# A New Parametrization of Correlation Matrices††thanks: We are grateful for
many valuable comments made by Immanuel Bomze, Bo HonorÃ©, Ulrich MÃŒller,
Georg Pflug, Werner Ploberger, Rogier Quaedvlieg, and Christopher Sims, as
well as many conference and seminar participants. __

Ilya Archakova and Peter Reinhard Hansenb

a _University of Vienna_
b _University of North Carolina & Copenhagen Business School _ Address:
University of North Carolina, Department of Economics, 107 Gardner Hall Chapel
Hill, NC 27599-3305

(_August 27, 2024_)

###### Abstract

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 $n\times n$
correlation matrix from any vector in $\mathbb{R}^{n(n-1)/2}$ is provided, and
we derive its numerical complexity.

Keywords: Correlation Matrix, Covariance Modeling, Fisher Transformation.

JEL Classification: C10; C22; C58

##  1 Introduction

We propose a new way to parametrize a covariance matrix that ensures positive
definiteness without imposing additional restrictions. The central element of
the parametrization is the matrix logarithmic transformation of the
correlations matrix, $\log C$, whose lower off-diagonal elements are stacked
into the vector $\gamma=\gamma(C)$. We show that this transformation defines a
one-to-one correspondence between the set of $n\times n$ non-singular
correlation matrices and $\mathbb{R}^{n(n-1)/2}$, and we propose a fast
algorithm for the computation of the inverse mapping.111Code for this
algorithm (Julia, Matlab, Ox, Python, and R) is provided in the Web Appendix.
In the bivariate case, $n=2$, $\gamma(C)$ is identical to the Fisher
transformation, and simulation results suggest that $\gamma(C)$ inherits some
of the attractive properties of the Fisher transformation when $n>2$.

Our results show that a non-singular $n\times n$ covariance matrix can be
expressed as a unique vector in $\mathbb{R}^{n(n+1)/2}$ that consists of the
$n$ log-variances and $\gamma$. This facilitates the modeling of covariance
matrices in terms of an unrestricted vector in $\mathbb{R}^{n(n+1)/2}$. In
models with dynamic covariance matrices, such as multivariate GARCH models and
stochastic volatility models, the parametrization offers a new way to
structure multivariate volatility models. The vector representation offers new
ways to regularizing large covariance matrices by imposing structure on
$\gamma$. The new parametrization can also be used to specify distributions on
the space of non-singular correlation matrices and covariance matrices. This
could be useful in multivariate stochastic volatility models and Bayesian
analysis.

It is convenient to reparametrize a covariance matrix as a vector that is
unrestricted in $\mathbb{R}^{d}$, and the literature has proposed several
methods to this end, see Pinheiro & Bates (1996). These methods include the
Cholesky decomposition, the spherical trigonometric transformation,
transformations based on partial correlation vines, and methods based on the
spectral representation, such as the matrix logarithm, see e.g. Kurowicka &
Cooke (2003). The matrix logarithm has been used in the modeling of covariance
matrices in Leonard & Hsu (1992) and Chiu et al. (1996). In GARCH and
stochastic volatility models it was used in Kawakatsu (2006), Ishihara et al.
(2016), and Asai & So (2015), and Bauer & Vorkink (2011) used the matrix
logarithm for modeling and forecasting of realized covariance matrices. The
transformation also emerges as a special case of the Box-Cox transformation,
see Weigand (2014) for an application to realized covariance matrices.

We do not apply the matrix logarithm to covariance matrices, but to
correlation matrices. Modeling the correlation matrix separately from the
individual variances is commonly done in multivariate GARCH models, see e.g.
Bollerslev (1990), Engle (2002), Tse & Tsui (2002), and Engle & Kelly (2012).
The new parametrization can be used to define a new family of multivariate
GARCH models, that need not impose additional restrictions beyond positivity.
Additional structure can be imposed, if so desired, and we provide examples of
this in Section 4. The new parametrization can also be used in dynamic models
of multivariate volatility that make use of realized measures of volatility.
Such as those in Liu (2009), Chiriac & Voev (2011), Golosnoy et al. (2012),
Bauwens et al. (2012), Noureldin et al. (2012), Hansen et al. (2014), and
Gorgi et al. (2019).

The paper is organized as follows. We introduce and motivate the new
parametrization of correlation matrices in Section 2 by relating it to the
Fisher transformation. We present the main theoretical results in Section 3,
auxiliary results in Section 4, and analyze the algorithm for evaluating the
inverse mapping, $C(\gamma)$, in Section 5. We conclude and summarize in
Section 6. All proofs are given in the Appendix, and additional results and
computer code are collected in the Web Appendix, see Archakov & Hansen (2020).

##  2 Motivation

We motivate the proposed method by considering a non-singular $2\times 2$
covariance matrix, with variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ and
the correlation $\rho=\sigma_{12}/(\sigma_{1}\sigma_{2})\in(-1,1)$. This
matrix can be reparametrized as the vector
$v=(\log\sigma_{1},\log\sigma_{2},\digamma(\rho))^{\prime}$, where
$\digamma(\rho)=\tfrac{1}{2}\log\tfrac{1+\rho}{1-\rho}$ is the Fisher
transformation. Because any $v\in\mathbb{R}^{3}$ maps to a unique non-singular
covariance matrix this defines a one-to-one mapping between the non-singular
covariance matrices and $\mathbb{R}^{3}$. The vector parametrization is
convenient because a positive definite covariance matrix is guaranteed without
imposing additional restrictions.

We seek a similar parametrization of covariance matrices when $n>2$.
Specifically, a mapping so that 1) Any non-singular covariance matrix,
$\Sigma$, maps to a unique vector $v=\nu(\Sigma)\in\mathbb{R}^{d}$; 2) Any
vector $v\in\mathbb{R}^{d}$ maps to a unique covariance matrix
$\Sigma=\nu^{-1}(v)$; 3) The parametrization, $v=\nu(\Sigma)$, is “invariant”
to the ordering of the variables that define $\Sigma$; and 4) the elements of
$v$ are easily interpretable.

The parametrization,
$v=(\log\sigma_{1},\log\sigma_{2},\tfrac{1}{2}\log\frac{1+\rho}{1-\rho})^{\prime}$,
has all these above properties. The Cholesky representation is not invariant
to the ordering of variables. The matrix logarithm transformation of
covariance matrix, $\log\Sigma$, satisfies the first three three properties,
but the resulting elements are difficult to interpret, because they depend
non-linearly on all elements of $\Sigma$. For $n>2$ one could consider the
element-wise Fisher transformations of every correlation, but this will not
satisfy the second property.222For instance, the inverse Fisher transformation
of, $-2$, $0$, and $\tfrac{1}{2}$ will result in three correlations that,
combined, will produce a “correlation matrix” with a negative eigenvalue.

Returning to the case with a $2\times 2$ correlation matrix. We observe that
the Fisher transformation appears as the off-diagonal elements when we take
the matrix-logarithm of an $2\times 2$ correlation matrix:

$\log\left(\begin{array}[]{cc}1&\rho\\\
\rho&1\end{array}\right)=\left(\begin{array}[]{cc}\frac{1}{2}\log(1-\rho^{2})&\frac{1}{2}\log\tfrac{1+\rho}{1-\rho}\\\
\frac{1}{2}\log\tfrac{1+\rho}{1-\rho}&\frac{1}{2}\log(1-\rho^{2})\end{array}\right).$

In this paper, we propose to parametrize correlation matrices using the off-
diagonal elements of $\log C$, so that an $n\times n$ covariance matrix,
$\Sigma$, is parametrized by the $n$ log-variances and the $n(n-1)/2$ off-
diagonal elements of $\log C$, denoted by $\gamma$. We will show that this
parametrization satisfies the first three objectives stated above. The fourth
objective is partly satisfied, because $n$ elements of $v$ will correspond to
the $n$ individual variances, whereas the remaining elements parametrize the
underlying correlation matrix. The Fisher transformation has attractive finite
sample properties (variance stabilizing and skewness reducing) and $\gamma$ is
identical to the Fisher transformation when $n=2$. Simulation results in the
Web Appendix suggest that the off-diagonal elements of $\log C$ inherit some
of these properties when $n>2$.

