Covariance Matrix Estimation under Total Positivity for Portfolio Selection

Abstract

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 (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 MTP 2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.

Optimal Rates for Estimation of Two-Dimensional Totally Positive Distributions

“…Observe that h i,j ( δi,j ) is a continuous function of δi,j as a consequence of the bounded convergence theorem. Hence, the intermediate value theorem allows us to conclude (6.4.3) if we can show that h i,j (δ) ≤ H and h i,j (C 3 δ) ≥ H. The first inequality h i,j (δ) ≤ H follows from the fact that exp z 1 z 2 + δi,j g i,j (z) − exp(z 1 z 2 ) ≤ exp 1 + δi,j g i,j (z) − exp (1) and changing the limits of the integral. The second inequality h i,j (C 3 δ) ≥ H follows from the following estimates.…”

“…Smooth MTP 2 distributions have long been studied in the literature. Examples include, but are not limited to, (1) pairwise marginals of Gaussian latent tree models [13], such as Brownian motion tree models and 1 factor analysis models, (2) joint distributions of pairs of time points of a strong Markov process on the real line with continuous paths [23], such as a diffusion process, and (3) MTP 2 transelliptical distributions, such as MTP 2 multivariate t-distributions, which are commonly used in finance [1].…”

“…As a stepping stone to this problem, we also consider the following discrete version of MTP 2 . A distribution on the grid [n 1 ] × [n 2 ] is MTP 2 if its probability mass function (PMF) p, which is an n 1 × n 2 matrix, fulfills (1). Thus, MTP 2 says that all the 2 × 2 minors of p are non-negative:…”

Covariance estimation with nonnegative partial correlations

“…Note that the precision matrix having nonpositive off-diagonal entries Θ * jk is equivalent to nonnegative partial correlations −Θ * jk / Θ * jj Θ * kk (Bølviken, 1982). Examples of practical covariance estimation problems with nonnegative partial correlations abound (see, e.g., Lake and Tenenbaum, 2010;Slawski and Hein, 2015;Agrawal, Roy and Uhler, 2019). More generally, Karlin and Rinott (1983) showed that for the normal distribution the condition that the precision matrix belongs to M p×p is equivalent to multivariate total positivity of order two (MTP 2 ).…”