Factor models for high-dimensional functional time series II: Estimation and forecasting

Abstract

This article is the second one in a set of two laying the theoretical foundations for a high-dimensional functional factor model approach in the analysis of large cross-sections (panels) of functional time series (FTS). Part I establishes a representation result by which, under mild assumptions on the covariance operator of the cross-section, any FTS admits a canonical representation as the sum of a common and an idiosyncratic component; common components are driven by a finite and typically small number of scalar factors loaded via functional loadings, while idiosyncratic components are only mildly cross-correlated. Building on that representation result, Part II is dealing with the identification of the number of factors, their estimation, the estimation of their loadings and the common components, and the resulting forecasts. We provide a family of information criteria for identifying the number of factors, and prove their consistency. We provide average error bounds for the estimators of the factors, loadings, and common components; our results encompass the scalar case, for which they reproduce and extend, under weaker conditions, well-established similar results. Under slightly stronger assumptions, we also provide uniform bounds for the estimators of factors, loadings, and common components, thus extending existing scalar results. Our consistency results in the asymptotic regime where the number $N$ of series and the number $T$ of time points diverge thus extend to the functional context the ‘blessing of dimensionality’ that explains the success of factor models in the analysis of high-dimensional (scalar) time series. We provide numerical illustrations that corroborate the convergence rates predicted by the theory, and provide a finer understanding of the interplay between $N$ and $T$ for estimation purposes. We conclude with an application to forecasting mortality curves, where our approach outperforms existing methods.