  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.
