A simple realized factor-based portfolio: improving minimum variance portfolio performance by incorporating low-frequency betas

In many engineering applications, estimation of covariance and precision matrices is of great importance, helping researchers understand the dependency and conditional dependency between variables of interest. Among various matrix estimation methods, the modified Cholesky decomposition is a commonly used technique. It has the advantage of transforming the matrix estimation task into solving a sequence of regression models. Moreover, the sparsity on the regression coefficients implies certain sparse structure on the covariance and precision matrices. In this chapter, we first overview the Cholesky-based covariance and precision matrices estimation. It is known that the Cholesky-based matrix estimation depends on a prespecified ordering of variables, which is often not available in practice. To address this issue, we then introduce several techniques to enhance the Cholesky-based estimation of covariance and precision matrices. These approaches are able to ensure the positive definiteness of the matrix estimate and applicable in general situations without specifying the ordering of variables. The advantage of Cholesky-based estimation is illustrated by numerical studies and several real-case applications.

Estimation of the covariance matrix and its inverse, the precision matrix, in high-dimensional situations is of great interest in many applications. In this paper, we focus on the estimation of a class of sparse precision matrices which are assumed to be approximately inversely closed for the case that the dimensionality p can be much larger than the sample size n, which is fundamentally different from the classical case that p <; n. Different in nature from state-of-the-art methods that are based on penalized likelihood maximization or constrained error minimization, based on the truncated Neumann series representation, we propose a computationally efficient precision matrix estimator that has a computational complexity of O (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ). We prove that the proposed estimator is consistent in probability and in L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> under the spectral norm. Moreover, its convergence is shown to be rate-optimal in the sense of minimax risk. We further prove that the proposed estimator is model selection consistent by establishing a convergence result under the entry-wise ∞-norm. Simulations demonstrate the encouraging finite sample size performance and computational advantage of the proposed estimator. The proposed estimator is also applied to a real breast cancer data and shown to outperform existing precision matrix estimators.

Estimation of sparse covariance matrices and their inverse subject to positive definiteness constraints has drawn a lot of attention in recent years. The abundance of high-dimensional data, where the sample size (n) is less than the dimension (d), requires shrinkage estimation methods since the maximum likelihood estimator is not positive definite in this case. Furthermore, when n is larger than d but not sufficiently larger, shrinkage estimation is more stable than maximum likelihood as it reduces the condition number of the precision matrix. Frequentist methods have utilized penalized likelihood methods, whereas Bayesian approaches rely on matrix decompositions or Wishart priors for shrinkage. In this paper we propose a new method, called the Bayesian Covariance Lasso (BCLASSO), for the shrinkage estimation of a precision (covariance) matrix. We consider a class of priors for the precision matrix that leads to the popular frequentist penalties as special cases, develop a Bayes estimator for the precision matrix, and propose an efficient sampling scheme that does not precalculate boundaries for positive definiteness. The proposed method is permutation invariant and performs shrinkage and estimation simultaneously for non-full rank data. Simulations show that the proposed BCLASSO performs similarly as frequentist methods for non-full rank data.

Estimating large covariance and precision (inverse covariance) matrices has become increasingly important in high dimensional statistics because of its wide applications. The estimation problem is challenging not only theoretically due to the constraint of its positive definiteness, but also computationally because of the curse of dimensionality. Many types of estimators have been proposed such as thresholding under the sparsity assumption of the target matrix, banding and tapering the sample covariance matrix. However, these estimators are not always guaranteed to be positive-definite, especially, for finite samples, and the sparsity assumption is rather restrictive. We propose a novel two-stage adaptive method based on the Cholesky decomposition of a general covariance matrix. By banding the precision matrix in the first stage and adapting the estimates to the second stage estimation, we develop a computationally efficient and statistically accurate method for estimating high dimensional precision matrices. We demonstrate the finite-sample performance of the proposed method by simulations from autoregressive, moving average, and long-range dependent processes. We illustrate its wide applicability by analyzing financial data such S&amp;P 500 index and IBM stock returns, and electric power consumption of individual households. The theoretical properties of the proposed method are also investigated within a large class of covariance matrices.

ABSTRACTWe propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.

In this paper, we consider the estimation of a high dimensional precision matrix of Gaussian graphical model. Based on the re-parameterized likelihood, we obtain the full conditional distribution of all parameters in Cholesky factor. Furthermore, by imposing the prior information, we obtain the shrinkage Bayesian estimator of large precision matrix, and establish the asymptotic distribution of all parameters in the Cholesky factor. At last, we demonstrate our method through the simulation study and an application to telephone call center data.

Purpose: The study evaluates the performance of alternative variance-covariance estimators as a fundamental ingredient to portfolio optimization.&#x0D; Methodology: The study estimates eleven covariance matrices on the data of Pakistan stock exchange's non-financial sector firms covering the period from July 2006 to June 2020. The accuracy and efficiency of covariance estimators are assessed through two evaluation parameters: root mean square error and minimum variance portfolios (risk behavior).&#x0D; Main findings: Empirical findings based on evaluation parameters suggest that more complex covariance estimators in the equity market of Pakistan yield no additional financial gains than the equally weighted portfolio of estimators.&#x0D; Application of the study: As the estimation of the variance-covariance matrix is one of the essential elements of portfolio construction, this study guides investor(s) on selecting an appropriate covariance estimator among eleven estimators endorsed by literature.&#x0D; Novelty/ originality of the study: Based on detailed analysis, the study documents that investor(s) of the Pakistan stock exchange cannot gain any additional benefit from more complex and tricky methods of variance-covariance estimators compared to a portfolio of estimators for the non-financial sector. Investors are advised to consider the equally weighted portfolio of estimators when formulating their investment strategy.