##  3 Theoretical Framework and Main Results

We need to introduce some useful notation and terminology. The operator,
$\mathrm{diag}(\cdot)$, is used in two ways. When the argument is a vector,
$v=(v_{1},\ldots,v_{n})^{\prime}$, then $\mathrm{diag}(v)$ denotes the
$n\times n$ diagonal matrix with $v_{1},\ldots,v_{n}$ along the diagonal, and
when the argument is a square matrix, $A\in\mathbb{R}^{n\times n}$, then
$\mathrm{diag}(A)$ extracts the diagonal of $A$ and returns it as a column
vector, i.e.
$\mathrm{diag}(A)=(a_{11},\ldots,a_{nn})^{\prime}\in\mathbb{R}^{n}$. The
matrix exponential is defined by $e^{A}=\sum_{k=0}^{\infty}\tfrac{A^{k}}{k!}$
for any matrix $A$. For any symmetric matrix, $A$, we have
$e^{A}=Q\mathrm{diag}(e^{\lambda_{1}},\ldots,e^{\lambda_{n}})Q^{\prime}$,
where $A=Q\Lambda Q^{\prime}$, with $Q$ being an orthonormal matrix, i.e.
$Q^{\prime}Q=I$, and $\Lambda=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{n})$
where $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $A$. The general
definition of the matrix logarithm is more involved, see Higham (2008), but
for a symmetric positive definite matrix, we have that $\log A=Q\log\Lambda
Q^{\prime}$, where
$\log\Lambda=\mathrm{diag}(\log\lambda_{1},\ldots,\log\lambda_{n})$.

We use $\mathrm{vecl}(A)$ to denote the vectorization operator of the lower
off-diagonal elements of $A$. For a non-singular correlation matrix, $C$, we
let $G=\log C$ denote the logarithmically transformed correlation matrix, and
let $F$ be the matrix of element-wise Fisher transformed correlations (whose
diagonal is unspecified). The vector of correlation coefficients is denoted by
$\varrho=\mathrm{vecl}C$, and the corresponding elements of $G$ and $F$ are
denoted by $\gamma=\mathrm{vecl}G$ and $\phi=\mathrm{vecl}F$, respectively.

######  Definition 1 (New Parametrization of Correlation Matrices).

For a non-singular correlation matrix, $C$, we introduce the following
parametrization: $\gamma(C):=\mathrm{vecl}(\log C)$.

Because $\gamma(C)$ discards the diagonal elements of $\log C$, it is relevant
to ask: Can $C$ be reconstructed from $\gamma$ alone? If so: Is the
reconstructed correlation matrix unique for all $\gamma$? To formalize this
inversion problem, we introduce the following operator. For an $n\times n$
matrix, $A$, and any vector $x\in\mathbb{R}^{n}$ we let $A[x]$ denote the
matrix $A$ where $x$ has replaced its diagonal. So it follows that
$\mathrm{vecl}(A)=\mathrm{vecl}(A[x])$ and that $x=\mathrm{diag}(A[x])$.

###  3.1 Main Theoretical Results

######  Theorem 1.

For any real symmetric matrix, $A\in\mathbb{R}^{n\times n}$, there exists a
unique vector, $x^{\ast}\in\mathbb{R}^{n}$, such that $e^{A[x^{\ast}]}$ is a
correlation matrix.

This shows that any vector in $\mathbb{R}^{n(n-1)/2}$ maps to a unique
correlation matrix, so that $\gamma(C)$ is a one-to-one correspondence between
$\mathcal{C}_{n}$ and $\mathbb{R}^{n(n-1)/2}$, where $\mathcal{C}_{n}$ denotes
the set of non-singular correlation matrices.333Singular correlation matrices
with known null space can be parametrized applying the transformation to a
full rank principal sub-matrix. We do not explore this topic in this paper.
The inverse mapping, denoted $C(\gamma)$, is therefore well defined.

Next, we outline the structure of the proof of Theorem 1, because it provides
intuition for the algorithm that is used to reconstruct $C$ from $\gamma$.

Consider the mapping $g:\mathbb{R}^{n}\curvearrowright\mathbb{R}^{n}$,
$g(x)=x-\log\mathrm{diag}(e^{A[x]})$, where the logarithm is applied element-
wise to vector of diagonal elements. Because $e^{A[x]}$ is a correlation
matrix if and only if all diagonal elements are equal to one, the requirement
is simply $g(x^{\ast})=x^{\ast}$. So Theorem 1 is equivalent to the statement
that $g$ has a unique fixed-point for any matrix $A$. This follows by showing
the following result and applying Banach fixed-point theorem.

######  Lemma 1.

The mapping $g$ is a contraction for any symmetric matrix $A$.

The proof of Lemma 1 entails deriving the Jacobian for $g$, denoted $\nabla
g$, and showing that all its eigenvalues are less than one in absolute value.
The largest eigenvalue of $\nabla g$ is, not surprisingly, key for the
algorithm that reconstructs $C$ from $\gamma$.

###  3.2 Invariance to Reordering of Variables

The mapping, $\gamma(C)$, is invariant to a reordering of variables that
define $C$, in the sense that a permutation of the variables that define $C$
will merely result in a permutation of the elements of $\gamma$. The formal
statement is as follows.

######  Proposition 1.

Suppose that $C_{x}=\mathrm{corr}(X)$ and $C_{y}=\mathrm{corr}(Y)$, where the
elements of $X$ is a permutation of the elements of $Y$. Then the elements of
$\gamma_{x}=\gamma(C_{x})$ is a permutation of the elements of
$\gamma_{y}=\gamma(C_{y})$.

###  3.3 An Algorithm for Computing $C(\gamma)$

Evidently, the solution, $x^{\ast}$, must be such that the diagonal elements
of the matrix, $e^{A[x^{\ast}]}$, are all equal to one. Equivalently,
$\log\mathrm{diag}(e^{A[x^{\ast}]})=0\in\mathbb{R}^{n}$, where the logarithm
is applied element-wise to the vector of diagonal elements. This observation
motivates the following iterative procedure for determining $x^{\ast}$:

######  Corollary 1.

Consider the sequence,

$\displaystyle x_{(k+1)}=x_{(k)}-\log\mathrm{diag}(e^{A[x_{(k)}]}),$
$\displaystyle\qquad k=0,1,2,\ldots$

with an arbitrary initial vector $x_{(0)}\in\mathbb{R}^{n}$. Then
$x_{(k)}\rightarrow x^{\ast}$, where $x^{\ast}$ is the solution in Theorem 1.

In practice we find that the simple algorithm, proposed in Corollary 1,
converges very fast. This is demonstrated in Section 5 for matrices with
dimension up to $n=100$. The result in Theorem 1 and the algorithm in
Corollary 1 are easily adapted to a covariance matrix with known diagonal
elements, as we show in Section 4.4.

###  3.4 Asymptotic Distribution of $\hat{\gamma}$

Next, we derive the asymptotic distributions of $\hat{\gamma}$ and the vector
of Fisher transformed correlations, $\hat{\phi}$, by deducing them from those
of the empirical correlation matrix.

