In many applications a statistical analysis is simultaneously done for groups of data. In this section a technique is presented that allows us to analyze group elements that have common PCs. From a statistical point of view, estimating PCs simultaneously in different groups will result in a joint dimension reducing transformation. This multi-group PCA, the so called common principle components analysis (CPCA), yields the joint eigenstructure across groups.
In addition to traditional PCA, the basic assumption of CPCA is that the space spanned by the eigenvectors is identical across several groups, whereas variances associated with the components are allowed to vary.
More formally, the hypothesis of common principle components can be stated in the
following way (Flury; 1988):
where
is a positive definite
population covariance matrix for every
,
is an orthogonal
transformation matrix and
diag
is the matrix of
eigenvalues. Moreover, assume that all
are distinct.
Let be the (unbiased) sample covariance matrix of an underlying
-variate normal
distribution
with sample size
.
Then the distribution of
has
degrees of freedom and is known as the
Wishart distribution (Muirhead; 1982, p. 86):
The density is given in (5.16).
Hence, for a given Wishart matrix
with sample size
, the likelihood function
can be written as
![]() |
(9.41) |
Assuming that holds, i.e., in replacing
by
, after some manipulations one obtains
As we know from Section 2.2, the vectors in
have to be
orthogonal. Orthogonality of the vectors
is achieved
using the Lagrange method,
i.e., we impose the
constraints
using the Lagrange
multipliers
and the remaining
constraints
for
using the multiplier
(Flury; 1988).
This yields
Taking partial derivatives with respect to all
and
, it can
be shown that the solution of the CPC model is given by the generalized
system of characteristic equations
Figure 9.9 shows the first three eigenvectors in a parallel coordinate plot. The basic structure of the first three eigenvectors is not altered. We find a shift, a slope and a twist structure. This structure is common to all maturity groups, i.e., when exploiting PCA as a dimension reducing tool, the same transformation applies to each group! However, by comparing the size of eigenvalues among groups we find that variability is decreasing across groups as we move from the short term contracts to long term contracts.
![]() |
Before drawing conclusions we should convince ourselves that the CPC model is truly a
good description of the data. This can be done by using a likelihood ratio test. The
likelihood ratio statistic for comparing a restricted (the CPC) model against the
unrestricted model (the model where all covariances are treated separately) is given by
The calculations yield
, which corresponds to the
-value
for the
distribution. Hence we cannot reject the CPC
model against the unrestricted model,
where PCA is applied to each maturity separately.
Using the methods in Section 9.3, we can estimate the amount of variability, ,
explained by the first
principle components: (only a few factors,
three at the most, are needed to capture a large amount of
the total variability present in
the data).
Since the model now captures the variability
in both the strike and maturity
dimensions, this is a suitable starting point
for a simplified VaR calculation for
delta-gamma neutral option portfolios using Monte Carlo methods, and is hence a valuable
insight in risk management.