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acfarma(phi,theta [,lag:L][,nfreq:Nfreq][,arsign:Arsign,\
  masign:Masign]), REAL vectors phi, theta, positive integer L, integer
  Nfreq != 0, Arsign and Masign either +1 or -1

Keywords: arima models, time domain, autocovariance
Macro acfarma() computes the theoretical autocovariance function (ACVF)
of an ARMA time series with given coefficients and innovation variance

gamma <- acfarma(phi, theta, lag:L), where phi and theta are REAL
vectors and L > 0 is an integer, returns the ACVF from 0 to L lags of
an ARMA time series with REAL coefficient vectors phi and theta.
gamma[1] is the variance and gamma[h+1] is the lag h autocovariance, h
= 1, ..., L.  The innovation variance is assumed to be 1.

To omit part of model, use phi = 0 or theta = 0.

'lag:L' can be omitted, in which case the default value is L = 100.

You can make an "impulse" plot of the first 50 lags of the ACFV by
  Cmd> tsplot(acfarma(phi, theta, lag:50),0,impulse:T,lines:F)

If you omit 'lines:F', lines are drawn between successive values.

The model assumed for series X is
  (1 + Arsign*sum(phi*B^run(p)))X[t] =
                    (1 + Masign*sum(theta*B^run(q)))Z[t].

where Arsign and Masign are either +1 or -1 and {Z[t]} is zero mean
white noise with standard deviation 1.

The default value for Arsign is variable ARSIGN if it exists and -1
otherwise.  The default value for Masign is variable MASIGN if it exists
and -1 otherwise.  Or you can include either or both keyword phrases
'arsign:Arsign' and 'masign:Masign' as arguments.  See topic 'MASIGN'
for an explanation.  If you want the convention used by Brockwell and
Davis, you should do the following before you start your analysis

  Cmd> MASIGN <- 1; ARSIGN <- -1

It is an error for ARSIGN, MASIGN or supplied values of Arsign and
Masign to be other than +1 or -1.

acfarma() uses the fast discrete Fourier transform to compute the ACVF.
The number of frequencies used is the smallest integer Nfreq >=
max(500,L+1) with no prime factors > 29.  Or you can specify Nfreq
by including nfreq:Nfreq as an argument, where Nfreq is non-zero
integer.  When Nfreq < 0, abs(Nfreq) is used without further checking;
when Nfreq > 0 it is an error when Nfreq has a prime factor > 29.

See also specarma(), tsplot().

Gary Oehlert 2003-01-15