High-Dimensional Functional Factor Models

In this paper, we set up theoretical foundations for high-dimensional functional factor models for the analysis of large panels of functional time series (FTS). We first establish a representation result stating that if the first $r$ eigenvalues of the covariance operator of the cross-section of $N$ FTS are unbounded as $N$ diverges and if the $(r+1)$th eigenvalue is bounded, then we can represent each FTS as a sum of a common component driven by $r$ factors, common to all the series, and a weakly cross-correlated idiosyncratic component (all the eigenvalues of the corresponding covariance operator bounded as $N\to\infty$). Our model and theory are developed in a general Hilbert space setting that allows panels mixing functional and scalar time series. We then turn to the estimation of the factors, their loadings, and the common components. We derive consistency results in the asymptotic regime where the number $N$ of series and the number $T$ of time observations diverge, thus exemplifying the "blessing of dimensionality" that explains the success of factor models in the context of high-dimensional (scalar) time series. Our results encompass the scalar factor models, for which they reproduce and extend, under weaker conditions, well-established results (Bai & Ng 2002). We also provide a family of information criteria for identifying the number of factors, and prove that they consistently estimate the correct number of factors as $N$ and $T$ diverge. We provide numerical illustrations that corroborate the convergence rates predicted by the theory, and provide finer understanding of the interplay between $N$ and $T$ for estimation purposes.