Selecting the optimal Markowitz portfolio 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 (MTP2). 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 30 years, we show that estimating the covariance matrix under MTP2 outperforms previous state-of-the-art methods including shrinkage estimators and factor models.
Many real-world decision-making tasks require learning casual relationships between a set of variables. Typical causal discovery methods, however, require that all variables are observed, which might not be realistic in practice. Unfortunately, in the presence of latent confounding, recovering casual relationships from observational data without making additional assumptions is an ill-posed problem. Fortunately, in practice, additional structure among the confounders can be expected, one such example being pervasive confounding, which has been exploited for consistent causal estimation in the special case of linear causal models. In this paper, we provide a proof and method to estimate causal relationships in the non-linear, pervasive confounding setting. The heart of our procedure relies on the ability to estimate the pervasive confounding variation through a simple spectral decomposition of the observed data matrix. We derive a DAG score function based on this insight, and empirically compare our method to existing procedures. We show improved performance on both simulated and real datasets by explicitly accounting for both confounders and non-linear effects.
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.
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