We will first check the data with the help of the ``Augmented
Dickey-Fuller`` Tests (ADF-Test - see (10.46)) for
stationarity. The null hypothesis of a unit root for the
individual VDAX sub-indices
cannot be
rejected at the 90% significance level. Obviously due to this
result the first differences
of the implied volatility indices will be used for further
analysis. Additional ADF tests support the assumption of
stationarity for the first differences.
SFEAdfKpss.xpl
|
Let be the respective sample mean of the first
differences
. Table 19.1 contains the
empirical covariance matrix
used as an estimator
for the
matrix
of the covariance
. With help of the Jordan
decomposition we obtain
. The
diagonal matrix
contains the eigenvalues
of
,
are the eigenvectors. Time series of the principal components can
be obtained with the help of
, where
represents the
matrix of the centered first
differences
. The
matrix
contains the principal
components.
How accurately the first principal components have already
determined the process of the centered first differences can be
measured using the proportion of variance
with respect
to the total variance of the data. The proportion of explained
variance corresponds to the relative proportion of the
corresponding eigenvalue, i.e.,
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(20.2) |
where
are the eigenvalues of the true covariance matrix
. An estimator
for
is
|
By displaying the eigenvalues in a graph, a form with a strong
curvature at the second principal component is shown. In
accordance with the well known ``elbow'' criterion, using the
first two principal components with an explanation power of over
80% of the total variance is considered to be sufficient in
describing the data set. The remaining variance can be interpreted
for analytical purposes as the effect of an unsystematic error
term. Figure 19.3 contains the factor loading of the
first two principal components. Based on the orthogonality of the
components the loading factors can be estimated using the least
squares regression of the individual equations
Based on the factor loadings it is clear that a shock to the first factor would affect the implied volatility of all times to maturity considered in a similar way, or would cause a non-parallel shift in the maturities' structure. A shock to the second principal component, on the other hand, causes a tilt of the structure curve: while at short times to maturity it causes a positive change, the longer time to maturities are influenced negatively. The absolute size of the effect of a shock decreases in both factors with the time to maturity.