estimating high dimensional covariance matrices and its….pdf

ANNALS OF ECONOMICS AND FINANCE12-2, 199–215 (2011) Estimating High Dimensional Covariance Matrices and its Applications Jushan Bai * Department of Economics, Columbia University, New York, NY 10027 CEMA, the Central University of Finance and Economics, Beijing, China and Shuzhong Shi Department of Finance, Guanghua School of Management, Beijing, China Estimating covariance matrices is an important part of portfolio selection, risk management, and asset pricing. This paper reviews the recent develop- ment in estimating high dimensional covariance matrices, where the number of variables can be greater than the number of observations. The limitations of the sample covariance matrix are discussed. Several new approaches are presented, including the shrinkage method, the observable and latent factor method, the Bayesian approach, and the random matrix theory approach. For each method, the construction of covariance matrices is given. The relation- ships among these methods are discussed. Key Words: Factor analysis; ponents; Singular value posi- tion; Random matrix theory; Empirical Bayes; Shrinkage method; Optimal portfo- lios; CAPM; APT; GMM. JEL Classi?cation Numbers: C33, C38. 1. INTRODUCTION Estimating covariance matrices is an important part of portfolio selec- tion, risk management, and asset pricing. The sample covariance matrix is often used for these purposes, but the sample covariance matrix has a number of undesirable properties when the dimension of the matrix is large. *Financial support from the NSF (grants SES-0551275 and SES-0962410 ) is acknowl- edged. 199 1529-7373/2011 All rights of reproduction in any form reserved. 200 JUSHAN BAI AND SHUZHONG SHI First, when the number of assets (N) is larger than the number of obser- vations (T), the sample covariance matrix is not of full rank, so its inverse will not exist. Second, even if the sample covariance matrix is invertible, the expected value of its inverse is a biased estimator for the theoretical inve