Suppose that $\sqrt{T}(\hat{C}-C)\overset{d}{\rightarrow}N(0,\Omega)$, as
$T\rightarrow\infty$. The asymptotic covariance matrix,
$\Omega=\mathrm{avar}(\text{vec}(\hat{C}))$, will be singular because
$\hat{C}$ is symmetric and has constant diagonal elements. Convenient closed-
form expressions for $\Omega$ is available in special cases, see e.g.
Neudecker & Wesselman (1990), Nel (1985), and Browne & Shapiro (1986).

For the vector of correlation coefficients,
$\hat{\varrho}=\text{vecl}(\hat{C})$, it follows that
$\sqrt{T}(\hat{\varrho}-\varrho)\overset{d}{\rightarrow}N(0,\Omega_{\varrho})$,
as $T\rightarrow\infty$, where $\Omega_{\varrho}=E_{l}\Omega E_{l}^{\prime}$
and $E_{l}$ is an elimination matrix, characterized by
$\mathrm{vecl}[M]=E_{l}\mathrm{vec}[M]$ for any $n\times n$ matrix $M$. For
the element-wise Fisher transform, the asymptotic distribution reads

$\sqrt{T}(\hat{\phi}-\phi)\overset{d}{\rightarrow}N(0,\Omega_{\phi}),\qquad\Omega_{\phi}=D_{c}E_{l}\Omega
E_{l}^{\prime}D_{c},$ (1)

where
$D_{c}=\text{diag}\Bigl{(}\frac{1}{1-c_{i}^{2}},\frac{1}{1-c_{2}^{2}},\ldots,\frac{1}{1-c_{d}^{2}}\Bigl{)}$
and $c_{i}$ is an $i$-th element of $c=\text{vecl}(C)\in\mathbb{R}^{d}$ with
$d=n(n-1)/2$, whereas the asymptotic distribution of the new parametrization
of correlation matrices, can be shown to be

$\sqrt{T}(\hat{\gamma}-\gamma)\overset{d}{\rightarrow}N(0,\Omega_{\gamma}),\qquad\Omega_{\gamma}=E_{l}A^{-1}\Omega
A^{-1}E_{l}^{\prime},$ (2)

where $A$ is a Jacobian matrix, such that
$\partial\mathrm{vec}(C)=A\,\partial\mathrm{vec}(\log C)$. The expression for
$A$ is given in the Appendix, see (A.1)-(A.2), and is taken from Linton &
McCrorie (1995).

In a classical setting where $\hat{C}$ is computed from i.i.d. random vectors,
the diagonal elements of $\Omega_{\phi}$ are all equal to one. This
demonstrates the variance stabilizing property of the Fisher transformation.
The transformation $\gamma(C)$ is, evidently, not variance stabilizing when
$n>2$, except in special cases. However, it does appear to reduce skewness,
which is another attribute of the Fisher transformation.

The two expressions for the asymptotic variances, $\Omega_{\phi}$ and
$\Omega_{\gamma}$, are not easily compared unless $\Omega$ is known. Here we
will compare them in the situation where $\hat{C}$ is computed from
$X_{i}\sim\text{iid}N_{3}(0,\Sigma)$, for four different choices for $\Sigma$.
Scaling the elements of $X_{i}$ does not affect the limit distributions for
$\hat{\varrho}$, $\hat{\phi}$, and $\hat{\gamma}$. So we can, without loss of
generality, focus on the case where $\Sigma=C$.

$\Sigma=C$ | $\text{avar}(\hat{\varrho})$ | $\text{avar}(\hat{\phi})=$ | $\text{avar}(\hat{\gamma})$ | $\text{acorr}(\hat{\gamma})$
---|---|---|---|---
|  | $\text{acorr}(\hat{\varrho})=\text{acorr}(\hat{\phi})$ |  |
$\left(\begin{array}[]{ccc}1&\bullet&\bullet\\\ 0&1&\bullet\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0&1.000&\bullet\\\ 0&0&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0&1.000&\bullet\\\ 0&0&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0&1.000&\bullet\\\ 0&0&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0&1.000&\bullet\\\ 0&0&1.000\end{array}\right)$
$\left(\begin{array}[]{ccc}1&\bullet&\bullet\\\ 0.5&1&\bullet\\\ 0.25&0.5&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}0.562&\bullet&\bullet\\\ 0.316&0.879&\bullet\\\ 0.070&0.316&0.562\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.450&1.000&\bullet\\\ 0.125&0.450&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}0.966&\bullet&\bullet\\\ 0.018&0.962&\bullet\\\ 0.021&0.018&0.966\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.018&1.000&\bullet\\\ 0.021&0.018&1.000\end{array}\right)$
$\left(\begin{array}[]{ccc}1&\bullet&\bullet\\\ 0.9&1&\bullet\\\ 0.81&0.9&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}0.036&\bullet&\bullet\\\ 0.046&0.118&\bullet\\\ 0.015&0.046&0.036\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.698&1.000&\bullet\\\ 0.405&0.698&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}0.817&\bullet&\bullet\\\ 0.081&0.860&\bullet\\\ 0.093&0.081&0.817\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.097&1.000&\bullet\\\ 0.114&0.097&1.000\end{array}\right)$
$\left(\begin{array}[]{ccc}1&\bullet&\bullet\\\ 0.99&1&\bullet\\\ 0.98&0.99&1\end{array}\right)$ | $\tfrac{1}{10}\left(\begin{array}[]{ccc}0.004&\bullet&\bullet\\\ 0.006&0.016&\bullet\\\ 0.002&0.006&0.004\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.745&1.000&\bullet\\\ 0.490&0.745&1.000\end{array}\right)$ | $\left(\begin{array}[]{ccc}0.756&\bullet&\bullet\\\ 0.106&0.793&\bullet\\\ 0.134&0.106&0.756\end{array}\right)$ | $\left(\begin{array}[]{ccc}1.000&\bullet&\bullet\\\ 0.137&1.000&\bullet\\\ 0.178&0.137&1.000\end{array}\right)$
Table 1: Asymptotic covariance and correlation matrices for $\hat{\varrho}$,
$\hat{\phi}$ and $\hat{\gamma}$, for four different correlation matrices. The
diagonal elements of the asymptotic variance matrix for $\hat{\phi}$ are all
one, so it is also the asymptotic correlation matrix for $\hat{\phi}.$ Because
$\hat{\phi}$ is based on an element-by-element transformation of the
corresponding elements of $\hat{\varrho}$, it is also the asymptotic
correlation matrix for $\hat{\varrho}$.

The asymptotic variance and correlation matrices for the three vectors,
$\hat{\varrho}$, $\hat{\phi}$ and $\hat{\gamma}$, are reported in Table 1. The
true correlation matrix is given in the first column of Table 1. The
asymptotic variance of the correlation coefficient, $\hat{\varrho}_{j}$, is
$(1-\varrho_{j}^{2})^{2}$, which defines the diagonal elements of
$\Omega_{\varrho}$, and the element-wise Fisher transformation ensures that
$\mathrm{avar}(\hat{\phi}_{j})=1$ for all $j=1,\dots,n$. However, we observe a
high degree of correlation across the elements of $\hat{\phi}$. The asymptotic
correlation matrix for $\hat{\phi}$ is, in fact, identical to that of the
empirical correlations, $\hat{\varrho}$, because the Fisher transformation is
an element-by-element transformation. Its Jacobian,
$D_{c}=\partial\phi/\partial\varrho$, is therefore a diagonal matrix.
Consequently, the asymptotic correlations are unaffected by the element-wise
Fisher transformation, and
$\mathrm{acorr}(\hat{\varrho})=\mathrm{acorr}(\hat{\phi})$. While the diagonal
elements of $\Omega_{\phi}$ are invariant to $C$, this is not the case for the
diagonal elements of $\Omega_{\gamma}$, but it is interesting to note that the
asymptotic correlations between elements of $\hat{\gamma}$ tend to be
relatively small, and close to zero when the correlations in $C$ are small.

