# A Canonical Representation of Block Matrices with Applications to Covariance
and Correlation Matrices††thanks: The second author would like to thank the
Department of Statistics and Operations Research at University of Vienna for
their hospitality during a visit in early 2020.__

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 28, 2024_)

###### Abstract

We obtain a canonical representation for block matrices. The representation
facilitates simple computation of the determinant, the matrix inverse, and
other powers of a block matrix, as well as the matrix logarithm and the matrix
exponential. These results are particularly useful for block covariance and
block correlation matrices, where evaluation of the Gaussian log-likelihood
and estimation are greatly simplified. We illustrate this with an empirical
application using a large panel of daily asset returns. Moreover, the
representation paves new ways to regularizing large covariance/correlation
matrices and to test block structures in matrices.

Keywords: Block Matrices, Block Covariance Matrix, Block Correlation Matrix,
Equicorrelation, Covariance Regularization, Covariance Modeling, High
Dimensional Covariance Matrices, Matrix Logarithm

JEL Classification: C10; C22; C58

##  1 Introduction

We derive a canonical representation for a broad class of block matrices,
which includes block covariance matrices. A special case of particular
interest are block correlation matrices. The representation is a semi-spectral
decomposition of block matrices, that are diagonalized with the exception of a
single diagonal block, whose dimension is given by the number of blocks.

The canonical representation facilitates simple computations of several matrix
functions, such as the matrix inverse, the matrix exponential, and the matrix
logarithm. Consequently, the decomposition greatly simplifies the evaluation
of Gaussian log-likelihood functions when the covariance matrix, or the
correlation matrix, has a block structure.

We contribute to the literature on block correlation models by providing
simple expressions for the inverse of any (invertible) block correlation
matrix, as well as a simple expression for its determinant. This greatly eases
the computational burden in the evaluation of a Gaussian (quasi-) log-
likelihood function. The results apply to block correlation matrices with an
arbitrary number of blocks. For block correlation matrices with two blocks, an
expression for its inverse was obtained in Engle & Kelly (2012, lemma 2.3),
and related results can be found in Viana & Olkin (1997).

As a preview of some of the results in this paper, one can consider the
following $n\times n$ correlation matrix,

$C=\left[\begin{array}[]{cccc}1&\rho&\cdots&\rho\\\ \rho&1&\ddots&\vdots\\\
\vdots&\ddots&\ddots&\rho\\\ \rho&\cdots&\rho&1\end{array}\right].$

This correlation matrix is known as an equicorrelation matrix, and it is well
known that its eigenvalues are $1+\rho(n-1)$ and $1-\rho$, where the latter
has multiplicity $n-1$. This follows directly from the spectral decomposition,

$Q^{\prime}CQ=D=\left[\begin{array}[]{cc}1+\rho(n-1)&0\\\
0&(1-\rho)I_{n-1}\end{array}\right],$ (1)

where $Q$ is an orthonormal matrix, so that $Q^{\prime}Q=I_{n}$. Here $I_{n}$
denotes the $n\times n$ identity matrix. The matrix $Q$ is given by
$Q=(v_{n},v_{n\bot})$, where $v_{n}$ is the $n$-dimensional vector,
$v_{n}=(\tfrac{1}{\sqrt{n}},\ldots,\tfrac{1}{\sqrt{n}})^{\prime}$, and
$v_{n\bot}$ is an $n\times(n-1)$ matrix that is orthogonal to $v_{n}$, i.e.,
$v_{n\bot}^{\prime}v_{n}=0$, and orthonormal, i.e.,
$v_{n\bot}^{\prime}v_{n\bot}=I_{n-1}$.111When $n=1$, $v_{1\bot}$ is an
$1\times 0$ “matrix” and we use the conventions:
$v_{1\bot}^{\prime}v_{1\bot}=\emptyset$ (dimension $0\times 0$) and
$v_{1\bot}v_{1\bot}^{\prime}=0$ (dimension $1\times 1$). This ensures that our
expressions also hold in the special case, where one or more blocks has size
one. It can now be verified that $QQ^{\prime}=I$, so that $Q$ is orthonormal
and $C=QQ^{\prime}CQQ^{\prime}=QDQ^{\prime}$. In this example, the matrix,
$D$, is the canonical form of $C$, which is obtained via a rotation of $C$,
where the rotation does not depend on $\rho$. In this example, where $K=1$,
$D$ coincides with the diagonal matrix of eigenvalues in the spectral
decomposition of $C$.

In this paper, we derive a similar decomposition for a broad class of block
matrices that includes block covariance matrices and block correlation
matrices. In the general case with multiple blocks, $K\geq 2$, the canonical
representation does not fully disentangle all eigenvalues, and some
eigenvalues may be complex-valued. The canonical representation decomposes any
block matrix into a $K\times K$ matrix and $n-K$ real-valued eigenvalues,
where $K$ is the number of blocks. We can illustrate the general results with
a $2\times 2$ block correlation matrix,

$C=\left[\begin{array}[]{c|c}C_{\rho_{11}}&\rho_{21}\mathbf{1}_{n_{1}\times
n_{2}}\\\ \hline\cr\bullet&C_{\rho_{22}}\end{array}\right]$

where $C_{\rho_{11}}$ and $C_{\rho_{22}}$ are equicorrelation matrices with
correlations $\rho_{11}$ and $\rho_{22}$, respectively, and dimensions
$n_{1}\times n_{1}$ and $n_{2}\times n_{2}$ respectively, and
$\mathbf{1}_{n_{1}\times n_{2}}$ is the $n_{1}\times n_{2}$ whose elements are
all equal to one. Now define

$Q=\left[\begin{array}[]{cccc}v_{n_{1}}&0&v_{n_{1}\bot}&0\\\
0&v_{n_{2}}&0&v_{n_{2}\bot}\end{array}\right].$

For the equicorrelation matrix, we now have the following representation,

$Q^{\prime}CQ=\left[\begin{array}[]{cccc}1+\rho_{11}(n_{1}-1)&\rho_{12}\sqrt{n_{1}n_{2}}&0&0\\\
\rho_{12}\sqrt{n_{1}n_{2}}&1+\rho_{22}(n_{2}-1)&0&0\\\
0&0&(1-\rho_{11})I_{n_{1}-1}&0\\\
0&0&0&(1-\rho_{22})I_{n_{2}-1}\end{array}\right].$ (2)

We denote the upper-left $2\times 2$ matrix by $A$. In general, $A$ will be a
$K\times K$ matrix, whose eigenvalues are also eigenvalues of $C$. The general
result for block matrices with $K$ blocks will be presented in Theorem 1, with
a structure similar to that in (2). An important feature is that the matrix
$Q$ does not depend on the elements in block matrix, but is solely determined
by the block partition, $(n_{1},\ldots,n_{K})$, where $n=n_{1}+\cdots+n_{K}$.

The canonical representation is obtained for general block matrices that need
not be symmetric, nor positive semidefinite. In fact, our results are
applicable to non-square matrices. Block covariance matrices and block
correlation matrices are interesting special cases. For block correlation
matrices, the $A$-matrix, which emerges in (2), was previously established in
Huang & Yang (2010) and Cadima et al. (2010), as we will discuss in Section 3.
We derive additional results for block correlation matrices that simplify the
evaluation of the log-likelihood function.

