Selecting the optimal Markowitz porfolio depends on estimating the covariance matrix of the returns of $N$ assets from $T$ periods of historical data. Problematically, $N$ is typically of the same order as $T$, which makes the sample covariance matrix estimator perform poorly, both empirically and theoretically. While various other general purpose covariance matrix estimators have been introduced in the financial economics and statistics literature for dealing with the high dimensionality of this problem, we here propose an estimator that exploits the fact that assets are typically positively dependent. This is achieved by imposing that the joint distribution of returns be multivariate totally positive of order 2 ($\text{MTP}_2$). This constraint on the covariance matrix not only enforces positive dependence among the assets, but also regularizes the covariance matrix, leading to desirable statistical properties such as sparsity. Based on stock-market data spanning over thirty years, we show that estimating the covariance matrix under $\text{MTP}_2$ outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

中文翻译：

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总积极性下投资组合选择的协方差矩阵估计*

选择最佳的Markowitz组合取决于从历史数据的$ T $期间估算$ N $资产收益的协方差矩阵。有问题的是，$ N $通常与$ T $处于同一数量级，这使得样本协方差矩阵估计器在经验和理论上均表现不佳。尽管在金融经济学和统计文献中已经引入了各种其他通用协方差矩阵估计器来处理此问题的高维度，但我们在此提出了一种估计器，该估计器利用了资产通常是正相关的这一事实。这是通过强加收益的联合分布为阶数为2的完全为正的多元变量（$ \ text {MTP} _2 $）实现的。对协方差矩阵的这种约束不仅迫使资产之间产生正相关关系，而且还会规范协方差矩阵，从而产生理想的统计特性，例如稀疏性。根据三十年来的股票市场数据，我们显示，在$ \ text {MTP} _2 $下估计协方差矩阵的性能优于以前的最新方法，包括收缩估计器和因子模型。