Simulation results in the Web Appendix suggest that the elements of
$\hat{\gamma}$ tend to be weakly correlated, and that $\gamma(C)$ reduces
skewness, as is the case for the Fisher transformation. Empirical results in
Archakov et al. (2020) show that the empirical distribution of transformed
realized correlation matrices is well approximated by a Gaussian distribution.

##  4 Auxiliary Results and Properties

###  4.1 Structure for Certain Correlation Matrices

While the elements of $\gamma$ depend on the correlation matrix in a nonlinear
way, there are some interesting correlation structures that do carry over to
the matrix $G=\log C$, and hence $\gamma$. First, we consider the case with an
equicorrelation matrix and a block-equicorrelation matrix.

######  Proposition 2.

Suppose $C$ is an equicorrelation matrix with correlation parameter $\rho$.
Then, all the off-diagonal elements of matrix $G=\log C$ are identical and
equal to

$\gamma_{c}=-\tfrac{1}{n}\log\left(\tfrac{1-\rho}{1+\rho(n-1)}\right)=\tfrac{1}{n}\log(1+n\tfrac{\rho}{1-\rho})\in\mathbb{R},$
(3)

so that $\gamma=\gamma_{c}\iota$, where $\iota\in\mathbb{R}^{n(n-1)/2}$ is the
vector of ones, $\iota=(1,\ldots,1)^{\prime}$.

This result, in conjunction with Theorem 1, establishes that $\gamma_{c}$ is a
one-to-one correspondence from the set of non-singular equicorrelation
matrices to the real line, $\mathbb{R}$, and the inverse mapping is given in
closed-form by
$\rho(\gamma_{c},n)=\frac{1-e^{-n\gamma_{c}}}{1+(n-1)e^{-n\gamma_{c}}}$. It
follows that $\rho(\gamma_{c},n)$ is confined to the interval
$\bigl{(}-\frac{1}{n-1},1\bigl{)}$.

It is easy to verify that if $C$ is a block diagonal matrix, with
equicorrelation diagonal blocks and zero correlation across blocks, then $\log
C$ will have the same block structure, and (3) can be used to compute the
elements in $\gamma$. In the more general case where $C$ is a block
correlation matrix, then it can be shown that the logarithmic transformation
preserves the block structure. This is used in Archakov et al. (2020) in a
multivariate GARCH model. So that $\log C$ has the same block structure as
$C$, and this transformation provides a simple way to model block correlation
matrices. We illustrate this with the following example

$C=\left(\begin{array}[]{cccccc}1.0&{\color[rgb]{0,.5,.5}0.4}&{\color[rgb]{0,.5,.5}0.4}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}\\\
{\color[rgb]{0,.5,.5}0.4}&1.0&{\color[rgb]{0,.5,.5}0.4}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}\\\
{\color[rgb]{0,.5,.5}0.4}&{\color[rgb]{0,.5,.5}0.4}&1.0&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}\\\
{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&1.0&{\color[rgb]{0,0,1}0.6}&{\color[rgb]{0,0,1}0.6}\\\
{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{0,0,1}0.6}&1.0&{\color[rgb]{0,0,1}0.6}\\\
{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{.75,0,.25}0.2}&{\color[rgb]{0,0,1}0.6}&{\color[rgb]{0,0,1}0.6}&1.0\end{array}\right)\Leftrightarrow\log
C=\left(\begin{array}[]{cccccc}-.16&{\color[rgb]{0,.5,.5}.349}&{\color[rgb]{0,.5,.5}.349}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}\\\
{\color[rgb]{0,.5,.5}.349}&-.16&{\color[rgb]{0,.5,.5}.349}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}\\\
{\color[rgb]{0,.5,.5}.349}&{\color[rgb]{0,.5,.5}.349}&-.16&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}\\\
{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&-.36&{\color[rgb]{0,0,1}.553}&{\color[rgb]{0,0,1}.553}\\\
{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{0,0,1}.553}&-.36&{\color[rgb]{0,0,1}.553}\\\
{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{.75,0,.25}.104}&{\color[rgb]{0,0,1}.553}&{\color[rgb]{0,0,1}.553}&-.36\end{array}\right).$

Another interesting class of correlation matrices are the Toeplitz-correlation
matrices, which arise in some models, such as stationary time series models.
For this case, $\log C$ is a bisymmetric matrix.

###  4.2 The Inverse and other Powers of the Correlation Matrix

Since $C^{\alpha}=e^{\alpha G}$, it is possible to obtain powers of $C$ from
$\gamma$. For instance, the inverse covariance matrix is given by
$\Sigma^{-1}=\Lambda^{-1}e^{-G}\Lambda^{-1}$, where
$\Lambda=\mathrm{diag}(\sigma_{1},\ldots,\sigma_{n})$. The inverse is, for
instance, of interest for computing the partial correlation coefficients and
in portfolio choice problems. Some estimation methods impose sparsity on
$\Sigma^{-1}$. While it is not simple to impose sparsity on $\Sigma^{-1}$
through $\gamma$, the new parametrization facilitate new ways to impose a
parsimonious structure on $\Sigma$ or $\Sigma^{-1}$, by imposing sparsity (or
some other structure) on $\gamma$ directly.

###  4.3 The Jacobian $\partial\varrho/\partial\gamma$

Next we establish a result that shows that
$\partial\varrho/\partial\gamma=\partial\mathrm{vecl}[C]/\partial\mathrm{vecl}[G]$
has a relatively simple expression. This is convenient for inference, such as
computation of standard errors, and for the construction of dynamic GARCH-type
models, such as a score-driven model for $\gamma=\mathrm{vecl}G$, see Creal et
al. (2013), and for the construction of parameter stability tests, such as
that of Nyblom (1989).

######  Proposition 3.

We have
$\tfrac{\partial\varrho}{\partial\gamma}=E_{l}\Bigl{(}I-AE_{d}^{\prime}\Bigl{(}E_{d}AE_{d}^{\prime}\Bigl{)}{}^{-1}E_{d}\Bigl{)}A(E_{l}+E_{u})^{\prime}$,
where $A=\partial\mathrm{vec}C/\partial\mathrm{vec}G$ and the matrices
$E_{l}$, $E_{u}$ and $E_{d}$ are elimination matrices, such that
$\mathrm{vecl}M=E_{l}\mathrm{vec}M$,
$\mathrm{vecl}M^{\prime}=E_{u}\mathrm{vec}M$ and
$\mathrm{diag}M=E_{d}\mathrm{vec}M$ for any square matrix $M$ of the same size
as $C$.

The matrix, $A$, is the same matrix that appeared in the asymptotic
distribution for $\hat{\gamma}$, see (2). In the Web Appendix we compute
$\partial\varrho/\partial\gamma$ for two correlation matrices: A $10\times 10$
Toeplitz correlation matrix and one based on the empirical correlation matrix
for the 10 industry portfolios in the Kenneth R. French data library. The two
have a very similar structure.

###  4.4 Results for Covariance Matrices with Known Diagonal Elements

Some of our results for correlation matrices, apply equally to covariance
matrices with known diagonal elements, and these could be useful in some
applications that involve the matrix logarithm of covariance matrices. In
Corollary 2 we state the extensions to this situation.

######  Corollary 2.

For any real symmetric matrix, $A\in\mathbb{R}^{n\times n}$, and any vector,
$v\in\mathbb{R}^{n}$ with strictly positive elements, there exists a unique
vector, $x^{\ast}\in\mathbb{R}^{n}$, such that $\Sigma=e^{A[x^{\ast}]}$ is a
covariance matrix with diagonal $\mathrm{diag}(\Sigma)=v$. Moreover,
$x^{\ast}=\lim_{k\rightarrow\infty}x_{(k)}$, where $x_{(k+1)}=x_{(k)}+[\log
v-\log\mathrm{diag}(e^{A[x_{(k)}]})]$, for $k=0,1,2,\ldots$, with an arbitrary
initial vector $x_{(0)}\in\mathbb{R}^{n}$.

