Robust factor models for high-dimensional time series and their forecasting

Abstract

We first present some concepts and properties, which are used again and again below. ∥B∥ F , denoting the Frobenius norm of the matrix B, is equal to the positive square root of the sum of the squares for every entry of B. ∥B∥ 2 is the spectral norm of the matrix B, which is equal to the positive square root of the maximum eigenvalue of B T B or BB T .σ i (B) is ith largest singular value, and. Generally, the following properties hold.(1)(5) If B is an m × (n − 1) matrix obtained by deleting any column of an m × n matrix A, ), where r = min{m, n}.(6) For an n×n matrix A, if λ is an eigenvalue of A, then λ+σ 2 is an eigenvalue of A+σ 2 I; and 1/λ is an eigenvalue of its inverse matrix A −1 .(7) If an eigenvalue of a matrix A is λ, then the corresponding eigenvalue of the matrix polynomial(8) If there is p > 0 such that E(|ξ n | p ), n ≥ 1, is bounded, then ξ n = O P (1). In the properties above, (1) (7) can be found in Zhang (2017), and (8) in Jiang (2010). Now, we give proofs of Lemma and Theorem. We point out that the proof of Theorem 3.1 and 3.2 is quite similar to that of Theorem 1 of Lam et al. (2011) and Theorem 1 of Lam and Yao (2012).

A dimension reduction factor approach for multivariate time series with long-memory: a robust alternative method

“…Other data reduction methods were proposed in the 1980s see, for example, a review in (Fernández-Macho, 1997) and for more recent developments, see (Christou, 2020). Several recent surveys have been dedicated to dynamic factors models, including stationary and non-stationary times series processes in dierent areas of applications such as environmental, health, and nancial sciences see, for example, (Eichler et al, 2011;Lam et al, 2011;Toman, 2014;Bai and Wang, 2016;Chen et al, 2020;Fan et al, 2021;Lin et al, 2022;Bai and Zheng, 2023) to mention a few. Peña and Box (1987) have proposed a simple model to identify hidden factors in multivariate short-memory processes.…”

“…These authors have established asymptotic properties of the estimator of the number of factors and have discussed the method's usefulness in air pollution data. Bai and Zheng (2023) discussed the robust method introduced by Reisen et al (2019). The authors also suggested an algorithm for constructing bootstrap prediction intervals for the high-dimensional time series.…”