Factor modeling for high-dimensional time series: inference for the number of factors

By Clifford Lam, Qiwei Yao. 2012

548 citations recorded by the source corpus.

This paper deals with the factor modeling for high-dimensional time series based on a dimension-reduction viewpoint. Under stationary settings, the inference is simple in the sense that both the number of factors and the factor loadings are estimated in terms of an eigenanalysis for a nonnegative definite matrix, and is therefore applicable when the dimension of time series is on the order of a few thousands. Asymptotic properties of the proposed method are investigated under two settings: (i) the sample size goes to infinity while the dimension of time series is fixed; and (ii) both the sample size and the dimension of time series go to infinity together. In particular, our estimators for zero-eigenvalues enjoy faster convergence (or slower divergence) rates, hence making the estimation for the number of factors easier. In particular, when the sample size and the dimension of time series go to infinity together, the estimators for the eigenvalues are no longer consistent. However, our estimator for the number of the factors, which is based on the ratios of the estimated eigenvalues, still works fine. Furthermore, this estimation shows the so-called “blessing of dimensionality” property in the sense that the performance of the estimation may improve when the dimension of time series increases. A two-step procedure is investigated when the factors are of different degrees of strength. Numerical illustration with both simulated and real data is also reported.

Subfields: Artificial Intelligence, Computational Theory and Mathematics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Yuan Liao
2. Jianqing Fan
3. Lingzhou Xue
4. Clifford Lam
5. Peter M. Robinson
6. Rong Chen
7. Jiawei Yao
8. Jushan Bai

In Computational Theory and Mathematics

1. Jianqing Fan
2. Liangjun Su
3. Elynn Y. Chen
4. Yuan Liao
5. Qiwei Yao
6. Clifford Lam
7. Rong Chen
8. Matteo Barigozzi

In Statistics and Probability

1. Jianqing Fan
2. Liangjun Su
3. Yuan Liao
4. Clifford Lam
5. Lingzhou Xue
6. Jushan Bai
7. Qiwei Yao
8. Jiawei Yao

Citations in the corpus

• Sufficient Forecasting Using Factor Models (2015, background): “Recently, Lam and Yao (2012) and Ahn and Horenstein (2013) proposed a ratio-based estimator by maximizing the ratio of two adjacent eigenvalues of X (cid:48) X arranged in descending order, i.e. where ˆ λ 1 ≥ · · · ≥ ˆ λ T are the eigenvalues.”
• Generalized Dynamic Principal Components (2016, background): “Some of these models assume stationarity (Peña and Box 1987; Stock andWatson 1988, 2002; Bai andNg 2002, and Lam and Yao 2012, among others).”
• Principal component analysis for second-order stationary vector time series (2014, methodology): “An incomplete list of the publications includes Anderson (1963), Peña and Box (1987), Bai and Ng (2002), Theis, Meyer-Baese and Lang (2004), Stock and Watson (2005), Forni et al. (2005), Pan and Yao (2008), Lam, Yao and Bathia (2011), Lam and Yao (2012) and Chang, Guo and Yao (2015).”
• A network analysis of the volatility of high dimensional financial series (2017, background): “…the GDFM by Forni et al. (2000) and Forni and Lippi (2001) is the most general, whereas most other factor models that have been considered in the time series literature (Stock and Watson (2002), Bai and Ng (2002), Lam and Yao (2012) and Fan et al. (2013), to quote only a few) are particular cases.”
• Statistical Inference for High-Dimensional Matrix-Variate Factor Models (2020, methodology): “With respect to statistical analyses, Wang et al. (2019) and Chen et al. (2019) consider a similar model in the bilinear form (1.1), yet under a very di ff erent setting where E t is assumed to be white noise (Lam and Yao, 2012; Lam et al., 2011).”
• Constrained Factor Models for High-Dimensional Matrix-Variate Time Series (2017, methodology): “Lam and Yao (2012) used a two-step approach to facilitate the discovery of weak factors that may be masked by strong factors.”
• Estimating Number of Factors by Adjusted Eigenvalues Thresholding (2019, methodology): “) if s>1 + p p=(n 1); In other words, the threshold s= 1+ p p=(n 1) is optimal. We have conducted extensive simulations to compare our method with those in Bai and Ng (2002), Onatski (2005, 2010), Lam and Yao (2012), Ahn and Horenstein (2013). Simulation results 8 show that in most of our testing”
• Factor models for matrix-valued high-dimensional time series (2016, methodology): “In what follows, the subscripts mat and vec denote our approach and Lam and Yao (2012)’s method, respectively.”
• PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS (2014, methodology): “More recently, Ahn and Horenstein (2013) and Lam and Yao (2012) proposed a tuning parameter-free estimator, which selects the largest ratio of the adjacent eigenvalues of Y (cid:48) Y , based on the fact that the K largest eigenvalues of the sample covariance matrix grow unboundedly as p increases,…”
• Inverse Moment Methods for Sufficient Forecasting Using High-Dimensional Predictors (2016, methodology): “3 and letting Kmax = op(min{p, T }), Li et al. (2013) showed that the data-driven K̂ is a consistent estimator of K.”
• Long-Range Dependent Curve Time Series (2020, methodology): “Instead, we use a simple ratio method introduced by Lam and Yao (2012) to determine p1.”
• Identifying the number of factors from singular values of a large sample auto-covariance matrix (2014, methodology): “On the other hand, the one-step estimator ˜ k of Lam and Yao [16] tends to detect only one factor in each step due to the fact that the two factors are of different strength.”
• Efficient Estimation of Approximate Factor Models (2012, methodology): “In practice, r can be estimated from the data, and there has been a large literature addressing its consistent estimation, e.g., Bai and Ng (2002), Kapetanios (2010), Onatski (2010), Alessi et al. (2010), Hallin and Lǐska (2007), Lam and Yao (2012), among others.”
• Estimation of large dimensional factor models with an unknown number of breaks (2018, methodology): “Macroeconomic Data Set (Stock and Watson, 2009) to detect possible structural breaks in the underlying factor model. The data set consists of N = 108 monthly macroeconomic time-series variables including real economic activity measures, prices, interest rates, money and credit aggregates, stock pric”
• An Overview on the Estimation of Large Covariance and Precision Matrices (2015, unknown): “See, for instance, Stock and Watson (2002); Bai (2003); Bai and Ng (2002); Fan et al. (2008); Breitung and Tenhofen (2011); Onatski (2012); Lam and Yao (2012); Fan et al. (2013), among others.”