The rest of this paper is organized as follows. We present the main result in
Section 2, where the canonical representation is established for a broad class
of block matrices, along with some related results for the matrix exponential,
matrix logarithm of block matrices, and matrix powers, including the matrix
inverse. In Section 3, we consider the special case with block covariance
matrices and block correlation matrices. Many of these results are useful for
maximum likelihood estimation with a Gaussian log-likelihood function, as we
show in Section 4. In Section 5, we apply the results to estimation of block
covariance matrices for a very large panel of daily stock returns. We conclude
in Section 6 and all proofs are presented in the Appendix.

##  2 Canonical Representation of Block Matrices

Let $B$ be a square $n\times n$ matrix. The extension to rectangular matrices,
which is trivial, will be addressed in the end of this section. The matrix,
$B$, is called a block matrix with block partition, $n_{1},\ldots,n_{K}$, if
it can be expressed as:

$B=\left[\begin{array}[]{cccc}B_{[1,1]}&B_{[1,2]}&\cdots&B_{[1,K]}\\\
B_{[2,1]}&B_{[2,2]}\\\ \vdots&&\ddots\\\
B_{[K,1]}&&&B_{[K,K]}\end{array}\right],$

where $B_{[i,j]}$ is an $n_{i}\times n_{j}$ matrix with the following
structure

$B_{[i,i]}=\left[\begin{array}[]{cccc}d_{i}&b_{ii}&\cdots&b_{ii}\\\
b_{ii}&d_{i}&\ddots\\\ \vdots&\ddots&\ddots\\\
b_{ii}&&&d_{i}\end{array}\right]\quad\text{and }\quad
B_{[i,j]}=\left[\begin{array}[]{ccc}b_{ij}&\cdots&b_{ij}\\\ \vdots&\ddots\\\
b_{ij}&&b_{ij}\end{array}\right]\quad\text{if }i\neq j,$ (3)

for some constants, $d_{i}$ and $b_{ij}$, $i,j=1,\ldots,K$. So the diagonal
elements of the diagonal blocks, $B_{[i,i]}$, can take a different value than
the off-diagonal elements, whereas all elements in an off-diagonal block,
$B_{[i,j]}$, $i\neq j$, are identical.

We introduce the following notation that relates to orthogonal projections.
Let $P_{[i,j]}=v_{n_{i}}v_{n_{j}}^{\prime}$ be the $n_{i}\times n_{j}$ matrix
whose elements are all equal to $\frac{1}{\sqrt{n_{i}n_{j}}}$. It is simple to
verify that $P_{[i,k]}P_{[k,j]}=P_{[i,j]}$, and with $i=k=j$ it follows that
$P_{[i,i]}P_{[i,i]}=P_{[i,i]}$, so that $P_{[i,i]}$ is a projection matrix. It
then follows that $P_{[i,i]}^{\bot}=I_{n_{i}}-P_{[i,i]}$ is a projection
matrix, and it can be verified that
$P_{[i,i]}^{\bot}=v_{n_{i}\bot}v_{n_{i}\bot}^{\prime}$, where the matrix,
$v_{n\bot}$, was characterized in the introduction.

Finally, we define the $n\times n$ matrix

$Q=\left[\begin{array}[]{cccccccc}v_{n_{1}}&0&\cdots&&v_{n_{1}\bot}&0&\cdots&0\\\
0&v_{n_{2}}&&&0&v_{n_{2}\bot}&&\vdots\\\ \vdots&&\ddots&&&&\ddots\\\
0&\cdots&&v_{n_{K}}&0&\cdots&&v_{n_{K}\bot}\end{array}\right],$

and observe that $Q$ is an orthonormal matrix, characterized by the identity
$Q^{\prime}Q=I$. The first $K$ columns of $Q$ can be used to form averages
within each of the $K$ blocks, whereas the remaining columns of $Q$ capture
“differences” within each block. The two sets of columns span orthogonal
subspaces that correspond to distinct components of the block decomposition.
Note that $Q$ is solely defined by the block partition, $n_{1},\ldots,n_{K}$,
and it is therefore invariant to the actual values taken by the elements in
the block matrix.

######  Theorem 1.

Suppose that $B$ is a block matrix with block partition $n_{1},\ldots,n_{K}$.
Then

$B_{[i,j]}=a_{ij}P_{[i,j]}+1_{\\{i=j\\}}\lambda_{i}P_{[i,i]}^{\bot},\qquad\text{for}\quad
i,j=1,\ldots,K,$

where $a_{ij}=b_{ij}\sqrt{n_{i}n_{j}}$, for $i\neq j$,
$a_{ii}=d_{i}+(n_{i}-1)b_{ii}$, and $\lambda_{i}=d_{i}-b_{ii}$. Moreover,

$B=QDQ^{\prime},\qquad\text{with}\quad
D=\left[\begin{array}[]{cccc}A&0&\cdots&0\\\
0&\lambda_{1}I_{n_{1}-1}&\ddots&\vdots\\\ \vdots&\ddots&\ddots&0\\\
0&\cdots&0&\lambda_{K}I_{n_{K}-1}\end{array}\right].$ (4)

The matrix $Q$ rotates $B$ into its canonical form, $D$. The first $K$ columns
of $Q$ span an eigenspace of $B$, associated with the eigenvalues that $A$ and
$B$ have in common. The last $n-K$ columns of $Q$ are the remaining
eigenvectors of $B$.

Theorem 1 can be used to characterize properties of $B$ and simplifies the
computation of some transformations of $B$, including the matrix logarithm of
$B$, which is denoted by $\log B$. These results are stated in the following
corollary:

######  Corollary 1.

Suppose that $B$ is a block matrix as defined above. $(i)$ The eigenvalues of
$B$ are given by those of $A$ as well as $\lambda_{i}=d_{i}-b_{ii}$,
$i=1,\ldots,K$, so that $\det
B=\det(A)\lambda_{1}^{n_{1-1}}\cdots\lambda_{K}^{n_{K}-1}$. $(ii)$ $B$ is
invertible, if and only if $A$ is invertible and $d_{i}\neq b_{ii}$, for all
$i=1,\dots,K$. $(iii)$ The $q$-th power of the block matrix, $B^{q}$, is well-
defined whenever $A^{q}$ and $\lambda_{i}^{q}$, $i=1,\ldots,K$, are well-
defined, in which case $B^{q}$ has the same block structure as $B$, with
blocks given by

$B_{[i,j]}^{q}=a_{ij}^{(q)}P_{[i,j]}+1_{\\{i=j\\}}\lambda_{i}^{q}P_{[i,i]}^{\bot},$

where $a_{ij}^{q}$ is the $ij$-th element of $A^{q}$, for $i,j=1,\ldots,K$.
$(iv)$ The matrix exponential of $B$ has the same block structure as $B$, with
blocks given by

$\exp(B)_{[i,j]}=a_{ij}^{\exp}P_{[i,j]}+1_{\\{i=j\\}}e^{\lambda_{i}}P_{[i,i]}^{\bot},$

where $a_{ij}^{\exp}$ is the $ij$-th element of $\exp A$, for
$i,j=1,\ldots,K$. $(v)$ If $\log A$ and $\log\lambda_{i}$, $i=1,\ldots,K$,
exist, then $\log B$ has the same block structure as $B$, with blocks given by

$\log(B)_{[i,j]}=a_{ij}^{\log}P_{[i,j]}+1_{\\{i=j\\}}\log\lambda_{i}P_{[i,i]}^{\bot},$

where $a_{ij}^{\log}$ is the $ij$-th element of $\log A$.

