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

Abstract

In this article, which consists of two parts (Part I: representation results; Part II: estimation and forecasting methods), we set up the theoretical foundations for a high‐dimensional functional factor model approach in the analysis of large cross‐sections (panels) of functional time series (FTS). In Part I, we establish a representation result stating that, under mild assumptions on the covariance operator of the cross‐section, we can represent each FTS as the sum of a common component driven by scalar factors loaded via functional loadings, and a mildly cross‐correlated idiosyncratic component. Our model and theory are developed in a general Hilbert space setting that allows for mixed panels of functional and scalar time series. We then turn, in Part II, to the identification of the number of factors, and the estimation of the factors, their loadings, and the common components. 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 observations 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 we demonstrate that our approach outperforms existing methods.

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

“…Note that Assumptions A–D imply that the largest eigenvalues of diverge while the th one remains bounded (Lemma S2.15 in the Supporting information), hence the common and idiosyncratic components are asymptotically identified (Hallin et al ., 2023, Theorems 3.1 and 3.2).…”

“…Remark As mentioned earlier, Assumptions A–D imply that the common and idiosyncratic components are asymptotically identified, see (Hallin et al ., 2023, Theorems 3.1 and 3.2) and Lemma S2.15 of the Supporting information. The extra assumptions for consistent estimation of the factors' row space (and for the loadings and common component, see Theorems 3.6 and 3.7) are needed because the covariance is unknown, and its first eigenvectors must be estimated. Notice, in particular, that Theorem 3.3 holds for the case for all , where it coincides with Theorem 1 of Bai and Ng (2002).…”

“…This phenomenon is corroborated by our application to forecasting, see Section 5, and also Hallin et al . (2023, Section 3.2)…”

“…This article is the second one of a set of two in which we set up the theoretical foundations for a high-dimensional functional factor model approach to the analysis and forecasting of large cross-sections (panels) of functional time series (FTS). Part I (Hallin et al, 2023) is establishing a representation result (section 2.3) 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.…”

A journey from univariate to multivariate functional time series: A comprehensive review

“…Dynamic FPCA assumes a central role in various methods and applications, as illustrated by its utilization in the following examples: foreign exchange forecasts through dynamic FTS (Shang & Kearney, 2022), dynamic principal component regression (Shang, 2019), surface time series models (Martínez‐Hernández & Genton, 2023), as well as high‐dimensional analyses undertaken by Gao et al (2019), Tang and Shi (2021), Martínez‐Hernández et al (2022), and Hallin et al (2023).…”