Description Usage Arguments Details Value Method Warning Note References See Also Examples
Calculates the maximum likelihood estimators of the parameters of a fractionallydifferenced 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).
1 2 3 
x 
time series (numeric vector) for the ARIMA model 
nar 
number of autoregressive parameters p. 
nma 
number of moving average parameters q. 
ar 
initial autoregressive parameters. 
ma 
initial moving average parameters. 
dtol 
interval of uncertainty for d. If 
drange 
interval over which the likelihood function is to be maximized as a function of d. 
h 
size of finite difference interval for numerical derivatives.
By default (or if negative),
This is used to compute a finite difference approximation to the
Hessian, and hence only influences the cov, cor, and std.error
computations; use 
M 
number of terms in the likelihood approximation (see Haslett and Raftery 1989). 
trace 
optional integer, specifying a trace level. If positive, currently the “outer loop” iterations produce one line of diagnostic output. 
The fracdiff package has — for historical reason, namely,
Splus arima()
compatibility — used an unusual
parametrization for the MA part, see also the ‘Details’ section
in arima
(in standard R's stats package).
The ARMA (i.e., d = 0) model in fracdiff()
and
fracdiff.sim()
is
X[t]  a[1]X[t1]  …  a[p]X[tp] = e[t]  b[1]e[t1]  …  b[q]e[tq],
where e[i] are mean zero i.i.d., for fracdiff()
's
estimation, e[i] ~ N(0, s^2).
This model indeed has the signs of the MA coefficients b[j]
inverted, compared to other parametrizations, including
Wikipedia's
http://en.wikipedia.org/wiki/Autoregressive_movingaverage_model
and the one of arima
.
Note that NA
's in the initial values for ar
or ma
are replaced by 0's.
an object of S3 class
"fracdiff"
, which is
a list with components:
log.likelihood 
logarithm of the maximum likelihood 
d 
optimal fractionaldifferencing parameter 
ar 
vector of optimal autoregressive parameters 
ma 
vector of optimal moving average parameters 
covariance.dpq 
covariance matrix of the parameter estimates (order : d, ar, ma). 
stderror.dpq 
standard errors of the parameter estimates

correlation.dpq 
correlation matrix of the parameter estimates (order : d, ar, ma). 
h 
interval used for numerical derivatives, see 
dtol 
interval of uncertainty for d; possibly altered from input

M 
as input. 
hessian.dpq 
the approximate Hessian matrix H of 2nd order
partial derivatives of the likelihood with respect to the
parameters; this is (internally) used to compute

The optimization is carried out in two levels:
an outer univariate unimodal
optimization in d over the interval drange
(typically [0,.5]),
using Brent's fmin
algorithm), and
an inner nonlinear leastsquares optimization in the AR and MA parameters to
minimize white noise variance (uses the MINPACK subroutine lm
DER).
written by Chris Fraley (March 1991).
The variancecovariance matrix and consequently the standard errors
may be quite inaccurate, see the example in fracdiff.var
.
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 problems.
J. Haslett and A. E. Raftery (1989) Spacetime Modelling with Longmemory Dependence: Assessing Ireland's Wind Power Resource (with Discussion); Applied Statistics 38, 1–50.
R. Brent (1973) Algorithms for Minimization without Derivatives, PrenticeHall
J. J. More, B. S. Garbow, and K. E. Hillstrom (1980) Users Guide for MINPACK1, Technical Report ANL8074, Applied Mathematics Division, Argonne National Laboratory.
coef.fracdiff
and other methods for "fracdiff"
objects;
fracdiff.var()
for reestimation of variances or
standard errors;
fracdiff.sim
1 2 3 4 5 6 7 8 9 10 
Call:
fracdiff(x = ts.test$series, nar = length(ts.test$ar), nma = length(ts.test$ma))
Coefficients:
d ar ma
0.3285966 0.1482305 0.4118797
sigma[eps] = 0.9976798
a list with components:
[1] "log.likelihood" "n" "msg" "d"
[5] "ar" "ma" "covariance.dpq" "fnormMin"
[9] "sigma" "stderror.dpq" "correlation.dpq" "h"
[13] "d.tol" "M" "hessian.dpq" "length.w"
[17] "call"
2.5 % 97.5 %
d 0.29834999 0.3588433
ar 0.07827237 0.2181886
ma 0.46224100 0.3615185
dopt() debugging: dinit = 0 ==> xx = 0.190983, fx = pqopt(x[], xx) = 7107.09; min_fnorm = 70.8729
it.  uu  pqopt(uu)  delta 
.. DBG dopt() [ 2]: 0.309017  7083.81  1.909830e01 
.. DBG dopt() [ 3]: 0.381966  7086.82  1.180340e01 
.. DBG dopt() [ 4]: 0.328951  7083.33  5.301502e02 
.. DBG dopt() [ 5]: 0.328743  7083.33  1.972569e02 
.. DBG dopt() [ 6]: 0.32836  7083.33  3.823418e04 
.. DBG dopt() [ 7]: 0.328597  7083.33  2.363002e04 
.. DBG dopt() [ 8]: 0.328637  7083.33  2.363002e04 
.. DBG dopt() [ 9]: 0.328556  7083.33  4.070990e05 
[1] "Component \"call\": target, current do not match when deparsed"
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