High dimensional covariance matrix estimation using a factor model

By Jianqing Fan, Yingying Fan, Jinchi Lv. 2007

712 citations recorded by the source corpus.

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p tends to [infinity] as the sample size n increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K is allowed to grow with p. We investigate the impact of p and K on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied. The asymptotic results are supported by a thorough simulation study.

Subfields: Finance

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Citations in the corpus

Sufficient Forecasting Using Factor Models (2015, methodology): “Following Fan et al. (2015a), the unobserved latent vectors ft in the semi-parametric factor”
Improving Mean Variance Optimization through Sparse Hedging Restrictions (2015, background): “Fan, Fan, and Lv (2008) also report the advantage of employing a factor structure in optimal portfolio construction.”
OPTIMAL SPARSE VOLATILITY MATRIX ESTIMATION FOR HIGH-DIMENSIONAL ITÔ PROCESSES WITH MEASUREMENT ERRORS (2013, methodology): “Fan, Fan and Lv (2008) utilized factor models for estimating large covariance matrices.”
Modeling Dependence in High Dimensions With Factor Copulas (2015, background): “…to determine the degree of flexibility 1For recent work on high-dimensional covariance matrix estimation, see Engle, Shephard, and Sheppard (2008), Fan, Fan, and Lv (2008), Engle and Kelly (2012), Fan, Li, and Ke (2012), and Hautsch, Kyj, and Oomen (2012). based on the number of variables and the…”
Regularized estimation of large covariance matrices (2008, neutral): “19 09 v1 [ m at h. ST ] 1 3 M ar 2 00 8 The Annals of Statistics 2008, Vol. 36, No. 1, 199–227 DOI: 10.1214/009053607000000758 c© Institute of Mathematical Statistics, 2008”
Informative Estimation and Selection of Correlation Structure for Longitudinal Data (2012, methodology): “…the estimation of the covariance matrix rather than on the selection of correlation structure, including the Cholesky decomposition approach (Huang et al. 2006; Huang, Liu, and Liu 2007), the factor modeling approach (Fan, Fan, and Lv 2008), and the spectrum random matrix approach (El Karoui 2008).”
Econometric Analysis of Vast Covariance Matrices Using Composite Realized Kernels and Their Application to Portfolio Choice (2016, methodology): “The use of factor models is analyzed by Fan, Fan, and Lv (2008) and Fan, Li, and Yu (2012).”
Latent variable graphical model selection via convex optimization (2010, methodology): “Other techniques that have been studied include methods based on shrinkage [19], [26] and factor analysis [14].”
CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS (2008, background): “The monographs of Ramsay and Silverman (2005) and Ferraty and Vieu (2006) give detailed accounts of the applications of functional data approach to various problems in these fields.”
Theory and Practice of GVAR Modeling (2016, positive): “…proposed in the literature for estimation of high-dimensional covariance matrices, including Ledoit and Wolf (2004), Bickel and Levina (2008), Fan et al. (2008), Friedman et al. (2008), the shrinkage estimator considered in Dées et al. (2014), and the multiple testing approach by Bailey et…”
Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices (2012, background): “), Khare and Rajaratnam (201 1), Fan, Fan and Lv (2008) and the references therein.”
High-dimensional minimum variance portfolio estimation based on high-frequency data (2020, positive): “This is based on Fan et al. (2008). We use the S&P 500 ETF (SPY) as the factor. For each month, we use the daily returns during the past 12 months to construct the covariance matrix estimator Σ̂t, and plug its inverse into equation (2.3); (iv) Nonlinear shrinkage portfolio. This is based on Ledoit a”
Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks (2017, neutral): “• FM: The dominating portfolio of Frahm and Memmel (2010). The particular version we use is defined in their equation (10), where the reference portfolio wR is given by the equal-weighted portfolio.”
Covariance Regression Analysis (2017, neutral): “…role in many fields, including but not limited to: finance and risk management (Markowitz 1952; Jagannathan and Ma 2003), econometrics (Chen and Conley 2001; Fan, Fan, and Lv 2008), biostatistics (Tong and Wang 2007; Friedman, Hastie, and Tibshirani 2008), and machine learning (Bilmes 2000).”
Vast Portfolio Selection with Gross-exposure Constraints (2012, positive): “Fan, Fan, and Lv (2008) studied the covariance matrix estimated based on the factor model and demonstrated that the resulting allocation vector significantly outperforms the allocation vector based on the sample covariance.”