##  5 Properties of the Algorithm for the Inverse Mapping, $C(\gamma)$

The algorithm that reconstructs the correlation matrix, $C$, from $\gamma$
converges exponentially fast, and its complexity is of order $O(n^{3}\log n)$.
This follows, as we show below, from the fact that the number of required
iterations is of order $\log n$, and because each iteration entails a matrix
exponential evaluation which is of order $O(n^{3})$, see e.g. Lu (1998).

Let $K_{\delta}=\inf\\{k:||x_{(k+1)}-x_{(k)}||_{p}\leq\delta\\}$ be the number
of iterations required for convergence for some $p$-norm and some threshold
$\delta>0$. From the contraction property it follows that
$||x_{(k+1)}-x_{(k)}||_{p}\leq L||x_{(k)}-x_{(k-1)}||_{p}\leq
L^{k}||x_{(1)}-x_{(0)}||_{p}$, for $k=1,2,\ldots$, where $L\in[0,1)$ is the
Lipschitz constant given from the contraction. So the number of iterations $k$
can be bounded from above by $k\leq
c_{L}(\log||x_{(1)}-x_{(0)}||_{p}-\log||x_{(k)}-x_{(k-1)}||_{p})$, where
$c_{L}=-\tfrac{1}{\log L}>0$ depends on the Lipschitz constant. Since
$||x||_{p}\leq(n\cdot\max_{1\leq i\leq
n}|x^{(i)}|^{p})^{1/p}=n^{1/p}||x||_{\infty}$, we have

$K_{\delta}\leq c_{L}(\tfrac{\log
n}{p}+\log||x_{(1)}-x_{(0)}||_{\infty}-\log\delta)=O(\log n).$ (4)

Note that the number of required iterations may be more sensitive to the
structure of $C$ (through the Lipschitz constant) than the dimension of $C$.
The Lipschitz constant approaches one as $C$ approaches singularity. The
number of iterations is less sensitive to the choice of initial vector
$x_{(0)}$, but it is useful to know that the elements of $x^{\ast}$ are non-
positive.

######  Lemma 2.

The diagonal elements of $\log C$ are non-positive for any
$C\in\mathcal{C}_{n}$.

The result in (4) is illustrated in Figure 1 where we recover the correlation
matrix from $\gamma$ using the algorithm in Corollary 1. The true $C$ has a
Toeplitz structure, $C_{ij}=\rho^{|i-j|}$, $i,j=1,\ldots,n$, for
$n=3,\ldots,100$ and $\rho\in\\{0.5,0.9,0.99\\}$. The number of iterations
needed for $||x_{(k)}-x_{(k-1)}||_{2}<\delta=10^{-8}\sqrt{n}$ increases with
the dimension at a rate that is consistent with $\log n$. The number of
iterations is sensitive to the correlation structure. For instance, when $C$
is almost singular ($\rho=0.99$), the number of iterations is about five times
that of a moderately correlated correlation matrix ($\rho=0.5$). The reason is
that a near zero eigenvalue of $C$ translates into a Lipschitz constant close
to one. To illustrate the sensitivity to the starting value, we use 1,000
different starting values, $x_{(0)}$, where the elements of $x_{(0)}$ are
drawn independently from the negative half-normal distribution with scale
$\sigma=10$ (i.e. $-|Z|$ with $Z\sim N(0,100)$). The shaded bands depict the
dispersion in the number of iterations (average $\pm 2$ standard deviations).
The dispersion is relatively modest which verifies that the algorithm is
relatively insensitive to the initial value, $x^{(0)}$.

Figure 1: Number of iterations needed for convergence at threshold
$\delta=10^{-8}\sqrt{n}$, when $C_{ij}=\rho^{|i-j|}$ $i,j=1,\ldots,n$, for
$n=3,\ldots,100$, using random initial value, $x_{(0)}$. Black lines
correspond to the average number of iterations required for convergence, and
the shaded bands ($\pm$2 standard deviations) show the variation resulting
from the different starting values.

The results in Figure 1 are not specific to the Toeplitz structure for $C$. In
a second design, we generate 50,000 distinct correlation matrices for each of
the dimensions, $n\in\\{5,10,25\\}$. This is done by generating random
vectors, $\gamma$, where each element in $\gamma$ is uniformly distributed on
the interval $[-b_{n},b_{n}]$. The constant, $b_{n}$, is chosen to provide a
sufficiently wide range of the smallest eigenvalue of $C$, denoted
$\lambda_{\min}$, and the spectral radius of $\nabla g(x^{\ast})$, denoted
$\nu_{\max}$. The Lipschitz constant for the contraction, $g(x)$, is
approximately equal to $\nu_{\max}$, so we should expect
$-1/\log\nu_{\max}\simeq c_{L}$ to be linearly related to (the bound on) the
number of iterations.

The number of iterations needed for convergence is shown in Figure 2, for
$n=5$, $n=10$, and $n=25$, using scatter plots against three characteristics
of $C$. The starting value is $x_{(0)}=0\in\mathbb{R}^{n}$ in all simulations
and $\delta=10^{-8}\sqrt{n}$ was used as the tolerance level.

The left panels reveal a fairly tight linear relationship between the number
of iterations and $-1/\log\nu_{\max}$ ($\approx c_{L}$). Similarly,
$\lambda_{\max}$ and $\gamma_{\max}$, which are easier to compute, are also
related to the number of iterations, albeit not as tightly as $\nu_{\max}$.

Figure 2: The number of iterations needed for convergence plotted against
three characteristics of $C$. The left panels plots the number of iterations
against $-1/\log\nu_{\max}\simeq c_{L}$. The smallest eigenvalue of $C$
(middle panels) and the largest $|\gamma_{i}|$ (right panels) are also useful
indicators.

##  6 Concluding Remarks

In this paper, we have shown that the space of non-singular $n\times n$
correlation matrices is one-to-one with $\mathbb{R}^{n(n-1)/2}$. A non-
singular covariance matrix can therefore be parametrized by the $n$
(log-)variances and the vector, $\gamma(C)$, which has unrestricted domain in
$\mathbb{R}^{n(n-1)/2}$. This opens new ways to model correlation and
covariance matrices where positive definiteness is an intrinsic property. For
instance, in multivariate GARCH models, as explored in Archakov et al. (2020).
The transformation can be used to specify probability distributions on
correlation and covariance matrices. Any distribution on
$\mathbb{R}^{n(n-1)/2}$ induces a distribution on the space of positive
definite correlation matrices, $\mathcal{C}$. This could be used in
multivariate stochastic volatility modeling, and defines a new approach to
specifying Bayesian priors on $\mathcal{C}$.

We have derived results for the asymptotic distribution of $\gamma(\hat{C})$.
Much is known about the finite sample properties when $n=2$, because
$\gamma(C)$ is identical to the Fisher transformation in this case. The Fisher
transformation has variance stabilizing and skewness eliminating properties.
The variance stabilizing property does not carry over to the case $n>2$.
However, simulation results suggest that it continues to have skewness
reducing properties, and that the empirical distribution of $\gamma(\hat{C})$
(in a classical setting) is well approximated by a Gaussian distribution even
in small samples. Moreover, the elements of $\gamma(\hat{C})$ tend to be
weakly dependent, as suggested by the asymptotic results in Table 1. This
makes the transformation potentially useful for regularization, see Pourahmadi
(2011), and inference. These attributes tend to deteriorate as $C$ approaches
singularity. This is not unexpected, because it is also true for the Fisher
transformation when the correlation is close to $\pm 1$.

