

   ffrraaccddiiffff:: MMaaxxiimmuumm lliikkeelliihhoooodd ppaarraammeetteerr eessttiimmaatteess ffoorr

        fracdiff( x, nar = 0, nma = 0, dtol = <see below>, M = 100)

   AArrgguummeennttss::

          x: time series for the ARIMA model

        nar: number of autoregressive parameters

        nma: number of moving average parameters

       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.

          M: number of terms in the likelihood approximation
             (see Haslett and Raftery 1989)

   DDeessccrriippttiioonn::

        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 likeli-
        hood.  The likelihood is approximated using the fast
        and accurate method of Haslett and Raftery (1989).

   VVaalluuee::

        a list containing the following elements :

   log.likelihood: logarithm of the maximum likelihood

          d: optimal fractional-differencing parameter

         ar: vector of optimal autoregressive parameters

         ma: vector of optimal moving average parameters

   covariance.dpq: covarianvce matrix of the parameter esti-
             mates (order : d, ar, ma)

   stderror.dpq: standard errors of the parameter estimates
             (order : d, ar, ma)

   correlation.dpq: correlation matrix of the parameter esti-
             mates (order : d, ar, ma)

       dtol: interval of uncertainty for d

   MMeetthhoodd::

        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 `lm'DER).  written by
        Chris Fraley (March 1991)

   NNoottee::

        Ordinarily nar and nma should not be too large (say <
        10) to avoid degeneracy in the model.  The function
        `fracdiff.sim' is available for generating test prob-
        lems.

   RReeffeerreenncceess::

        J. Haslett and A. E. Raftery, "Space-time Modelling
        with Long-memory Dependence: Assessing Ireland's Wind
        Power Resource (with Discussion)", Applied Statistics,
        38, 1-50.

        R. Brent, Algorithms for Minimization without Deriva-
        tives, Prentice-Hall (1973).

        J. J. More, B. S. Garbow, and K. E. Hillstrom, Users
        Guide for MINPACK-1, Technical Report ANL-80-74,
        Applied Mathematics Division, Argonne National Labora-
        tory (August 1980).

   SSeeee AAllssoo::

        `fracdiff.sim'

   EExxaammpplleess::

        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))

