Large Covariance Estimation by Thresholding Principal Orthogonal Complements

By Jianqing Fan, Yuan Liao, Martina Mincheva. 2011

963 citations recorded by the source corpus.

The paper deals with the estimation of a high dimensional covariance with a conditional sparsity structure and fast diverging eigenvalues. By assuming a sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross‐sectional correlation even after taking out common but unobservable factors. We introduce the principal orthogonal complement thresholding method ‘POET’ to explore such an approximate factor structure with sparsity. The POET‐estimator includes the sample covariance matrix, the factor‐based covariance matrix, the thresholding estimator and the adaptive thresholding estimator as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the effect of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Subfields: Economics and Econometrics, Statistics and Probability

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Citations in the corpus

Principal Components and Regularized Estimation of Factor Models (2017, methodology): “A second approach is the POET estimator considered in Fan et al. (2013), which uses thresholding to enforce sparsity of the idiosyncratic errors of a spiked-covariance matrix.”
Econometric Analysis of Vast Covariance Matrices Using Composite Realized Kernels and Their Application to Portfolio Choice (2016, neutral): “Here, the diagonal elements of R̃ are all one, and so are identical to those of R̂ and the correlation structure is determined by the factor structure: see the related Fan, Liao, and Mincheva (2013) who approached things from a distinctive viewpoint.”
On the number of common factors with high‐frequency data (2017, background): “It is a continuous-time analogue of Assumption B in Bai & Ng (2002) and Assumption 1 in Fan et al. (2013).”
Autoregressive models for matrix-valued time series (2018, background): “, 2011; Lam and Yao, 2012; Fan et al., 2013; Wang et al., 2019, among others). In most of these studies, the multiple observations at each time point are treated as a vector. Although it has been conventional to treat multiple observations as a vector, often the inter-relationship among the time ser”
Semiparametric Ultra-High Dimensional Model Averaging of Nonlinear Dynamic Time Series (2018, methodology): “For example, the conditions B2 and B3 are similar to Assumptions A and B in Bai and Ng (2002), whereas the condition B4 is similar to the corresponding conditions in Assumption 3.4 in Fan, Liao, and Mincheva (2013).”
An Overview on the Estimation of Large Covariance and Precision Matrices (2015, positive): “It was then shown by Fan et al. (2013) that max i,j≤p |su,ij − σu,ij| = OP (ω̃T ).”
NETS: Network estimation for time series (2019, neutral): “the so–called idiosyncratic component zt is allowed to be mildly cross–correlated Forni and Lippi (2001). This model is a generalization of the approximate factor model by Bai (2003).”
PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS (2014, background): “See, for example, Cand`es and Recht (2009); Fan et al. (2011); Koltchinskii et al. (2011); Negahban and Wainwright (2011); Fan et al. (2013); Cai et al. (2013); Ma (2013).”
On factor models with random missing: EM estimation, inference, and cross validation (2020, methodology): “In addition, Foroni and Marcellino (2013) survey methods for handling mixed-frequency data, including dynamic factor models and alternative approaches; Stock and Watson (2016) summarize the advantage and disadvantage of the state space estimation for factor models with missing observations; Athey…”
Multitask Quantile Regression under the Transnormal Model (2016, background): “…Gaussian model in high-dimensional statistical problems, including linear or quadratic discriminant analysis (Lin and Jeon 2003; Mai and Zou 2015; Fan et al. 2013), graphical modeling (Liu, Lafferty, and Wasserman 2009; Liu et al. 2012; Xue and Zou 2012), covariance matrix estimation (Xue and…”
Understanding Systematic Risk: A High‐Frequency Approach (2020, neutral): “Fan, Liao, and Mincheva (2013) study an approximate factor structure with sparsity.”
Generalized dynamic factor models and volatilities: estimation and forecasting (2017, positive): “2Recently, Fan et al. (2013, 2015) improved on this specific aspect by allowing for sparsity in the idiosyncratic covariance matrix of returns; but then, they just discard the idiosyncrati contribution to total volatility. interpreted as driving “market volatility” but nothing canbe said about…”
Limiting laws for divergent spiked eigenvalues and largest nonspiked eigenvalue of sample covariance matrices (2017, background): “[4] and [21] assume that the leading eigenvalues of the pervasive factor model are of order p.”
Modeling and Forecasting Large Realized Covariance Matrices and Portfolio Choice (2017, neutral): “Several other papers have dealt with the estimation of vast covariance matrices, e.g. Bickel and Levina (2008), Levina et al. (2008), Fan et al. (2008), Wang and Zou (2010), Fan et al. (2011), Fan et al. (2012), Fan et al. (2012), Hautsch et al. (2012), Fan et al. (2013).”