The inverse mapping, $C(\gamma)$ is not given in closed-form when $n>2$,
except in some special cases. Instead, we proposed a fast algorithm to
evaluate $C(\gamma)$, and showed that its numerical complexity is of order
$O(n^{3}\log n)$, where $n\times n$ is the dimension of $C$.

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## Appendix of Proofs

We prove $g$ is a contraction by deriving its Jacobian, $J(x)$, and showing
that all its eigenvalues are less than one in absolute value. Since
$g(x)=x-\log\delta(x),$ where $\delta(x)=\text{diag}(e^{G[x]})$, an
intermediate step towards the Jacobian for $g$, is to derive the Jacobian for
$\delta(x)$. To simplify notation, we sometimes suppress the dependence on $x$
for some terms. For instance, we sometimes write $\delta_{i}$ to denote the
$i$-th element of the vector $\delta(x)$. It follows that
$[J(x)]_{i,j}=\frac{\partial[g(x)]_{i}}{\partial
x_{j}}=1_{\\{i=j\\}}-\frac{1}{\delta_{i}}\frac{\partial[\delta(x)]_{i}}{\partial
x_{j}},$ so that $J(x)=I-[D(x)]^{-1}H(x)$, where
$D(x)=\mathrm{diag}(\delta_{1},\ldots,\delta_{n})$ is a diagonal matrix and
$H(x)$ is the Jacobian matrix of $\delta(x)$, derived below.

Let $G[x]=Q\Lambda Q^{\prime}$, where $\Lambda$ is the diagonal matrix with
the eigenvalues, $\lambda_{1},\ldots,\lambda_{n}$, of $G[x]$ and $Q$ is an
orthonormal matrix (i.e. $Q^{\prime}=Q^{-1}$) with the corresponding
eigenvectors. From Linton & McCrorie (1995), we have
$\mathrm{d}\text{vec}\,e^{G[x]}=A(x)\,\mathrm{d}\text{vec}G[x]$, where

$A(x)=(Q\otimes Q)\Xi\bigl{(}Q\otimes Q\bigl{)}^{\prime},$ (A.1)

is and $n^{2}\times n^{2}$ matrix and $\Xi$ is the $n^{2}\times n^{2}$
diagonal matrix with elements given by

$\xi_{ij}=\Xi_{(i-1)n+j,(i-1)n+j}=\begin{cases}e^{\lambda_{i}},&\text{if}\qquad\lambda_{i}=\lambda_{j}\\\
\tfrac{e^{\lambda_{i}}-e^{\lambda_{j}}}{\lambda_{i}-\lambda_{j}},&\text{if}\qquad\lambda_{i}\neq\lambda_{j}\end{cases}$
(A.2)

for $i=1,\ldots,n$ and $j=1,\ldots,n$. Evidently, we have $\xi_{ij}=\xi_{ji}$,
for all $i$ and $j$. Moreover, $A(x)$ is a symmetric positive definite matrix,
because all the diagonal elements of $\Xi$ are strictly positive.

Our interest concerns $\delta(x)=\text{diag}[e^{G[x]}]$ (a subset of the
elements of $\text{vec}[e^{G[x]}]$) so the Jacobian of $\delta(x)$, denoted
$H(x)$, is a principal sub-matrix of $A(x)$, defined by the elements
$[A(x)]_{l,m}$, $l,m=(i-1)n+i$, for $i=1,\ldots,n$. Thus

$\displaystyle[H(x)]_{i,j}$
$\displaystyle=\frac{\partial\delta(x)_{i}}{\partial x_{j}}=(e_{i}\otimes
e_{i})^{\prime}\bigl{(}Q\otimes Q\bigl{)}\Xi\bigl{(}Q\otimes
Q)^{\prime}(e_{j}\otimes e_{j})$ $\displaystyle=\bigl{(}e_{i}^{\prime}Q\otimes
e_{i}^{\prime}Q\bigl{)}\Xi\bigl{(}Q^{\prime}e_{j}\otimes
Q^{\prime}e_{j}\bigl{)}=\bigl{(}Q_{i,.}\otimes
Q_{i,.}\bigl{)}\Xi\bigl{(}Q_{j,.}\otimes Q_{j,.}\bigl{)}^{\prime}$ (A.3)
$\displaystyle=\sum_{k=1}^{n}\sum_{l=1}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl},$

where $e_{i}$ is a $n\times 1$ unit vector with one at the $i$-th position and
zeroes otherwise and $Q_{i,.}$ denotes the $i$-th row of $Q$.

Interestingly, the Jacobian of $g$ is such that $J(x)\iota=0$, so that the
vector of ones, $\iota$, is an eigenvector of $J(x)$ associated with the
eigenvalue $0$, i.e. $J(x)$ has reduced rank. Because the $i$-th row of $J(x)$
times $\iota$ reads

$1-\sum_{j=1}^{n}\frac{1}{\delta_{i}}\sum_{k=1}^{n}\sum_{l=1}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl}=1-\frac{1}{\delta_{i}}\sum_{k=1}^{n}\sum_{l=1}^{n}q_{ik}q_{il}\xi_{kl}\sum_{j=1}^{n}q_{jk}q_{jl}=1-\frac{1}{\delta_{i}}\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}=0,$

due to $\sum_{k=1}^{n}q_{ik}q_{jk}=1_{\\{i=j\\}}$.

### Proof that $g$ is a Contraction: Lemma 1

We now want to prove that the mapping $g(x)$ is a contraction. In order to
show this, it is sufficient to demonstrate that all eigenvalues of the
corresponding Jacobian matrix $J(x)$ are below one in absolute values for any
real vector $x$. First we establish a number of intermediate results.

######  Lemma A.1.

$(i)$ $e^{y}-y-1>0$ for all $y\neq 0$, and $(ii)$
$1+e^{y}-\frac{2}{y}(e^{y}-1)>0$ for $y\neq 0$.

###### Proof.

The first and second derivatives of $f(y)=e^{y}-y-1$ show that $f$ is strictly
convex with unique minimum, $f(0)=0$, which proves $(i)$. Next we prove
$(ii)$. Now let $f(y)=1+e^{y}-\frac{2}{y}(e^{y}-1)$. Its first derivative is
given by $f^{\prime}(y)=e^{y}y^{-2}g(y)$, where $g(y)=y^{2}-2y+2-2e^{-y}$, so
that $f^{\prime}(y)<0$ for $y<0$ and $f^{\prime}(y)>0$ for $y>0$. Since
$\lim_{y\rightarrow 0}f(y)=0$ (by l’Hospital’s rule) the result follows. ∎

From the definition, (A.2), it follows that $\xi_{ij}=\xi_{ii}=\xi_{jj}$
whenever $\lambda_{i}=\lambda_{j}$. When $\lambda_{i}\neq\lambda_{j}$ we have
the following results for the elements of $\Xi$:

######  Lemma A.2.

If $\lambda_{i}<\lambda_{j}$, then $\xi_{ii}<\xi_{ij}<\xi_{jj}$ and
$2\xi_{ij}<\xi_{ii}+\xi_{jj}$.

