Road casualties and the seat belt law (DK §8.2)
===============================================

The log of monthly car drivers killed or seriously injured in Great
Britain, January 1969 to December 1984 (Harvey and Durbin 1986). The basic
structural model

.. math::

   y_t = \mu_t + \gamma_t + \varepsilon_t, \qquad \mu_{t+1} = \mu_t + \xi_t,

with a trigonometric seasonal :math:`\gamma_t`, then the same with the log
real petrol price and a level intervention for the seat belt law of
February 1983 as regressors.

Results
-------

==============================  ==============  ==============
quantity                        DK              this library
==============================  ==============  ==============
:math:`\sigma^2_\varepsilon`    0.00341598      0.00341596
:math:`\sigma^2_\xi`            0.000935852     0.000935879
:math:`\sigma^2_\omega`         5.01096e-7      5.0098e-7
petrol (rmse)                   -0.29140        -0.29140
                                (0.09832)       (0.09832)
seat belt law (rmse)            -0.23773        -0.23774
                                (0.04632)       (0.04632)
==============================  ==============  ==============

The variances agree to 3-4 digits; the likelihood is flat in the seasonal
variance. The law reduced the number of drivers killed or seriously injured
by :math:`1 - e^{-0.238} = 21\%`.

Two printed numbers follow other conventions:

* DK's **prediction error variance**, 0.00586717, is the steady-state
  :math:`\bar F` (DK §4.3.4), 0.0062583, times (n - d)/n = 180/192.
* DK's **log likelihood**, 435.295, uses a constant we could not identify.
  Ours, 168.859, is the diffuse log likelihood (DK §7.2.2) and equals
  statsmodels'.

Program
-------

.. literalinclude:: ../../example/dk_8_2_seatbelt.f90
   :language: fortran

Output
------

.. literalinclude:: output/dk_8_2_seatbelt.txt
   :language: text