Dynamic factor analysis of the yield curve (DK §8.6)
====================================================

The dynamic Nelson-Siegel model (Nelson and Siegel 1987; Diebold and Li
2006), a dynamic factor model (DK §3.7):

.. math::

   y_t = \Lambda(\lambda) f_t + \varepsilon_t, \quad
   \varepsilon_t \sim N(0, \sigma^2 I), \qquad
   f_{t+1} = \Phi f_t + \eta_t, \quad \eta_t \sim N(0, \Sigma_\eta),

with level, slope and curvature factors and the Nelson-Siegel loadings for
each maturity. The model is written as an extension of ``ssm_model_t``,
the way to define models that the components do not cover.

Data
----

DK use the yields of Diebold and Li, 17 maturities from licensed data. This
example uses the public-domain constant-maturity yields of the Federal
Reserve for the same months, January 1985 to December 2000, at 8
maturities (``data/us_yields.csv``), so its estimates differ from DK's.

Results
-------

The estimated decay :math:`\hat\lambda = 0.079` is close to DK's 0.078, and
the persistence of the factors, :math:`\hat\Phi_{11} = 0.994` and
:math:`\hat\Phi_{22} = 0.941`, to DK's 0.994 and 0.939. Collapsing the
8 observations to the 3 factors (DK §6.5), as DK do, gives the same log
likelihood and smoothed factors to 1e-12.

Program
-------

.. literalinclude:: ../../example/dk_8_6_yield_curve.f90
   :language: fortran

Output
------

.. literalinclude:: output/dk_8_6_yield_curve.txt
   :language: text