Generate five sequences of 1000 observations from five different ARMA
processes with
parameters of your choice, using different small values for p
and q and different ARMA parameters for each of the five sequences.
You can use the routines
available in the ts library of
R to generate
the necessary sequences of observations, or you can generate them
based on the definition of an ARMA process.
(Alternatively, use "real" data sets of your choice instead of one
or more of the five sequences.)
Write a program to estimate the
parameters of the ARMA process using the Kalman filter. Programs
for evaluating the likelihood of an ARMA process
are available in ~s244/as/as154.f, and can be accessed either by
compiling them directly, or using the s244 library (-ls244).
Your program should be written to be able to switch between at least
two different
maximization algorithms from among Powell's
method, the downhill simplex (Nelder and Mead), and other algorithms.
The
programs for Powell's method can be found in the directory ~s244/nr ,
on the class webpage,
and are also available via -ls244. (These programs are direct
transcriptions from the book "Numerical Recipes in C", except that
floats have been converted to doubles,
the one-based arrays have been converted to zero-based arrays, and
function arguments have been updated to the ANSI standard.)
Compare the
performance of the minimization routines you choose, especially when the
program is presented with bad guesses for the initial values, with
regard to execution time and number of function evaluations.
Using one of the data sets which you generated in part 1, calculate
the first 20 autocovariances for the sequence, first by the
basic formula, i.e.:
C(k)=
1
N
N-k å
t=1
(Xt-
-
X
) (Xt+k-
-
X
) k=1,¼,20
and then by using the Fast Fourier Transform (FFT).
Estimate the
relative execution times you would expect for the two methods based
on an analysis of the algorithms, and then compare the actual
execution times. How would the relative efficiencies change if you
desired to calculate the first 50 autocovariances instead of the first
20? You can determine this either by analysis of the algorithms, or by
computer timing.
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