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 of series and the number 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 and for estimation purposes. We conclude with an application to forecasting mortality curves, where our approach outperforms existing methods.

Factor models for high‐dimensional functional time series I: Representation results

“…This estimation problem is covered in the companion article Tavakoli et al . (2023), where consistency requires the more traditional asymptotic scheme, yielding double asymptotics under which both and tend to infinity.…”

“…This article is Part I of a set of two, and proposes a new paradigm for the analysis of high‐dimensional time series with functional (and possibly also scalar) components. The corresponding estimation and forecasting methods are developed in Part II (Tavakoli et al ., 2023). Our approach is based on a new concept of (high‐dimensional) functional factor model which, in the particular case of purely scalar series reduces to the well‐established concepts studied by (Stock and Watson, 2002a, 2002b) and Bai and Ng (2002), albeit under weaker assumptions.…”

“…We tacitly assume that all 's are random elements with mean zero and finite second‐order moments defined on some common probability space ; we also assume that constitutes the finite realization of some second‐order ‐stationarydouble‐indexed process . The mean zero assumption is not needed – neither here nor in the estimation and forecasting methods developed in Part II (Tavakoli et al ., 2023) of this article, but simplifies the exposition: see the online supplement to Part II for a justification.…”

“…Further important issues are related with the estimation and forecasting problems: they are discussed in the concluding remarks at the end of Part II of this article (Tavakoli et al ., 2023) and its online supplement.…”

“…Drawing inspiration from Stock and Watson (2002a), Stock and Watson (2002b), and Bai and Ng (2002), we then develop, in Part II (Tavakoli et al ., 2023), an estimation theory providing consistent estimators for the unobserved factors, loadings, and common components, along with rates of convergence. This is laying the theoretical foundations for an analysis of high‐dimensional FTS.…”