We study the accuracy of estimating the covariance and the precision matrix of a $D$-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation accuracy depends almost exclusively on the components of the distribution that correspond to desired subspaces or directions. This is relevant and important for problems where the behavior of data along a lower-dimensional space is of specific interest, such as dimension reduction or structured regression problems. We also show that estimation of precision matrices is almost independent of the condition number of the covariance matrix. The presented applications include direction-sensitive eigenspace perturbation bounds, relative bounds for the smallest eigenvalue, and the estimation of the single-index model. For the latter, a new estimator, derived from the analysis, with strong theoretical guarantees and superior numerical performance is proposed.

This paper considers the ridge-type shrinkage estimation of the large dimensional precision matrix. The asymptotic optimal shrinkage coefficients and the theoretical loss are derived. Data-driven estimators for the shrinkage coefficients are also conducted based on the asymptotic results from random matrix theory. The new method is distribution-free and no assumption on the structure of the covariance matrix or the precision matrix is required. The proposed method also applies to situations where the dimension is larger than the sample size. Finally, numerical studies using simulated data and real data demonstrate that the proposed estimator performs better than the existing competitors in a wide range of settings.

In this article, we consider the problem of estimating the precision matrix when the sample data contains cellwise contamination. For the widely employed methodologies (e.g. Graphical Lasso), using the sample covariance matrix as an input matrix potentially deteriorates the precision matrix estimation performance in the presence of outliers. We propose several robust alternatives for the covariance matrix, which are constructed by combining robust correlation estimators with robust variation measures. Through extensive numerical studies we demonstrate the robust performance of our proposed approaches compared to the standard methods based on the sample covariance matrix. Further, we apply our proposals to a real data application, aimed at studying the optimal portfolio allocations in Shanghai Stock Exchange Composite index. The results show that the proposed alternatives provide desirable out-of-sample performance.

We introduce nonparametric regularization of the eigenvalues of a sample covariance matrix through splitting of the data (NERCOME), and prove that NERCOME enjoys asymptotic optimal nonlinear shrinkage of eigenvalues with respect to the Frobenius norm. One advantage of NERCOME is its computational speed when the dimension is not too large. We prove that NERCOME is positive definite almost surely, as long as the true covariance matrix is so, even when the dimension is larger than the sample size. With respect to the Stein’s loss function, the inverse of our estimator is asymptotically the optimal precision matrix estimator. Asymptotic efficiency loss is defined through comparison with an ideal estimator, which assumed the knowledge of the true covariance matrix. We show that the asymptotic efficiency loss of NERCOME is almost surely 0 with a suitable split location of the data. We also show that all the aforementioned optimality holds for data with a factor structure. Our method avoids the need to first estimate any unknowns from a factor model, and directly gives the covariance or precision matrix estimator, which can be useful when factor analysis is not the ultimate goal. We compare the performance of our estimators with other methods through extensive simulations and real data analysis.

The determination of the covariance matrix and its inverse, the precision matrix, is critical in the statistical analysis of cosmological measurements. The covariance matrix is typically estimated with a limited number of simulations at great computational cost before inversion into the precision matrix; therefore, it can be ill-conditioned and overly noisy when the sample size n used for estimation is not much larger than the data vector dimension. In this work, we consider a class of methods known as shrinkage estimation for the precision matrix, which combines an empirical estimate with a target that is either analytical or stochastic. These methods include linear and non-linear shrinkage applied to the covariance matrix (the latter represented by the so-called NERCOME estimator), and the direct linear shrinkage estimation of the precision matrix which we introduce in a cosmological setting. By performing Bayesian parameter inference and using metrics like matrix loss functions, the Kullback–Leibler divergence and the eigenvalue spectrum, we compare their performance against the standard sample estimator with varying sample size n. We have found the shrinkage estimators to significantly improve the posterior distribution at low n, especially for the linear shrinkage estimators either inverted from the covariance matrix or applied directly to the precision matrix, with an empirical target constructed from the sample estimate. Our results are particularly relevant to the analyses of Stage-IV spectroscopic galaxy surveys such as the Dark Energy Spectroscopic Instrument and Euclid, whose statistical power can be limited by the computational cost of obtaining an accurate precision matrix estimate.

A note on covariance estimation in the unbiased estimator of risk framework

End-to-end large portfolio optimization for variance minimization with neural networks through covariance cleaning

Journal Article Nonparametric estimation of large covariance matrices of longitudinal data Get access Wei Biao Wu, Wei Biao Wu Search for other works by this author on: Oxford Academic Google Scholar Mohsen Pourahmadi Mohsen Pourahmadi Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 90, Issue 4, December 2003, Pages 831–844, https://doi.org/10.1093/biomet/90.4.831 Published: 01 December 2003