###### Proof.

From the definition, (A.2),
$\xi_{ij}-\xi_{ii}=\tfrac{e^{\lambda_{j}}-e^{\lambda_{i}}}{\lambda_{j}-\lambda_{i}}-e^{\lambda_{i}}=e^{\lambda_{i}}(\tfrac{e^{\lambda_{j}-\lambda_{i}}-1}{\lambda_{j}-\lambda_{i}}-1)=e^{\lambda_{i}}\tfrac{e^{\lambda_{j}-\lambda_{i}}-1-(\lambda_{j}-\lambda_{i})}{\lambda_{j}-\lambda_{i}}>0$,
where the numerator is positive by Lemma A.1 ($i$). So are $e^{\lambda_{i}}$
and $\lambda_{j}-\lambda_{i}$, which proves $\xi_{ij}>\xi_{ii}$. Analogously,
$\xi_{jj}-\xi_{ij}=e^{\lambda_{j}}-\tfrac{e^{\lambda_{j}}-e^{\lambda_{i}}}{\lambda_{j}-\lambda_{i}}=e^{\lambda_{j}}(1-\tfrac{1-e^{\lambda_{i}-\lambda_{j}}}{\lambda_{j}-\lambda_{i}})=e^{\lambda_{j}}\tfrac{-(\lambda_{i}-\lambda_{j})-1+e^{\lambda_{i}-\lambda_{j}}}{\lambda_{j}-\lambda_{i}}>0$,
because all terms are positive, where we again used Lemma A.1 ($i$). Next,
$\xi_{ii}+\xi_{jj}-2\xi_{ij}=e^{\lambda_{i}}+e^{\lambda_{j}}-2\tfrac{e^{\lambda_{i}}-e^{\lambda_{j}}}{\lambda_{i}-\lambda_{j}}=e^{\lambda_{i}}(1+e^{\lambda_{j}-\lambda_{i}}-2\tfrac{e^{\lambda_{j}-\lambda_{i}}-1}{\lambda_{j}-\lambda_{i}})>0$,
where the inequality follows by Lemma A.1.$ii$, because
$\lambda_{i}\neq\lambda_{j}$. ∎

######  Lemma A.3.

$J(x)$ and $\tilde{J}(x)=I-D^{-\frac{1}{2}}HD^{-\frac{1}{2}}$ have the same
eigenvalues, where

$\tilde{J}(x)=\sum_{k=1}^{n-1}\sum_{l=k}^{n}\varphi_{kl}\Bigl{(}D^{-\frac{1}{2}}u_{kl}u_{kl}^{\prime}D^{-\frac{1}{2}}\Bigl{)},$

with $u_{kl}=Q_{\cdot,k}\odot Q_{\cdot,l}\in\mathbb{R}^{n}$ and
$\varphi_{kl}=\xi_{kk}+\xi_{ll}-2\xi_{kl}$.

###### Proof.

For a vector $y$ and a scalar $\nu$, $Jy=\nu y\Leftrightarrow\tilde{J}w=\nu
w$, where $y=D^{-\frac{1}{2}}w$, because
$J=I-D^{-1}H=D^{-\frac{1}{2}}(I-D^{-\frac{1}{2}}HD^{-\frac{1}{2}})D^{\frac{1}{2}}=D^{-\frac{1}{2}}\tilde{J}D^{\frac{1}{2}}$.
Next, we turn to the expression for $\tilde{J}$. First, note that
$\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}=\sum_{k=1}^{n}q_{ik}^{2}e^{\lambda_{k}}=Q_{i,\cdot}e^{\Lambda}Q_{i,\cdot}^{\prime}=[e^{Q\Lambda
Q^{\prime}}]_{ii}=[e^{G}]_{ii}=\delta_{i}$. So diagonal elements of
$\tilde{J}$ are given by

$\displaystyle\tilde{J}_{ii}=1-\frac{H_{ii}}{\delta_{i}}$
$\displaystyle=\frac{1}{\delta_{i}}\Bigl{(}\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}-\sum_{k=1}^{n}\sum_{l=1}^{n}q_{ik}^{2}q_{il}^{2}\xi_{kl}\Bigl{)}=\frac{1}{\delta_{i}}\Bigl{(}\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}-\sum_{k=1}^{n}q_{ik}^{2}q_{ik}^{2}\xi_{kk}-2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}\xi_{kl}\Bigl{)}$
$\displaystyle=\frac{1}{\delta_{i}}\Bigl{(}\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}(1-q_{ik}^{2})-2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}\xi_{kl}\Bigl{)}=\frac{1}{\delta_{i}}\Bigl{(}\sum_{k=1}^{n}q_{ik}^{2}\xi_{kk}\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{n}q_{il}^{2}-2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}\xi_{kl}\Bigl{)}$
$\displaystyle=\frac{1}{\delta_{i}}\Bigl{(}\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}(\xi_{kk}+\xi_{ll})-2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}\xi_{kl}\Bigl{)}=\frac{1}{\delta_{i}}\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}^{2}q_{il}^{2}\varphi_{kl},$

where we used (A.3). Similarly for the off-diagonal elements we have

$\displaystyle\tilde{J}_{ij}=-\frac{H_{ij}}{\sqrt{\delta_{i}\delta_{j}}}$
$\displaystyle=-\frac{1}{\sqrt{\delta_{i}\delta_{j}}}\sum_{k=1}^{n}\sum_{l=1}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl}=-\frac{1}{\sqrt{\delta_{i}\delta_{j}}}\Bigl{(}\sum_{k=1}^{n}q_{ik}^{2}q_{jk}^{2}\xi_{kk}+2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl}\Bigl{)}$
$\displaystyle=-\frac{1}{\sqrt{\delta_{i}\delta_{j}}}\Bigl{(}\sum_{k=1}^{n}q_{ik}q_{jk}\Big{(}-\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{n}q_{il}q_{jl}\Big{)}\xi_{kk}+2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl}\Bigl{)}$
$\displaystyle=-\frac{1}{\sqrt{\delta_{i}\delta_{j}}}\Bigl{(}-\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}q_{jk}q_{il}q_{jl}(\xi_{kk}+\xi_{ll})+2\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}q_{jk}q_{il}q_{jl}\xi_{kl}\Bigl{)}$
$\displaystyle=\frac{1}{\sqrt{\delta_{i}\delta_{j}}}\sum_{k=1}^{n-1}\sum_{l=k}^{n}q_{ik}q_{jk}q_{il}q_{jl}\varphi_{kl}.$

In the derivations above we used that
$\sum_{k=1}^{n}q_{ik}q_{jk}=1_{\\{i=j\\}}$, since $Q^{\prime}Q=QQ^{\prime}=I$.
∎

Proof of Lemma 1. Because $A(x)$ is symmetric and positive definite, then so
is the principal sub-matrix, $H(x)$. Consequently,
$M=D^{-\frac{1}{2}}H(x)D^{-\frac{1}{2}}$ is symmetric and positive definite.
Thus, any eigenvalue, $\mu$ of $M$ is strictly positive. So if $\nu$ is an
eigenvalue of $\tilde{J}(x)=I-D^{-\frac{1}{2}}HD^{-\frac{1}{2}}$, then
$\nu=1-\mu$ where $\mu$ is an eigenvalue of $M$, from which it follows that
all eigenvalues of $\tilde{J}$ are strictly less than 1.

Consider a quadratic form of $\tilde{J}$ with an arbitrary vector
$z\in\mathbb{R}^{n}$. Using Lemma A.3, it follows that any quadratic form is
bounded from below by

$z^{\prime}\tilde{J}z=\sum_{k=1}^{n-1}\sum_{l=k}^{n}\varphi_{kl}\Bigl{(}z^{\prime}D^{-\frac{1}{2}}u_{kl}u_{kl}^{\prime}D^{-\frac{1}{2}}z\Bigl{)}=\sum_{k=1}^{n-1}\sum_{l=k}^{n}\varphi_{kl}\Bigl{(}z^{\prime}D^{-\frac{1}{2}}u_{kl}\Bigl{)}^{2}\geq
0,$

because $\varphi_{kl}>0$ by Lemma A.2. Hence, $\tilde{J}$ is positive semi-
definite and $\nu_{i}\geq 0$, for all $i=1,\ldots,n$.