It follows that $B^{q}$ is well-defined for all positive integers of $q$, and
the matrix inverse, $B^{-1}$, exists whenever $A$ is invertible and
$\lambda_{i}\neq 0$, for all $i=1,\ldots,K$, in which case $B^{d}$ is also
well-defined for other negative integers of $d$. The logarithms, $\log A$ and
$\log(d_{k}-b_{kk})$, exist provided that $A$ is invertible and
$d_{k}-b_{kk}\neq 0$. This may result in a complex-valued solution to the
matrix logarithm. If a real-valued solution is required, then the conditions
are that $A$ is positive definite and that $d_{k}-b_{kk}>0$ for all
$k=1,\ldots,K$.

###  2.1 Block Matrices with Kronecker Representation

Many of the expressions can be simplified further, in the special case, where
all block sizes are identical, so that $n_{1}=n_{2}=\ldots=n_{K}=n$, with
$n=N/K$. In this situation, we have $B=A\otimes P+\Lambda\otimes P_{\bot}$,
where $P$ is the $n\times n$ matrix with $1/n$ in all entries,
$P_{\bot}=I_{n}-P$, and
$\Lambda=\mathrm{diag}(\lambda_{1},\ldots,\lambda_{K})$. In this case, it
follows that $h(B)=h(A)P+h(\Lambda)P_{\bot}$, where $h(\cdot)$ represents the
matrix inverse, the matrix exponential, or the matrix logarithm, provided
these are well-defined.

###  2.2 Rectangular Block Matrices

Suppose that $B$ has blocks, $B_{[i,j]}\in\mathbb{R}^{n_{i}\times n_{j}}$, as
specified in (3), where $i=1,\ldots,K_{1}$ and $j=1,\ldots,K_{2}$, and
$K_{1}\neq K_{2}$, so that $B$ is a non-square matrix. Set
$K=\max(K_{1},K_{2})$ and suppose that $K_{1}>K_{2}$. Then, by appending
blocks with zero elements to $B$, we obtain a square matrix,
$\tilde{B}=(B,0)$, which is a block matrix with block partition,
$n_{1},\ldots,n_{K}$. Our results apply to $\tilde{B}$, so that it has the
canonical form $\tilde{B}=QDQ^{\prime}$ and $B=QD\tilde{Q}^{\prime}$, where
$\tilde{Q}$ is made up of the first $n_{1}+\cdots+n_{K_{2}}$ columns of $Q$.
If $K_{2}>K_{1}$, we can instead define $\tilde{B}=(B^{\prime},0)^{\prime}$,
and the results follow similarly.

##  3 Block Correlation Matrices

A block correlation matrix is characterized by the correlation coefficients
that form a block structure, so that the correlation between two variables is
solely determined by the blocks to which the two variables belong. This
results in a correlation matrix with a common correlation coefficient within
each block.

Block correlation matrices offer a way to parameterize large covariance
matrices in a parsimonious manner, and can be used to impose economically
relevant structures that reduce the complexity of the covariance matrix. This
structure is used in some multivariate GARCH models, see Engle & Kelly (2012)
and Archakov et al. (2020).

An $n\times n$ block correlation matrix, $C$, with $K$ blocks, is a symmetric
block matrix with blocks,

$C_{[i,i]}=\left[\begin{array}[]{cccc}1&\rho_{ii}&\cdots&\rho_{ij}\\\
\rho_{ii}&1&\ddots\\\ \vdots&\ddots&\ddots\\\
\rho_{ii}&&&1\end{array}\right]\quad\text{and for }i\neq j,\quad
C_{[i,j]}=\left[\begin{array}[]{ccc}\rho_{ij}&\cdots&\rho_{ij}\\\
\vdots&\ddots\\\ \rho_{ij}&&\rho_{ij}\end{array}\right],$ (5)

where $\rho_{ii}$ is within-block correlations, and $\rho_{ij}=\rho_{ji}$,
$i\neq j$, are between-block correlations, for $i,j=1,\ldots,K$. For $C$ to be
a correlation matrix, we obviously need $\rho_{ij}\in[-1,1]$ for all
$i,j=1,\ldots,K$. However, this alone is not sufficient to produce a valid
correlation matrix, because negative eigenvalues can arise with some
combinations of correlation coefficients, even if these area all strictly
smaller than one.

The case with block equicorrelation matrices corresponds to the case where the
diagonal elements of all diagonal blocks, $B_{[kk]}$ equal $d_{k}=1$, for all
$k=1,\ldots,K$. So Theorem 1 fully characterizes the set of correlation
coefficients that yields a positive (semi-) definite correlation matrix. We
formulate this result as a separate Corollary. Note that the canonical form,
(4), for $C$ in (5), is such that $A$ is symmetric with elements given by
$a_{ij}=\rho_{ij}\sqrt{n_{i}n_{j}}$, for $i\neq j$,
$a_{ii}=1+\rho_{ii}(n_{i}-1)$, and $\lambda_{i}=1-\rho_{ii}$.

######  Corollary 2 (Block correlation matrices).

Let $C$ be a block correlation matrix. Then

$\det C=\det A\cdot\prod_{i=1}^{K}(1-\rho_{ii})^{n_{i}-1},$

so that $C$ is a non-singular block correlation matrix if and only if $A$ is
positive definite and $|\rho_{ii}|<1$. In this case, both the inverse
correlation matrix, $C^{-1}$, and the matrix logarithm, $\log C$, have the
same block structure as $C$, with blocks given by

$C_{[i,j]}^{-1}=a_{ij}^{\\#}P_{[i,j]}+1_{\\{i=j\\}}\tfrac{1}{1-\rho_{ii}}P_{[i,i]}^{\bot},$

and

$\log(C)_{[i,j]}=\tilde{a}_{ij}P_{[i,j]}+1_{\\{i=j\\}}\log(1-\rho_{ii})P_{[i,i]}^{\bot},$

respectively, where $a_{ij}^{\\#}$ is the $ij$-th element of $A^{-1}$ and
$\tilde{a}_{ij}$ is the $ij$-th element of $\log A$.

The conditions for $C$ in (5) to be a (possibly singular) correlation matrix
is that $A$ is positive semi-definite and $|\rho_{ii}|\leq 1$ . So, Corollary
2 characterizes the set of positive definite block equicorrelation matrices,
where the additional requirements are that $A$ is positive definite and
$|\rho_{ii}|<1$.

