.1002/asmb.2443 RESEARCH ARTICLE Time series of functional data with application to yield curves Rituparna Sen 1 Claudia Klüppelberg 2 1 Applied Statistics Division, Indian Statistical Institute, Kolkata, India 2 Lehrstuhl fr Mathematische Statistik, Technische Universität München, Munich, Germany Correspondence Rituparna Sen, Applied Statistics Division, Indian Statistical Institute, Kolkata, India. Email: rsen@isichennai.res.in Abstract 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 func- tional data and subsequently modeling the principal component scores as vector autoregressive moving averag (VARMA) process. We justify the method by show- ing 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. KEYWORDS asymptotics, functional principal component, functional regression, prediction, vector ARMA 1 INTRODUCTION Functional data analysis (FDA, see the work of Ramsay and Silverman 1 for an overview) is an extension of multivariate data analysis to functional data, where each observation is a curve, rather than a vector in R n . An important feature of FDA is its ability to handle dependencies within each observation, especially smoothness, ordering, and neighborhood. Actual observations can be at discrete and irregular points within the curve. The first step of FDA is to replace these actual observations by a simple functional representation. Spline-based approximation is the most commonly used method. Kernel or wavelet-based approximations are also used. FDA has been successfully applied to real-life problems such as analysis of child size evolution, 1 climatic variation, 2 handwriting in Chinese, 3 medical research, 4 behavioral sciences, 5 spectrometry data, 6 etc. An important tool of FDA is functional principal component analysis (FPCA, see the works of Castro et al 7 and Rice and Silverman 8 ). Functional processes can be characterized by their mean function and the eigenfunctions of the autocovari- ance operator. This is a consequence of the Karhunen-Loève representation of the functional process. The components of this representation can be estimated. Individual trajectories are then represented by their functional principal component scores, which are available for subsequent statistical analysis. This often leads to substantial dimension reduction. Most of the development in FDA has been with independent and identical replications of function valued data. This permits the use of information from multiple curves to identify patterns. However, in certain situations, it is unrealistic to 1028 © 2019 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/asmb Appl Stochastic Models Bus Ind. 2019;35:1028–1043.