The redistributive effects of monetary policy

We introduce a model of the economy as a social network. Two agents are linked to the extent that they transact with each other. This generates well-defined topological notions of location, neighborhood and closeness. We investigate the implications of our model for monetary economics. When a central bank increases the money supply, it must inject the money somewhere in the economy. We demonstrate that the agent closest to the location where money is injected is better off, and the one furthest is worse off. This redistribution channel is independent from the ones previously noted in the literature. Symmetrically, any decrease in the money supply redistributes purchasing power in the other direction. We also outline the testable implications of our model.Money, redistribution, policy, central bank, social network, topology

Numerical Implementation of the QuEST Function

This paper deals with certain estimation problems involving the covariance matrix in large dimensions. Due to the breakdown of finite-dimensional asymptotic theory when the dimension is not negligible with respect to the sample size, it is necessary to resort to an alternative framework known as large-dimensional asymptotics. Recently, Ledoit and Wolf (2015) have proposed an estimator of the eigenvalues of the population covariance matrix that is consistent according to a mean-square criterion under large-dimensional asymptotics. It requires numerical inversion of a multivariate nonrandom function which they call the QuEST function. The present paper explains how to numerically implement the QuEST function in practice through a series of six successive steps. It also provides an algorithm to compute the Jacobian analytically, which is necessary for numerical inversion by a nonlinear optimizer. Monte Carlo simulations document the effectiveness of the code.Comment: 35 pages, 8 figure

Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.Comment: 40 pages, 8 figures, 5 tables, University of Zurich, Department of Economics, Working Paper No. 105, Revised version, July 201

Nonlinear shrinkage estimation of large-dimensional covariance matrices

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Honey, I shrunk the sample covariance matrix

The central message of this paper is that nobody should be using the sample covariance matrix for the purpose of portfolio optimization. It contains estimation error of the kind most likely to perturb a mean-variance optimizer. In its place, we suggest using the matrix obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients towards more central values, thereby systematically reducing estimation error where it matters most. Statistically, the challenge is to know the optimal shrinkage intensity, and we give the formula for that. Without changing any other step in the portfolio optimization process, we show on actual stock market data that shrinkage reduces tracking error relative to a benchmark index, and substantially increases the realized information ratio of the active portfolio manager.Covariance matrix, Markovitz optimization, shrinkage, tracking error

Improved estimation of the covariance matrix of stock returns with an application to portofolio selection

This paper proposes to estimate the covariance matrix of stock returns by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method is generally known as shrinkage, and it is standard in decision theory and in empirical Bayesian statistics. Our shrinkage estimator can be seen as a way to account for extra-market covariance without having to specify an arbitrary multi-factor structure. For NYSE and AMEX stock returns from 1972 to 1995, it can be used to select portfolios with significantly lower out-of-sample variance than a set of existing estimators, including multi-factor models.Covariance matrix estimation, factor models, portofolio selection, shrinkage

The coexistence of commodity money and fiat money

In reaction to the monetary turmoil created by the financial crisis of September 2008, both legislative and constitutional reforms have been proposed in different Countries to introduce Commodity Money longside existing National Fiat Currency. A thorough evaluation of the Economic consequences of these new proposals is warranted. This paper surveys some of the existing knowledge in Monetary and Financial Economics for the purpose of answering the significant Economic questions raised by these new political initiatives.Currency competition, commodity money, fiat money, gold, safe haven, search models

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite non-zero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.Concentration asymptotics, equality test, sphericity test

Carbon portfolio management

The aim of the European Union's Emissions Trading Scheme (EU ETS) is that by 2020, emissions from sectors covered by the EU ETS will be 21% lower than in 2005. In addition to large CO 2 emitting companies covered by the scheme, other participants have entered the market with a view of using emission allowances for the diversification of their investment portfolios. The performance of this asset as a stand alone investment and its portfolio diversification implications will be investigated in this paper. Our results indicate that the market views Phases 1, 2, and 3 European Union allowance futures as unattractive as stand alone investments. In a portfolio context, in Phase 1, once the short-selling option is added, there are considerable portfolio benefits. However, our results indicate that these benefits only existed briefly during the pilot stage of the EU ETS. There is no evidence to suggest portfolio diversification benefits exist for Phase 2 or the early stages of Phase 3