Factor Modelling for High‐Dimensional Time Series: Inference and Model Selection

Abstract

Analysis of high‐dimensional time series data is of increasing interest among different fields. This article studies high‐dimensional time series from a dimension reduction perspective using factor modelling. Statistical inference is conducted using eigen‐analysis of a certain non‐negative definite matrix related to autocovariance matrices of the time series, which is applicable to fixed or increasing dimension. When the dimension goes to infinity, the rate of convergence and limiting distributions of estimated factors are established. Using the limiting distributions of estimated factors, a high‐dimensional final prediction error criterion is proposed to select the number of factors. Asymptotic properties of the criterion are illustrated by simulation studies and real applications.

On determination of the number of factors in an approximate factor model

“…Many references on various factor models for economic or financial data with large numbers of cross-sectional units and a large sample of time series observations have appeared in literature. These factor models could be classified into three categories: Static approximate factor models proposed by Chamberlain and Rothschild (2014) and followed by Connor and Korajzcyk (1993), Bai and Ng (2002), Stock and Watson (2002), Onatski (2010), Alessi, Barigozzi, and Capasso (2010), Fan, Liao, and Mincheva (2013), Ahn and Horenstein (2013), Caner and Han (2014), Li, Li, and Shi (2017); Factor models for multivariate time series, see Pan and Yao (2008), Li and Pan (2008), Lam, Yao, and Bathia (2011), Lam and Yao (2012), Xia, Xu, and Zhu (2015), Chan, Lu, and Yau (2017), Xia, Liang, Wu, and Wong (2018) and Xia, Wong, Shen, and He (2022); and So-called dynamic factor models, see Forni, Wallin, Lippi, and Reichlin (2000), Hallin and Liska (2007), Amengual and Watson (2007), Bai and Ng (2007) and Onatski (2009), among others. Some of these references assume that factors are given and subsequently estimate factor loadings.…”

On high-dimensional wavelet eigenanalysis

“…The literature on random matrices under dependence as well as on high-dimensional stochastic processes has been expanding at a fast pace (e.g., Basu and Michailidis (2015), Chakrabarty et al (2016), Merlevède and Peligrad (2016), Che (2017), Steland and von Sachs (2017), Taylor and Salhi (2017), Wang et al (2017), Zhang and Wu (2017), Merlevède et al (2019)). In applications, models of the form (1.2) appear in neuroscience (see Liu et al (2015), Section 3.4), in particular as related to fMRI or M/EEG imaging (Ciuciu et al (2012)), and also, for example, in econometrics (e.g., Brown (1989), Forni and Lippi (1999), Lam and Yao (2012), Chan et al (2017)).…”

Factor analysis for high‐dimensional time series: Consistent estimation and efficient computation

“…and there is a rising number of literature focusing on this topic. For example, [30,11,41,10], all considered a decomposition such that the dynamic part is fully driven by r common factors and the static part is a vector of white noise. Recent developments of the latent factor model also include [36,37] for the structure constrained factor models, where the column space of the factor loading matrix is assumed to be constrained by a prespecified low rank matrix.…”