In this context with block correlation matrices, the expression for $A$ was
previously obtained in Huang & Yang (2010, proposition 5) and in Cadima et al.
(2010, theorem 3.1). The focus in Huang & Yang (2010) was on computational
issues, which might explain that their paper is overlooked in much of the
literature.222We were, until recently, also unaware of the results in Huang &
Yang (2010) and Cadima et al. (2010). An anonymous referee (on a different
paper than the present one) directed us to Roustant & Deville (2017) and we
subsequently discovered the more detailed results in Huang & Yang (2010) and
Cadima et al. (2010). Some of their results, e.g., Huang & Yang (2010, eq. 6),
were rediscovered in Roustant & Deville (2017), who do not cite Huang & Yang
(2010) or Cadima et al. (2010). In fact, none of the papers, Cadima et al.
(2010), Huang & Yang (2010), Engle & Kelly (2012), and Roustant & Deville
(2017) cite any of the other papers listed here. Their results add valuable
insight about the block-DECO model by Engle & Kelly (2012). For instance,
their results provide a simple way to evaluate if a block matrix a positive
definite (or semidefinite) correlation matrix. The expression for the
determinant of a correlation matrix in Corollary 2 is a simple implication of
the eigenvalues derived in Huang & Yang (2010) and Cadima et al. (2010),
whereas the expressions for the inverse and logarithmically transformed
correlation matrices are new.

###  3.1 Parametrization of Block Correlation Matrices

A new parametrization of correlation matrices was introduced in Archakov &
Hansen (2020). The new parametrization consists of the elements below the
diagonal of $\log C$ (matrix logarithm of $C$). Let $\varrho$ denote the
vector with these $n(n-1)/2$ elements, where $n$ is the dimension of $C$
($n\times n$). Archakov & Hansen (2020) showed that the $\varrho=\varrho(C)$
is a one-to-one mapping between the set of non-singular correlation matrices
and $\mathbb{R}^{n(n-1)/2}$.

For a block equicorrelation matrix, $C$, it follows (from Corollary 2) that
$\log C$ has the same block structure as $C$. So, for $i\neq j$, all elements
in $[\log C]_{i,j}$ are identical and given by
$\frac{\tilde{a}_{ij}}{\sqrt{n_{i}n_{j}}}$, and the off-diagonal elements of
the diagonal blocks, $[\log C]_{k,k}$, $k=1,\ldots,K$, are all equal to
$\frac{\tilde{a}_{kk}-\log(1-\rho_{kk})}{n_{k}}$, where $\tilde{a}_{ij}$ are
the elements of $\log A$. Thus, the unique elements of $[\log C]$ are

$\Lambda_{n}^{-1}[\log
A-\log\Lambda_{1-\rho}]\Lambda_{n}^{-1}=\Lambda_{n}^{-1}[\log(\Lambda_{n}R\Lambda_{n}+\Lambda_{1-\rho})-\log\Lambda_{1-\rho}]\Lambda_{n}^{-1},$

where

$R=\left[\begin{array}[]{ccc}\rho_{11}&\cdots&\rho_{1K}\\\ \vdots&\ddots\\\
\rho_{K1}&&\rho_{KK}\end{array}\right]\qquad\Lambda_{n}=\left[\begin{array}[]{ccc}\sqrt{n_{1}}&&0\\\
&\ddots\\\
0&&\sqrt{n_{K}}\end{array}\right],\quad\Lambda_{1-\rho}=\left[\begin{array}[]{ccc}1-\rho_{11}&&0\\\
&\ddots\\\ 0&&1-\rho_{KK}\end{array}\right].$

##  4 Applications of the Canonical Representation to Gaussian Log-Likelihood

In this section, we focus on covariance and correlation matrices for normally
distributed random variables. We derive simplified expressions for the
corresponding log-likelihood functions, that greatly reduce the computational
burden when $n$ is large relative to $K$. We derive the maximum likelihood
estimators, and provide a simple expression for the first derivatives of the
log-likelihood function with respect to the unknown parameters (the scores).

We will follow the conventional notation for covariances and variances, we
write $\sigma_{ij}$ in place of $b_{ij}$, $i,j=1,\ldots,K$, and
$\sigma_{k}^{2}$ in place of $d_{k}$, $k=1,\ldots,K$. Similarly, for
correlation matrices we write $\rho_{ij}$ in place of $b_{ij}$, and observe
that $d_{k}=1$.

The density function for the multivariate Gaussian distribution with mean zero
and an $n\times n$ covariance matrix, $\Sigma$, is
$f(x)=(2\pi)^{-\frac{n}{2}}(\det\Sigma)^{-\frac{1}{2}}\exp(-\tfrac{1}{2}x^{\prime}\Sigma^{-1}x)$.
Suppose that $\Sigma$ has the block structure given by $(n_{1},\ldots,n_{K})$,
so that it can be expressed as $\Sigma=QDQ^{\prime}$, using the canonical
representation. The corresponding log-likelihood function (multiplied by $-2$)
can now be expressed as

$-2\ell=n\log 2\pi+\log\det D+X^{\prime}QD^{-1}Q^{\prime}X,$

where
$D=\mathrm{diag}(A,\lambda_{1}I_{n_{1}-1},\ldots,\lambda_{K}I_{n_{K}-1})$,
with $\lambda_{i}=\sigma_{i}^{2}-\sigma_{i,i}$.

$a_{ij}=\begin{cases}\sigma_{i}^{2}+(n_{i}-1)\sigma_{i,i}&\text{for }i=j,\\\
\sigma_{i,j}\sqrt{n_{i}n_{j}}&\text{for }i\neq j.\end{cases}$

So, if we define
$Y=(y_{0}^{\prime},y_{1}^{\prime},\ldots,y_{K}^{\prime})^{\prime}=Q^{\prime}X$,
where $y_{0}$ is $K$-dimensional and $y_{k}$ is $n_{k}-1$ dimensional,
$k=1,\ldots,K$, then it follows that

$-2\ell=n\log 2\pi+\log\det
A+y_{0}^{\prime}A^{-1}y_{0}+\sum_{k=1}^{K}\left((n_{k}-1)\log\lambda_{k}+\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}}\right).$
(6)

This expression of the log-likelihood function shows that the block structure
yields a considerable simplification in the evaluation of the log-likelihood.
Instead of inverting the $n\times n$ matrix $\Sigma$ and computing
$\det\Sigma$, it suffices to invert the smaller $K\times K$ matrix, $A$, and
evaluate its determinant. Moreover, the maximum likelihood estimator based on
a random sample, $X_{1},\ldots,X_{N}$, is easily expressed in terms of the
transformed variables, $Y_{1}=Q^{\prime}X_{1},\ldots,Y_{N}=Q^{\prime}X_{N}$,
as formulated in the following Theorem.

######  Theorem 2.

Suppose that $X_{1},\ldots,X_{N}$ are independent and identically distributed
as $N(0,\Sigma)$, where $\Sigma$ is a block covariance matrix with block
partition, $n_{1},\ldots,n_{K}$. Define the transformed variables,
$Y_{s}=Q^{\prime}X_{s}$, $s=1,\ldots N$, where
$Y_{s}=(y_{0,s}^{\prime},y_{1,s}^{\prime},\ldots,y_{K,s}^{\prime})^{\prime}$
and where $y_{0,s}$ is $K$-dimensional and $y_{k,s}$ is $n_{k}-1$ dimensional,
$k=1,\ldots,K$.

