Econometrica: Jul 2021, Volume 89, Issue 4

A New Parametrization of Correlation Matrices

https://doi.org/10.3982/ECTA16910
p. 1699-1715

Ilya Archakov, Peter Reinhard Hansen

We introduce a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property. This parametrization can be viewed as a generalization of Fisher's Z‐transformation to higher dimensions and has a wide range of potential applications. An algorithm for reconstructing the unique n × n correlation matrix from any vector in is provided, and we derive its numerical complexity.



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Supplemental Material

Supplement to "A New Parametrization of Correlation Matrices"

This zip file contains the replication files for the manuscript.

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Supplement to "A New Parametrization of Correlation Matrices"

This is a web appendix with supplementary material for the paper “A New Parametrization of Correlation Matrices” by Archakov and Hansen (2020). Here, we present four sets of results: 1) The finite sample properties of γˆ when C has a Toeplitz structure. 2) The finite sample properties of γˆ when C has general (randomly generated) structure. 3) The Jacobian ∂ρ/∂γ for two correlation matrices. One with a Toeplitz structure, and one based on the empirical correlation matrix for returns on 10 industry portfolios. 4) Software implementations of C(γ), that reconstructs C from γ for Julia, Matlab, Ox, Python, and R.

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