Finally, since $J(x)$ and $\tilde{J}(x)$ have the same eigenvalues, it follows
that all eigenvalues of $J(x)$ lie within the interval $[0,1)$, which proves
that $g(x)$ is a contraction. $\square$

Proof of Theorem 1. The Theorem is equivalent to the statement that for any
symmetric matrix $G$, there always exists a unique solution to $g(x)=x$. This
follows from Lemma 1 and Banach’s fixed point theorem. $\square$

Proof of Proposition 1. We have $Y=PX$, for some permutation matrix, $P$, so
that $C_{y}=PC_{x}P^{\prime}$. Let $C_{x}=Q\Lambda Q^{\prime}$ be the spectral
decomposition of $C_{x}$, such that $\log C_{x}=Q\log\Lambda_{x}Q^{\prime}$,
where $Q^{\prime}Q=I$ and $\Lambda$ is a diagonal matrix. So
$C_{y}=PC_{x}P^{\prime}=PQ\Lambda Q^{\prime}P^{\prime}$, where
$Q^{\prime}P^{\prime}PQ=Q^{\prime}Q=I$. The first equality uses the fact that
$P$ is a permutation matrix. Therefore, $C_{y}=(PQ)\Lambda(PQ)^{\prime}$ is
the spectral decomposition of $C_{y}$ and $\log
C_{y}=(PQ)\log\Lambda(PQ)^{\prime}=P[\log C_{x}]P^{\prime}$.

Next, let the $i$-th and $j$-th rows of $P$ be the $r$-th and $s$-th unit
vectors, $e_{r}^{\prime}$ and $e_{s}^{\prime}$, respectively. Then we have
$[\log C_{y}]_{ij}=[\log C_{x}]_{rs}$ and, by symmetry,

$[\log C_{y}]_{ij}=[\log C_{y}]_{ji}=[\log C_{x}]_{rs}\mathbf{=}[\log
C_{x}]_{sr},$

which shows that $\gamma_{y}$ is simply a permutation of the elements in
$\gamma_{x}$. $\square$

Proof of Proposition 2. An equicorrelation matrix can be written as
$C=(1-\rho)I_{n}+\rho U_{n}$, where $I_{n}\in\mathbb{R}^{n\times n}$ is the
identity matrix and $U_{n}\in\mathbb{R}^{n\times n}$ is a matrix of ones.
Using the Sherman–Morrison formula, we can obtain the inverse,
$C^{-1}=\frac{1}{1-\rho}(I_{n}-\frac{\rho}{1+(n-1)\rho}U_{n})$, so that

$G=\log
C=-\log(C^{-1})=-\log(\tfrac{1}{1-\rho}I_{n})-\log(I_{n}-\tfrac{\rho}{1+(n-1)\rho}U_{n}).$
(A.4)

Because the first term is a diagonal matrix, the off-diagonal elements of $G$
are determined only by the second term, which equals

$-\log(I_{n}-\varphi U_{n})=-\sum_{k=1}^{\infty}(-1)^{k+1}\tfrac{(-\varphi
U_{n})^{k}}{k}=-\left[\frac{1}{n}\sum_{k=1}^{\infty}(-1)^{k+1}\tfrac{(-n\varphi)^{k}}{k}\right]U_{n}=-\frac{1}{n}\log(1-n\varphi)U_{n},$
(A.5)

where $\varphi=\rho/(1+(n-1)\rho)$ and we have used the fact that
$U_{n}^{k}=n^{k-1}U_{n}$. It now follows that

$G_{ij}=\gamma_{c}=-\frac{1}{n}\log(1-\tfrac{n\rho}{1+(n-1)\rho})=-\frac{1}{n}\log\tfrac{1-\rho}{1+\rho(n-1)}=-\frac{1}{n}\log\tfrac{1-\rho}{1+\rho(n-1)},\quad\text{for
all }i\neq j$

for all $i$ and $j$, such that $i\neq j$. $\square$

Proof of Proposition 3. From Theorem 1 it follows that the diagonal,
$x=\mathrm{diag}G$, is fully characterized by the off-diagonal elements,
$y=\mathrm{vecl}G=\mathrm{vecl}G^{\prime}$, and we may write $x=x(y)$. For the
off-diagonal elements of the correlation matrix,
$z=\mathrm{vecl}C\mathrm{=vecl}C^{\prime}$, we have $z=z(x,y)=z(x(y),y)$,
since $C=e^{G}$, and it follows that

$\frac{dz(x,y)}{dy}=\frac{\partial z(x,y)}{\partial
x}\frac{dx(y)}{dy}+\frac{\partial z(x,y)}{\partial y}.$ (A.6)

With $A(x,y)=d\mathrm{vec}C/d\mathrm{vec}G$ and the definitions of $E_{l}$ and
$E_{u}$, the second term is given by:

$\frac{\partial z(x,y)}{\partial
y}=E_{l}A(x,y)E_{l}^{\prime}+E_{l}A(x,y)E_{u}^{\prime}.$ (A.7)

The expression has two terms because a change in an element of $y$ affects two
symmetric entries in the matrix $G$. Similarly, for the first part of the
first term in (A.6) we have,

$\frac{\partial z(x,y)}{\partial x}=E_{l}A(x,y)E_{d}^{\prime},$ (A.8)

and what remains is to determine $\frac{dx(y)}{dy}$. For this purpose we
introduce $D(x,y)=\mathrm{diag}[e^{G(x,y)}]-\iota$ which implicitly defines
the relation between $x$ and $y$. The requirement that $e^{G}$ is a
correlation matrix, is equivalent to $D(x,y)=0$. Next, let $\frac{\partial
D}{\partial x}$ and $\frac{\partial D}{\partial y}$ denote the Jacobian
matrices of $D(x,y)$ with respect to $x$ and $y$, respectively. These Jacobian
matrices have dimensions $n\times n$ and $n\times n(n-1)/2$, respectively, and
can also be expressed in terms of matrix $A(x,y)$, as follows

$\displaystyle\frac{\partial D}{\partial x}=E_{d}A(x,y)E_{d}^{\prime},$
$\displaystyle\qquad\frac{\partial D}{\partial
y}=E_{d}A(x,y)E_{l}^{\prime}+E_{d}A(x,y)E_{u}^{\prime}.$

Note that $\frac{\partial D}{\partial x}$ is a principal sub-matrix of
positive definite matrix $A$ and, hence, is an invertible matrix. Therefore,
from the Implicit Function Theorem it follows

$\frac{dx(y)}{dy}=-\Bigl{(}\frac{\partial D}{\partial
x}\Bigl{)}{}^{-1}\frac{\partial D}{\partial
y}=-\Bigl{(}E_{d}A(x,y)E_{d}^{\prime}\Bigl{)}{}^{-1}\Bigl{(}E_{d}A(x,y)E_{l}^{\prime}+E_{d}A(x,y)E_{u}^{\prime}\Bigl{)}.$
(A.9)

The results now follows by inserting (A.7), (A.8) and (A.9) into (A.6).
$\square$

Proof of Lemma 2. We have $G=Q\log\Lambda Q^{\prime}$, where $C=Q\Lambda
Q^{\prime}$ is the spectral decomposition of the correlation matrix. Thus a
generic element of $G$ can be written as
$G_{ij}=\sum_{k=1}^{n}q_{ik}q_{jk}\log\lambda_{k}$. By Jensen’s inequality it
follows that
$G_{ii}=\sum_{k=1}^{n}q_{ik}^{2}\log\lambda_{k}\leq\log\left(\sum_{k=1}^{n}q_{ik}^{2}\lambda_{k}\right)$,
where we used that $\sum_{k=1}^{n}q_{ik}q_{jk}=1_{\\{i=j\\}}$, because
$Q^{\prime}Q=I$. Finally, since
$\sum_{k=1}^{n}q_{ik}^{2}\lambda_{k}=C_{ii}=1$, it follows that
$G_{ii}\leq\log 1=0$. $\square$