The Common Principal Components model (CPC) in the multivariate setting can be motivated
as the model for similarity of the covariance matrices in the -sample problem, Flury (1988).
Having
random vectors,
the CPC-Model can be written as:
Using the normality assumption, the sample covariance matrices
, are Wishart-distributed:
As shown in Section 5.4, using the functional basis expansion, the FPCA and SPCA are basically implemented via the spectral decomposition of the ``weighted" covariance matrix of the coefficients. In view of the minimization property of the FG algorithm, the diagonalization procedure optimizing the criterion (5.13) can be employed. However, the obtained estimates may not be maximum likelihood estimates.
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Using this procedure for the IV-strings of 1M and 2M maturity we get ``common" smoothed eigenfunctions.
The first three common eigenfunctions
(
,
)
are displayed in Figures
5.6-5.8. The solid blue curve represents the estimated eigenfunction for the
1M maturity, the finely dashed green
curve for the 2M maturity and the dashed black curve is the common eigenfunction estimated by
the FG-algorithm.
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Assuming that
are centered for
and
(we subtract the sample mean of corresponding group from the estimated functions),
we may use the obtained weight functions in the factor model
of the IV dynamics of the form:
In addition, an econometric approach, successfully employed by Fengler, Härdle, and Mammen (2004)
can be employed. It consists of fitting an appropriate model to the time series of the
estimated principal component scores,
,
as displayed in Figure 5.9. Note that
are centered again
(sample means are zero). The fitted time series model can be used for forecasting future IV functions.
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There are still some open questions related to this topic.
First of all, the practitioner would be interested in a good automated choice of the parameters of our method
(dimension of the truncated functional basis and smoothing parameter
).
The application of the Fourier coefficients in this framework seems to be reasonable for the volatility smiles
(U-shaped strings), however for the volatility smirks (typically monotonically decreasing strings) the performance
is rather bad. In particular, the variance of our functional objects and the shape of our weight functions
at the boundaries is affected.
The application of regression splines in this setting seems to be promising, but it increases the number of smoothing parameters by the number and the choice of the knots - problems which are
not generally easy to deal with.
The next natural question, which is still open concerns the statistical properties of the technique and the testing procedure for the Functional Common PCA model.
Finally, using the data for a longer time period one may also analyze the longer maturities
like 3 months or 6 months.