A useful step in multivariate volatility modeling is to reparameterize 
by making use of its symmetric property. We consider two reparameterizations.
10.3.1 Use of Correlations
The first reparameterization of 
is to use the conditional correlation coefficients and variances of 
. Specifically, we write 
as
where 
is the conditional correlation matrix of 
, and 
is a k × k diagonal matrix consisting of the conditional standard deviations of elements of 
(i.e., 
).
Because 
is symmetric with unit diagonal elements, the time evolution of 
is governed by that of the conditional variances σii, t and the elements ρij, t of 
, where j < i and 1 ≤ i ≤ k. Therefore, to model the volatility of 
, it suffices to consider the conditional variances and correlation coefficients of ait. Define the k(k + 1)/2-dimensional vector
10.8 
where 
is a k(k − 1)/2-dimensional vector obtained by stacking columns of the correlation matrix 
, but using only elements below the main diagonal. Specifically, for a k-dimensional return series,
To illustrate, for k = 2, we have 
and
which is a three-dimensional vector, and for k = 3, we have 
and
10.10 
which is a six-dimensional random vector.
If 
is a bivariate normal random variable, then 
is given in Eq. (10.9) and the conditional density function of 
given 
is
where
The log probability density function of 
relevant to the maximum-likelihood estimation is
10.11 
This reparameterization is useful because it models covariances and correlations directly. Yet the approach has several weaknesses. First, the likelihood function becomes complicated when k ≥ 3. Second, the approach requires a constrained maximization in estimation to ensure the positive definiteness of 
. The constraint becomes complicated when k is large.
10.3.2 Cholesky Decomposition
The second reparameterization of 
is to use the Cholesky decomposition; see Appendix A of Chapter 8. This approach has some advantages in estimation as it requires no parameter constraints for the positive definiteness of 
; see Pourahmadi (1999). In addition, the reparameterization is an orthogonal transformation so that the resulting likelihood function is extremely simple. Details of the transformation are given next.
Because 
is positive definite, there exist a lower triangular matrix 
with unit diagonal elements and a diagonal matrix 
with positive diagonal elements such that
This is the well-known Cholesky decomposition of 
. A feature of the decomposition is that the lower off-diagonal elements of 
and the diagonal elements of 
have nice interpretations. We demonstrate the decomposition by studying carefully the bivariate and three-dimensional cases. For the bivariate case, we have
where gii, t > 0 for i = 1 and 2. Using Eq. (10.12), we have
Equating elements of the prior matrix equation, we obtain
Solving the prior equations, we have
However, consider the simple linear regression
where b2t denotes the error term. From the well-known least-squares theory, we have
Furthermore, the error term b2t is uncorrelated with the regressor a1t. Consequently, using Eq. (10.14), we obtain
where 
denotes no correlation. In summary, the Cholesky decomposition of the 2 × 2 matrix 
amounts to performing an orthogonal transformation from 
to 
such that
where q21, t = β is obtained by the linear regression (10.15) and Cov(
) is a diagonal matrix with diagonal elements gii, t. The transformed quantities q21, t and gii, t can be interpreted as follows:
1. The first diagonal element of 
is simply the variance of a1t.
2. The second diagonal element of 
is the residual variance of the simple linear regression in Eq. (10.15).
3. The element q21, t of the lower triangular matrix 
is the coefficient β of the regression in Eq. (10.15).
The prior properties continue to hold for the higher dimensional case. For example, consider the three-dimensional case in which
From the decomposition in Eq. (10.12), we have
Equating elements of the prior matrix equation, we obtain
or, equivalently,
These quantities look complicated, but they are simply the coefficients and residual variances of the orthogonal transformation
where βij are the coefficients of least-squares regressions
In other words, we have qij, t = βij, gii, t = Var(bit) and 
for i ≠ j.
Based on the prior discussion, using Cholesky decomposition amounts to doing an orthogonal transformation from 
to 
, where b1t = a1t, and bit, for 1 < i ≤ k, is defined recursively by the least-squares regression
where qij, t is the (i, j)th element of the lower triangular matrix 
for 1 ≤ j < i. We can write this transformation as
10.17 
where, as mentioned before, 
is also a lower triangular matrix with unit diagonal elements. The covariance matrix of 
is the diagonal matrix 
of the Cholesky decomposition because
The parameter vector relevant to volatility modeling under such a transformation becomes
which is also a k(k + 1)/2-dimensional vector.
The previous orthogonal transformation also dramatically simplifies the likelihood function of the data. Using the fact that 
, we have
If the conditional distribution of 
given the past information is multivariate normal 
, then the conditional distribution of the transformed series bt is multivariate normal N(0Gt, and the log-likelihood function of the data becomes extremely simple. Indeed, we have the log probability density of 
as
where for simplicity the constant term is omitted and gii, t is the variance of bit.
Using the Cholesky decomposition to reparameterize 
has several advantages. First, from Eq. (10.19), 
is positive definite if gii, t > 0 for all i. Consequently, the positive-definite constraint of 
can easily be achieved by modeling ln(gii, t) instead of gii, t. Second, elements of the parameter vector 
in Eq. (10.18) have nice interpretations. They are the coefficients and residual variances of multiple linear regressions that orthogonalize the shocks to the returns. Third, the correlation coefficient between a1t and a2t is
which is time varying if q21, t ≠ 0. In particular, if q21, t = c ≠ 0, then 
, which continues to be time-varying provided that the variance ratio σ11, t/σ22, t is not a constant. This time-varying property applies to other correlation coefficients when the dimension of 
is greater than 2 and is a major difference between the two approaches for reparameterizing 
.
Using Eq. (10.16) and the orthogonality among the transformed shocks bit, we obtain
where qvv, t = 1 for v = 1, … , k. These equations show the parameterization of 
under the Cholesky decomposition.