The maximum likelihood estimator of $\Sigma$ is given by
$\hat{\Sigma}=Q\hat{D}Q^{\prime}$, where
$\hat{D}=\mathrm{diag}(\hat{A},\hat{\lambda}_{1}I_{n_{1}-1},\allowbreak\ldots,\allowbreak\hat{\lambda}_{K}I_{n_{K}-1})$
with

$\hat{A}=\frac{1}{N}\sum_{s=1}^{N}y_{0,s}y_{0,s}^{\prime}\quad\text{and}\qquad\hat{\lambda}_{k}=\frac{1}{N}\sum_{s=1}^{N}\tfrac{y_{k,s}^{\prime}y_{k,s}}{n_{k}-1},\quad
k=1,\ldots,K.$

The maximum likelihood estimates of the individual parameters can be obtained
directly from $\hat{A}$ and $\hat{\lambda}_{k}$, $k=1,\ldots,K$. For $i\neq
j$, it follows from the definition of $A$ that
$\hat{\sigma}_{i,j}=\hat{a}_{ij}/\sqrt{n_{i}n_{j}}$. For $i=j$, we have
$\hat{\sigma}_{i,i}=(\hat{a}_{ii}-\hat{\lambda}_{i})/n_{i}$ and
$\hat{\sigma}_{i}^{2}=\hat{\lambda}_{i}+\hat{\sigma}_{i,i}=\tfrac{1}{n_{i}}\hat{a}_{ii}+\tfrac{n_{i}-1}{n_{i}}\hat{\lambda}_{i}$.

In the special case where a block has size one, we have
$\Sigma_{[k,k]}=\sigma_{k}^{2}$ and $\sigma_{k,k}$ is obviously undefined. In
this situation, the corresponding variables, $y_{k,s}$, $s=1,\ldots,N$, are
also undefined, and hence, so is $\hat{\lambda}_{k}$. Yet the expressions for
the maximum likelihood estimators continue to be valid, including the
expression for $\hat{\Sigma}$ in Theorem 2. If $n_{k}=1$, then
$\hat{\sigma}_{k}^{2}=\hat{a}_{kk}$, while the expression for
$\hat{\sigma}_{k,k}$ is undefined and can be ignored.

Estimation when the correlation matrix is assumed to have a block structure,
as opposed to the covariance matrix, is similar. However, a block correlation
matrix is entirely given by the $A$-matrix, and computation of the eigenvalues
$\lambda_{1},\ldots,\lambda_{K}$ is redundant.

######  Corollary 3.

Suppose that $X_{1},\ldots,X_{N}$ are independent and identically distributed
as $N_{n}(0,\Sigma)$, where $\Sigma=\Lambda_{\sigma}C\Lambda_{\sigma}$ with
$\Lambda_{\sigma}=\mathrm{diag}(\sigma_{1},\ldots,\sigma_{n})$ and $C$ is a
block correlation matrix with block partition, $n_{1},\ldots,n_{K}$. The
maximum likelihood estimates of the diagonal elements of $\Sigma$ are given by
$\hat{\sigma}_{i}^{2}=N^{-1}\sum_{s=1}^{N}X_{i,s}^{2}$, for $i=1,\ldots,n$.
Next define $\tilde{X}_{i,s}=X_{i,s}/\hat{\sigma}_{i}$, and introduce the
transformed variables, $\tilde{Y}_{s}=Q^{\prime}\tilde{X}_{s}$, $s=1,\ldots
N$, where
$\tilde{Y}_{s}=(\tilde{y}_{0,s}^{\prime},\tilde{y}_{1,s}^{\prime},\ldots,\tilde{y}_{K,s}^{\prime})^{\prime}$
and where $\tilde{y}_{0,s}$ is $K$-dimensional and $\tilde{y}_{k,s}$ is
$n_{k}-1$ dimensional, $k=1,\ldots,K$.

The maximum likelihood estimator of $C$ is given by
$\hat{C}=Q\tilde{D}Q^{\prime}$, where
$\tilde{D}=\mathrm{diag}(\tilde{A},\tilde{\lambda}_{1}I_{n_{1}-1},\allowbreak\ldots,\allowbreak\tilde{\lambda}_{K}I_{n_{K}-1})$
with

$\tilde{A}=\frac{1}{N}\sum_{s=1}^{N}\tilde{y}_{0,s}\tilde{y}_{0,s}^{\prime}\quad\text{and}\qquad\tilde{\lambda}_{k}=\frac{n_{k}-\tilde{a}_{kk}}{n_{k}-1},\quad
k=1,\ldots,K.$

So the estimate of $D$ can be obtained solely from $\tilde{A}$. For the
individual correlations we have
$\hat{\rho}_{i,j}=\tilde{a}_{ij}/\sqrt{n_{i}n_{j}}$, for $i\neq j$, and for
$i=j$, we have $\hat{\rho}_{i,i}=(\tilde{a}_{ii}-1)/n_{i}$.

The score of the log-likelihood function is often of separate interest. For
instance, the score is used for the computation of robust standard errors, in
Lagrange multiplier tests, in tests for structural breaks, see e.g., Nyblom
(1989), and in dynamic models with time-varying parameters (the so-called
score-drive models), see Creal et al. (2013). So we provide the expressions
for the score in this context with a block covariance matrix.

Suppose that $\Sigma$ is a block covariance matrix, and consider its canonical
representation $\Sigma=QDQ^{\prime}$. Since $Q$ is entirely given by the block
partition $(n_{1},\ldots,n_{K})$, and does not depend on the unknown
parameters in $\Sigma$, the expressions for the partial derivatives are
relatively simple.

######  Proposition 1.

Let $\Sigma=QDQ$ be the canonical representation of $\Sigma$. Then
$\partial(-2\ell)/\partial A=M=A^{-1}-A^{-1}y_{0}y_{0}^{\prime}A^{-1}$ and
for, $k=1,\ldots,K$, we have

$\displaystyle\frac{\partial(-2\ell)}{\partial\sigma_{k}^{2}}$
$\displaystyle=$ $\displaystyle
M_{k,k}+\left(\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}}\right)$
$\displaystyle\frac{\partial(-2\ell)}{\partial\sigma_{kk}}$ $\displaystyle=$
$\displaystyle(n_{k}-1)M_{k,k}-\left(\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}}\right),$

and, for $i\neq j$, we have
$\frac{\partial(-2\ell)}{\partial\sigma_{ij}}=2\sqrt{n_{i}n_{j}}M_{i,j}.$

The hessian could be derived similarly. In some applications, it might be
preferable to parametrize the block covariance matrix with $A$ and
($\lambda_{1},\ldots,\lambda_{K})$. In this case, one can use
$\partial(-2\ell)/\partial A=M$, and
$\partial(-2\ell)/\partial\lambda_{k}=\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}}$,
for $k=1,\ldots,K$.

##  5 Empirical Estimation of Block Correlation Matrices

We proceed to illustrate how high-dimensional covariance matrices with a block
structure are straightforward to estimate in practice. We estimated block
structures for a large panel of assets for two calendar years, 2008 and 2013,
using daily returns. We included all stocks in the CRSP database that could be
matched with a unique ticker symbol, and which did not have any missing
observations. This resulted in 3958 assets in 2008 and 2998 assets in 2013.
The objective of this empirical application is demonstrate that high-
dimensional covariance matrices can be estimated with relatively few
observation once block structures are imposed, and that the canonical
representation makes it simple to evaluate the log-likelihood function and to
obtain the maximum likelihood estimates. Given the well-known variation in
conditional variances and covariances, our estimated covariance matrices
should be viewed as estimates of the average covariance matrix for 2008 and
2013, rather than an accurate description of the data generating process.

