Box-Jenkins analysis of Internet use (DK §8.4)
==============================================

ARMA(p, q) models, p, q = 0, ..., 5, for the first differences of the
number of users logged on to an Internet server each minute (Makridakis,
Wheelwright and Hyndman 1998), in state space form (DK §3.4), compared by
AIC; then again with 14 observations treated as missing, and forecasts
with 50% intervals from the ARMA(1, 1) model.

Results
-------

The log likelihoods agree with statsmodels' SARIMAX to four decimals, with
and without the missing observations; for example -254.1497 and -225.7704
for the ARMA(1, 1). Higher-order models can have several local optima,
where different optimizers stop at different points.

DK's tables use a scale for the AIC that we could not identify: for the
complete series they are close to :math:`n^{-1}[-\log L + 2(p + q)]`, but
no simple formula fits the table with missing observations, and DK report
optimizer failures for several models. This page uses DK's own formula
(DK §7.4), :math:`n^{-1}[-2\log L + 2w]`. By it, ARMA(3, 0), ARMA(1, 1) and
some higher-order models lie within 0.03 of each other, the conclusion DK
draw.

Program
-------

.. literalinclude:: ../../example/dk_8_4_internet.f90
   :language: fortran

Output
------

.. literalinclude:: output/dk_8_4_internet.txt
   :language: text