TITLE(fracdiff @@ fracdiff: Maximum likelihood parameter estimates for  )
          fractionally-differenced ARIMA (p,d,q) models
PARA BOLD(DESCRIPTION)
Calculates the maximum likelihood estimators of the parameters
of a fractionally-differenced ARIMA (p,d,q) model, together (if possible)
with their estimated covariance and correlation matrices and
standard errors, as well as the value of the maximized likelihood.
The likelihood is approximated using the fast and accurate method
of Haslett and Raftery (1989).
USAGE(
fracdiff( x, nar = 0, nma = 0, dtol = <see below>, M = 100)

)
ARGUMENTS(
ARG(x@@)
time series for the ARIMA model
ARG(nar@@)
number of autoregressive parameters 
ARG(nma@@)
number of moving average parameters
ARG(dtol@@)
interval of uncertainty for d
If dtol is less than zero,
the fourth root of machine precision will be used.
dtol will be altered if necessary by the program.
ARG(M@@)
number of terms in the likelihood approximation (see Haslett and Raftery 1989)
)
PARA BOLD(VALUE)
a list containing the following elements :
ARG(log.likelihood@@)
logarithm of the maximum likelihood
ARG(d@@)
optimal fractional-differencing parameter
ARG(ar@@)
vector of optimal autoregressive parameters 
ARG(ma@@)
vector of optimal moving average parameters
ARG(covariance.dpq@@)
covarianvce matrix of the parameter estimates 
(order : d, ar, ma)
ARG(stderror.dpq@@)
standard errors of the parameter estimates 
(order : d, ar, ma)
ARG(correlation.dpq@@)
correlation matrix of the parameter estimates 
(order : d, ar, ma)
ARG(dtol@@)
interval of uncertainty for d
PARA BOLD(NOTES)
Ordinarily nar and nma should not be too large (say < 10) 
to avoid degeneracy in the model. 
The function LANG(fracdiff.sim) is available for generating test problems.
PARA BOLD(METHOD)
The optimization is carried out in two levels : an outer univariate unimodal
optimization in d over the interval [0,.5] (uses Brent's fmin algorithm), and
an inner nonlinear least-squares optimization in the AR and MA parameters to
minimize white noise variance (uses the MINPACK subroutine LANG(lm)DER).
written by Chris Fraley (March 1991)
PARA BOLD(REFERENCES)
J. Haslett and A. E. Raftery, "Space-time Modelling with Long-memory
Dependence: Assessing Ireland's Wind Power Resource (with Discussion)",
\(\backslash\)fIApplied Statistics, \(\backslash\)fR38, 1-50.
R. Brent, \(\backslash\)fIAlgorithms for Minimization without Derivatives, \(\backslash\)fRPrentice-Hall 
(1973).
J. J. More, B. S. Garbow, and K. E. Hillstrom, 
\(\backslash\)fIUsers Guide for MINPACK-1, \(\backslash\)fRTechnical Report ANL-80-74, 
Applied Mathematics Division, Argonne National
Laboratory (August 1980).
PARA BOLD(SEE ALSO)
 LANG(fracdiff.sim)
BLANK

PARA BOLD( )
BLANK

EXAMPLES(
> ts.test <- fracdiff.sim( 5000, ar = .2, ma = -.4, d = .3)
> fracdiff( ts.test\$series, nar = length(ts.test\$ar), nma = length(ts.test\$ma))
)
PARA ITALIC(File automatically converted from S(-PLUS) help format)