We impose five nested structures on the correlation matrix, where the
equicorrelation structure ($K=1$) is the simplest and most restrictive model.
The remaining four correlation models use block structures defined by the
Sector, Group, Industry, and Sub-Industry categories, as classified by the
Global Industry Classification Standard (GICS) in 2013. The five
specifications correspond to $K=1$, $10$, 24, 67, and 151, respectively, in
2008, and the same number of blocks in 2013, except for Sub-Industry
categories, which had $K=146$.

We estimated the canonical correlation matrix using the results in Corollary
3. Thus, we first compute the estimates of the variances,
$\hat{\sigma}_{i}^{2}=\sum_{t=1}^{N}X_{i,t}^{2}$, for each of the individual
assets. Then we define the standardized variables,
$\tilde{X}_{i,t}=X_{i,t}/\hat{\sigma}_{i}$, and
$\tilde{y}_{0,t}\in\mathbb{R}^{K}$, whose $k$-th element is given by
$\tfrac{1}{\sqrt{n_{k}}}\sum_{i=n_{1}+\cdots+n_{k-1}+1}^{n_{1}+\cdots+n_{k}}\tilde{X}_{i,t}$.
From Corollary 3, we have $\hat{\rho}_{i,j}=\tilde{a}_{ij}/\sqrt{n_{i}n_{j}}$,
for $i\neq j$, and $\hat{\rho}_{i,i}=(\tilde{a}_{ii}-1)/n_{i}$, where
$\tilde{A}=\frac{1}{N}\sum_{t=1}^{N}\tilde{y}_{0,t}\tilde{y}_{0,t}^{\prime}$.
Thus, the entire $n\times n$ covariance matrix with a block correlation
structure is estimated by computing the estimates of the $n$ variances, and
the $K\times K$ matrix $\tilde{A}$. Given the high number of assets and just
over 250 daily returns, the unrestricted sample covariance matrix would be
singular, because most of its eigenvalues will be zero. Once a block structure
is imposed, we can compute the inverse covariance matrix from the invertible
$K\times K$ matrix, $\tilde{A}$, and it becomes simple to evaluate the log-
likelihood function.

The empirical results are summarized in Table 1. We report the range of
estimated correlations for each of the block structures, with the results for
2008 and 2013 are reported separately. The range of estimated correlations.
i.e., the interval between the smallest and the largest coefficient in the
correlation matrix, obviously increases with the number of blocks in the
correlation matrix, and the correlations are generally higher in 2008 than in
2013. These estimates are likely biased, because they entail cross-sectional
averaging within each sector/group/industry and time averaging, over a full
calendar year. We also report the value of the maximized log-likelihood
function (scaled by $-2/(nN)$) and the corresponding value of the Bayesian
Information Criterion (BIC). The minimum BIC is obtained with a block
structures based on Groups in both 2008 and 2013.333The BIC adds the penalty
$p\log(nN)$ to $-2\ell$, where $p$ is the number of free parameters. For
comparison, the AIC, which uses the penalty $2p$, selects the most general
specification in both years. It is well known that the AIC tends to favor more
heavily parametrized models. The last column reports the number $K(K+1)/2$ of
unique correlations within a block structure with $K$ blocks, and while this
number increases rapidly with $K$, the gains in the log-likelihood are
relatively modest. Consequently, the BIC increases substantially once the
number of blocks are defined by Industries and Sub-industries.

The estimated block structures are illustrated in Figures 1 and 2. The upper
panels of Figure 1 show the estimated correlation coefficients of assets
within and between sectors. The lower panels are the estimates for the 24
groups, with the actual estimates indicated by color coding. A darker shade of
red denotes a stronger correlation. The left panels are for 2008 and the right
panels are for 2013. Figure 2 presents the estimated correlations using a
block structure based on industries and sub-industries. We observe that the
correlations were generally higher in 2008 than in 2013, in part because of
the turmoil period leading up to, and following, the collapse of Lehman
Brothers in late 2008. The block structure is perhaps more visible in 2008,
which might be explained by the Global Financial Crises having a
differentiated impact on different sections. For instance, Figure 1 shows that
the correlations between the Energy (10), Materials (15), and Utilities (55)
sectors were relatively high, while Financials (40) were relatively
uncorrelated with other sectors in 2008. The partition by the GICS groups in
the lower panels of Figure 1 reveals additional details about the
correlations. For financials (40), we observe that the low correlations with
other sectors are largely driven by Banks (4010).444Note that we are using the
GICS classification as it was in 2013, where Real Estate (4040) was a part of
the Financials Sector. A separate Real Estate Sector (50) was added to the
GICS classification in 2016. Figure 2 presents the corresponding results for
industries and sub-industries. The number of blocks are too plentiful to be
listed individually, however the industries and sub-industries are placed
chronologically in the ascending order in Figure 2 according to their GICS
code.

##  6 Concluding Remarks

We have derived a canonical representation of block matrices, that is
particularly useful for covariance and correlation matrices. We derived a
number of expressions that greatly simplify the computation of the log-
likelihood function. We illustrated this in an empirical application, where we
estimated the covariance matrix for nearly 4000 stocks returns using daily
returns from a single calendar year, i.e., just over 250 observations.
Inverting the covariance matrix, and evaluating the log-likelihood is
straightforward once a block structure is imposed, where we used as many as
$K=151$ blocks, motivated by the Global Industry Classification Standard.

The canonical representation and the related results are potentially useful
for regularizing large covariance matrices. For instance, one could shrink the
sample correlation matrix towards a block correlation matrix, analogous to the
way Ledoit & Wolf (2004) proposed to shrink towards the equicorrelation
matrix. The canonical representation also paves new way to testing block
structures in covariance and correlation matrices. This predominantly amounts
to testing a large number of zero-restrictions in the canonical
representation. We identified a number of transformations that preserves the
block structures, so testing of block structures could be based on any of the
transformations, rather than the original matrix. For instance, block
structures in a correlation matrix $C$ would be tested on the canonical
representation for $\log C$. This is potentially interesting, because the
connection between logarithmically transformed correlation matrix and the
Fisher transformation, see Archakov & Hansen (2020). Finally, the group
assignments, and hence $K$, will be unknown in many empirical applications.
The literature has therefore proposed various techniques that aim to determine
the most appropriate block structure. It is possible that the canonical
representation will be useful for this type of model selection problem.

