Approximate Factor Models for Functional Time Series

Abstract

A functional dynamic factor model for time-dependent functional data is proposed. We decompose a functional time series into a predictive low-dimensional common component consisting of a finite number of factors and an infinite-dimensional idiosyncratic component that has no predictive power. The conditions under which all model parameters, including the number of factors, become identifiable are discussed. Our identification results lead to a simple-to-use two-stage estimation procedure based on functional principal components. As part of our estimation procedure, we solve the separation problem between the common and idiosyncratic functional components. In particular, we obtain a consistent information criterion that provides joint estimates of the number of factors and dynamic lags of the common component. Finally, we illustrate the applicability of our method in a simulation study and to the problem of modeling and predicting yield curves. In an out-of-sample experiment, we demonstrate that our model performs well compared to the widely used term structure Nelson-Siegel model for yield curves.

Forecasting Environmental Data: An example to ground-level ozone concentration surfaces

“…Finally, the projections x l,t = X t − µ, ψ l are used as an estimator of factor x l,t . As shown in Otto and Salish (2022), Theorem 2 these estimators provide consistent estimates. For practical implementation of this step, we refer a reader to "fda" package of Ramsay and Silverman (2005).…”

“…This set of assumptions allows us to identify latent components of the model as functional principal components extracted from the global covariance function of {X t } defined as c(s, r) := lim Otto and Salish, 2022 for the theoretical discussion of this result). This, in turn, indicates that all latent components of model ( 5) can be easily estimated.…”

“…The flexibility of the model comes from the fact that, after the factors are estimated, any suitable multivariate forecasting technique can be used to obtain a forecast of the factors and therefore of the surface X T +1 (•). As shown by the identification results in Otto and Salish (2022) the behaviour of the factors is restricted only by weak stationarity (with some moment restrictions) indicating that nonlinear multivariate forecasting techniques can potentially be employed. Therefore, in this paper we propose two specifications of the adapted DFFM model to surface time series: one that accounts for a linear behaviour of the factors (using the vector autoregressive model) and the other for nonlinear (using K-nearest neighbours framework).…”

“…To this end, we adapt the dynamic functional factor model (DFFM) to the settings of the surface time series. The DFFM is proven to be useful in many applications to onedimensional manifold times series (See e.g., Hays, Shen and Huang, 2012, Liebl, 2013, Aue, Norinho and Hörmann, 2015and Otto and Salish, 2022. The advantage of this forecasting approach over existing methods is the simplicity of its application and the flexibility of the modelling framework.…”

“…The DFFM model allows decomposing rather complicated surface time series into a finite number of scalar factors that have predictable power and a reminder term (see Section 3 for a more detailed description). As discussed and shown in Aue, Norinho and Hörmann (2015) and Otto and Salish (2022) the simplicity of the application lies in the fact that estimates of the factors are obtained via standard functional principal component analysis. (For instance, R and Matlab packages for FDA by Ramsay and Silverman (2005) can be used for this purpose).…”