Abstract
The estimation of the large and high-dimensional covariance matrix and precision matrix is a fundamental problem in modern multivariate analysis. It has been widely applied in economics, finance, biology, social networks and health sciences. However, the traditional sample estimators perform poorly for large and high-dimensional data. There are many approaches to improve the covariance matrix estimation. The large dynamic conditional correlation model based on the nonlinear shrinkage and its application in portfolio selection attract increasing attention. In the estimation of the unconditional covariance matrix, the graphical lasso is more robust than the nonlinear shrinkage model, and the leptokurtic and fat tail characteristics of the asset returns are also more obvious. This article proposes improved large dynamic covariance matrix estimation based on the graphical lasso models under the multivariate normal distribution (glasso) and t distribution (tlasso), and the corresponding dynamic conditional correlation glasso and tlasso approaches are developed. To verify the effectiveness and robustness of the proposed methods, we conduct simulations and then apply the models to the classic Markowitz portfolio selection problem. Simulations and empirical results show that the combined dynamic conditional correlation glasso and tlasso approaches outperform the current dynamic covariance matrix estimators.
Highlights
As an essential input to many financial models, the covariance matrix plays a vital role in asset allocation and risk management
Numerical results showed that portfolios constructed by the glasso method can significantly reduce the risk of out-of-samples compared to the nonlinear shrinkage estimation
The bold red curve is a normal distribution with a mean value of 0.00025 and a standard deviation of 0.0169
Summary
As an essential input to many financial models, the covariance matrix plays a vital role in asset allocation and risk management. Ledoit and Wolf [4] proposed the classical nonlinear shrinkage estimation method based on random matrix theory, and subsequently introduced a non-random multivariate function, the quantized eigenvalues sampling transform, to the nonlinear contraction to improve the estimation of the large-dimensional covariance matrix [12]. An optimal nonlinear shrinkage estimation method for large-dimensional covariance matrix under Stein’s loss is proposed [14]. Considering that the glasso method in [26] and [27] is significantly better than the nonlinear shrinkage in estimating the unconditional covariance matrix, and that asset returns do not obey the normal distribution [5], [11], we apply the glasso and tlasso sparse estimation methods to estimate the unconditional covariance matrix Qin the DCC model. Using the sample covariance to construct a minimum variance portfolio will encounter the problem of large noise
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