## Appendix of Proofs

Proof of Theorem 1. For $i\neq j$, we have $B_{[i,j]}=a_{ij}P_{[i,j]}$ if
$a_{ij}=b_{ij}\sqrt{n_{i}n_{j}}$, since the elements of $P_{[i,j]}$ are all
equal to $\frac{1}{\sqrt{n_{i}n_{j}}}.$ For $i=j$, the diagonal elements
differs from off-diagonal elements by $\lambda_{i}=d_{i}-b_{ii}$, so that
$B_{[i,i]}=b_{ii}n_{i}P_{[i,i]}+(d_{i}-b_{ii})I_{n_{i}}$. Since
$I_{n_{i}}=P_{[i,i]}+P_{[i,i]}^{\bot}$, we have
$B_{[i,i]}=(b_{ii}n_{i}+d_{i}-b_{ii})P_{[i,i]}+(d_{i}-b_{ii})P_{[i,i]}^{\bot}=a_{ii}P_{[i,i]}+\lambda_{i}P_{[i,i]}^{\bot}$.
The canonical representation, (4), follows by verifying that $Q^{\prime}BQ$ is
equal to the block-diagonal matrix in (4). This follows from the identities:
$v_{n_{i}}^{\prime}P_{[i,j]}v_{n_{j}}=1$,
$v_{n_{i}}^{\prime}P_{[i,j]}v_{n_{j}\bot}=0$,
$v_{n_{i}\bot}^{\prime}P_{[i,j]}v_{n_{j}\bot}=0$,
$v_{n_{i}}^{\prime}P_{[i,i]}^{\bot}v_{n_{i}}=0$,
$v_{n_{i}}^{\prime}P_{[i,i]}^{\bot}v_{n_{i}\bot}=0$, and
$v_{n_{i}\bot}^{\prime}P_{[i,i]}^{\bot}v_{n_{i}\bot}=I_{n_{i}-1}$, and the
fact that $Q^{\prime}Q=I_{n}$, so that $Q^{-1}=Q^{\prime}$, and hence
$B=QQ^{\prime}BQQ^{\prime}$. This proves (4). $\square$

Proof of Corollary 1. The first result for the eigenvalues of $B$ and the
determinant of $B$, follows immediately from (4). The results for $f(B)$,
where $f$ denotes the $q$-th power of a matrix, the matrix exponential, or the
matrix logarithm, follow by $f(B)=Qf(D)Q^{\prime}$ and using the structure in
$Q$, such as $v_{n_{i}}v_{n_{j}}^{\prime}=P_{[ij]}$ and
$v_{n_{i\bot}}v_{n_{i}\bot}^{\prime}=P_{[ii]}^{\bot}$. This completes the
proof. $\square$

Proof of Corollary 2. It follows from Theorem 1 and Corollary 1 by setting
$d_{k}=1$ for all $k$. Some expressions can also be verified directly. For
instance, one can verify the expression for $C^{-1}$ , by noting that diagonal
blocks of $C^{-1}$ are given by

$(C^{-1})_{[i,i]}=\sum_{k=1}^{K}a_{ik}P_{[ik]}a_{ki}^{\\#}P_{[k,i]}+(1-\rho_{ii})P_{[i,i]}^{\bot}\tfrac{1}{1-\rho_{ii}}P_{[i,i]}^{\bot}=\sum_{k=1}^{K}a_{ik}a_{ki}^{\\#}P_{[i,i]}+P_{[i,i]}^{\bot}=I,$

where we used that $a_{ki}^{\\#}$ are the elements of the $A^{-1}$ so we have
$\sum_{k=1}^{K}a_{ik}a_{ki}^{\\#}=1$. Next, for $i\neq j$, we have

$(C^{-1})_{[i,j]}=\sum_{k=1}^{K}a_{ik}P_{[i,k]}a_{kj}^{\\#}P_{[k,j]}+\frac{a_{ij}}{b_{j}}P_{[i,j]}P_{[j,j]}^{\bot}+b_{i}a_{ij}^{\\#}P_{[i,i]}^{\bot}P_{[i,j]}=\sum_{k=1}^{K}a_{ik}a_{kj}^{\\#}P_{[i,j]}=0,$

where we used that $P_{[i,k]}P_{[k,j]}=P_{[i,j]}$ and
$P_{ij}P_{[j,j]}^{\bot}=P_{[i,j]}(I_{s_{j}}-P_{[j,j]})=0$, and that
$\sum_{k=1}^{K}a_{ik}a_{kj}^{\\#}=0$, for $i\neq j$. This completes the proof.
$\square$

Proof of Theorem 2. The expression, (6), shows that the log-likelihood
function is made up of two terms:

$-2N\left[\log\det
A+\mathrm{tr}\\{A^{-1}\tfrac{1}{N}\sum_{s=1}^{N}Y_{0,s}Y_{0,s}^{\prime}\\}\right],$

and

$-2N\sum_{k=1}^{K}(n_{k}-1)\left(\log\lambda_{k}+\frac{\tfrac{1}{N}\sum_{s=1}^{N}\frac{Y_{k,s}^{\prime}Y_{k,s}}{n_{k}-1}}{\lambda_{k}}\right).$

It is well known that
$\hat{A}=\frac{1}{N}\sum_{s=1}^{N}Y_{0,s}Y_{0,s}^{\prime}$ maximizes the first
term and that
$\hat{\lambda}_{k}=\frac{1}{N}\sum_{s=1}^{N}\frac{Y_{k,s}^{\prime}Y_{k,s}}{n_{k}-1}$
maximizes the elements of the second term. Since
$(A,\lambda_{1},\ldots,\lambda_{K})$ is merely a reparametrization of the
elements of the block covariance matrix $\Sigma$, it follows that
$\hat{\Sigma}=Q\hat{D}Q^{\prime}$ is the maximum likelihood estimator of
$\Sigma$. It is easy to verify that this result is also valid in the special
case, where one or more of the blocks are 1-dimensional. In this case,
$\sigma_{ii}$ is undefined, and so is $\hat{\lambda}_{k}$. In this case,
$\sigma_{i}^{2}$ is identified from the corresponding diagonal element of $A$,
since $\hat{a}_{ii}=\sigma_{i}^{2}$, when $n_{i}=1$. $\square$

Proof of Corollary 3. We have
$\det\Sigma=\det(\Lambda_{\sigma}C\Lambda_{\sigma})=\prod_{i=1}^{n}\sigma_{i}^{2}\det
C=\prod_{i=1}^{n}\sigma_{i}^{2}\det D$. So the log-likelihood function for the
observation, $X\in\mathbb{R}^{n}$, is

$-2\ell(\sigma_{1}^{2},\ldots,\sigma_{n}^{2},C)=n\log
2\pi+\sum_{j=1}^{n}\log\sigma_{j}^{2}+\log\det
C+X^{\prime}\Lambda_{\sigma}^{-1}C\Lambda_{\sigma}^{-1}X,$

and given a sample $X_{1},\ldots,X_{N}$, the first order condition for
$\sigma_{j}^{2}$:

$0=\tfrac{N}{\sigma_{j}^{2}}-\sum_{t=1}^{N}\tfrac{1}{\sigma_{j}^{4}}X_{t}^{\prime}e_{j}e_{j}^{\prime}Ce_{j}e_{j}^{\prime}X_{t}=\tfrac{N}{\sigma_{j}^{2}}-\tfrac{1}{\sigma_{j}^{4}}\sum_{t=1}^{N}X_{t}^{\prime}e_{j}1e_{j}^{\prime}X_{t}=\tfrac{N}{\sigma_{j}^{2}}-\tfrac{1}{\sigma_{j}^{4}}\sum_{t=1}^{N}X_{j,t}^{2},$

is invariant to $C$. So that
$\hat{\sigma}_{j}^{2}=\tfrac{1}{N}\sum_{t=1}^{N}X_{j,t}^{2}$ is the maximum
likelihood estimator of $\sigma_{j}^{2}$ regardless of the structure imposed
on $C$. The concentrated log-likelihood function for the observation
$X\in\mathbb{R}^{n}$ can be expressed as

