Algorithms and applications for Gaussian graphical models under the multivariate totally positive constraint of order 2

Abstract

We consider the problem of estimating an undirected Gaussian graphical model when the underlying distribution is multivariate totally positive of order 2 (MTP₂), a strong form of positive dependence. A large body of methods have been proposed for learning undirected graphical models without the MTP₂ constraint. A major limitation of these methods is that their consistency guarantees in the high-dimensional setting usually require a particular choice of a tuning parameter, which is unknown a priori in real world applications. We show that an undirected graphical model under MTP₂ can be learned consistently without any tuning parameters. We evaluate this new estimator on synthetic and real-world financial data sets, showing that it out-performs other methods in the literature with tuning parameters. We further explore applications of estimators in the MTP₂ setting to covariance estimation for finance. In particular, the very well-explored optimal Markowitz portfolio allocation problem requires a precise estimate of the covariance matrix of returns. By exploiting the fact that the returns of assets are typically positively dependent, we propose a new estimator based on MTP₂ estimation and show that this estimator outperforms (in terms of out-of-sample risk) baseline methods including shrinkage techniques and explicitly providing market factors on stock-market data spanning over thirty years.