$-2\ell(C)\equiv-2\ell(\hat{\sigma}_{1}^{2},\ldots,\hat{\sigma}_{n}^{2},C)=n\log
2\pi+\sum_{j=1}^{n}\log\hat{\sigma}_{j}^{2}+\log\det
D+\tilde{X}^{\prime}Q^{\prime}DQ\tilde{X},$

and it follows from the proof of Theorem 2 that minimizing $N\log\det
D+\sum_{t=1}^{N}\tilde{X}_{s}^{\prime}Q^{\prime}DQ\tilde{X}_{t}$ is solved by
the $D$-matrix whose elements are given by
$\tilde{A}=\frac{1}{N}\sum_{t=1}^{N}\tilde{y}_{0,t}\tilde{y}_{0,t}^{\prime}$
and
$\tilde{\lambda}_{k}=\frac{1}{N}\sum_{t=1}^{N}\tfrac{\tilde{y}_{k,t}^{\prime}\tilde{y}_{k,t}}{n_{k}-1}$,
for $k=1,\ldots,K$. For $Q\tilde{D}Q^{\prime}$ to be a correlation matrix
(have ones along its diagonal), we need
$\tilde{a}_{ii}=1+(n_{i}-1)\hat{\rho}_{ii}$ and
$\tilde{\lambda}_{i}=1-\hat{\rho}_{ii}$, which implies
$\tilde{\lambda}_{i}=\tfrac{n_{i}-\tilde{a}_{ii}}{n_{i}-1}$. $\square$

Proof of Proposition 1. Recall that
$a_{kk}=\sigma_{k}^{2}+(n_{k}-1)\sigma_{kk}$,
$a_{ij}=\sigma_{ij}\sqrt{n_{i}n_{j}}$, for $i\neq j$, and
$\lambda_{k}=\sigma_{k}^{2}-\sigma_{kk}$. It follows that

$\frac{\partial(\log\det A+y_{0}^{\prime}A^{-1}y_{0})}{\partial
a_{ij}}=\mathrm{tr}\\{A^{-1}(e_{i}e_{j}^{\prime})(I-A^{-1}y_{0}y_{0}^{\prime})=e_{j}^{\prime}(I-A^{-1}y_{0}y_{0}^{\prime})A^{-1}e_{i}=M_{j,i},$

where $M=A^{-1}-A^{-1}y_{0}y_{0}^{\prime}A^{-1}$. From the expression (6), we
find

$\displaystyle\frac{\partial(-2\ell)}{\partial\sigma_{k}^{2}}$
$\displaystyle=$ $\displaystyle\frac{\partial(\log\det
A+y_{0}^{\prime}A^{-1}y_{0})}{\partial
a_{kk}}+(\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}})=M_{k,k}+(\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}})$
$\displaystyle\frac{\partial(-2\ell)}{\partial\sigma_{kk}}$ $\displaystyle=$
$\displaystyle(n_{k}-1)\frac{\partial(\log\det
A+y_{0}^{\prime}A^{-1}y_{0})}{\partial
a_{kk}}-\left(\tfrac{n_{k}-1}{\lambda_{k}}-\tfrac{y_{k}^{\prime}y_{k}}{\lambda_{k}^{2}}\right)=n_{k}M_{k,k}-\frac{\partial(-2\ell)}{\partial\sigma_{k}^{2}},$

and, for $i\neq j$, we find that

$\frac{\partial(-2\ell)}{\partial\sigma_{ij}}=\sqrt{n_{i}n_{j}}\left(\frac{\partial(\log\det
A+y_{0}^{\prime}A^{-1}y_{0})}{\partial a_{ij}}+\frac{\partial(\log\det
A+y_{0}^{\prime}A^{-1}y_{0})}{\partial
a_{ji}}\right)=2\sqrt{n_{i}n_{j}}M_{i,j},$

where we used that $M$ is symmetric. $\square$

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## Tables and Figures

| Summary statistics of estimated block correlations |  |  | # blocks
---|---|---|---|---
Block structure | Mean | Std. | Min | $Q_{10\%}$ | $Q_{50\%}$ | $Q_{90\%}$ | Max | $-\frac{2\ell}{nN}$ |  $\frac{1}{nN}$BIC | $K$ | $\frac{K(K+1)}{2}$
| U.S. market in 2008 (3958 stocks and 253 days) |  |  |  |  |  |
Equicorrelation |  0.228 |  0 |  0.228 |  0.228 |  0.228 |  0.228 |  0.228 |  2.57829 |  2.57830 |  1 |  1
Sectors |  0.269 |  0.073 |  0.162 |  0.191 |  0.252 |  0.381 |  0.521 |  2.53748 |  2.53824 |  10 |  55
Groups |  0.253 |  0.059 |  0.119 |  0.177 |  0.253 |  0.321 |  0.521 |  2.52519 |  2.52933 |  24 |  300
Industries |  0.263 |  0.074 |  0.088 |  0.172 |  0.258 |  0.362 |  0.659 |  2.51123 |  2.54266 |  67 |  2278
Sub-industries |  0.273 |  0.095 |  -0.036 |  0.157 |  0.268 |  0.394 |  0.886 |  2.48261 |  2.64086 |  151 |  11476
| U.S. market in 2013 (2998 stocks and 252 days) |  |  |  |  |  |
Equicorrelation |  0.165 |  0 |  0.165 |  0.165 |  0.165 |  0.165 |  0.165 |  2.66583 |  2.01128 |  1 |  1
Sectors |  0.181 |  0.059 |  0.107 |  0.126 |  0.168 |  0.242 |  0.507 |  2.63742 |  1.99057 |  10 |  55
Groups |  0.174 |  0.045 |  0.092 |  0.126 |  0.166 |  0.231 |  0.507 |  2.62850 |  1.98715 |  24 |  300
Industries |  0.183 |  0.060 |  0.065 |  0.118 |  0.173 |  0.260 |  0.712 |  2.61095 |  2.00065 |  67 |  2278
Sub-industries |  0.182 |  0.067 |  -0.040 |  0.108 |  0.175 |  0.268 |  0.810 |  2.58182 |  2.09290 |  146 |  10731
Table 1: Summary statistics for the estimated block correlation matrices.

Figure 1: Estimated correlations for a block structure based on GICS sectors
(upper panels) and GICS groups (lower panels). Left panels are the estimates
based on 253 daily returns in 2008, and right panels are the estimated based
on 252 daily returns from 2013. The numbers to the left and below each plot
are GICS codes for sectors or groups.

Figure 2: Estimated correlations for a block structure based on GICS
industries (upper panels) and GICS sub-industries (lower panels). Left panels
are the estimates based on 253 daily returns in 2008, and right panels are the
estimated based on 252 daily returns from 2013. Industries and sub-industries
are listed chronologically according to their GICS code, that are too
plentiful to list individually.