fracdiff
ML Estimates for FractionallyDifferenced ARIMA (p,d,q) models
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).
 Keywords
 ts
Usage
fracdiff(x, nar = 0, nma = 0, ar = rep(NA, max(nar, 1)), ma = rep(NA, max(nma, 1)), dtol = NULL, drange = c(0, 0.5), h, M = 100, trace = 0)
Arguments
 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
dtol
is negative or NULL, the fourth root of machine precision will be used.dtol
will be altered if necessary by the program.  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),
h = min(0.1, eps.5 * (1+ abs(cllf)))
, whereclff := log. max.likelihood
(as returned) andeps.5 := sqrt(.Machine$double.neg.eps)
(typically 1.05e8).This is used to compute a finite difference approximation to the Hessian, and hence only influences the cov, cor, and std.error computations; see also
fracdiff.var
.  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.
Details
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
$$X_t  a_1X_{t1}  \cdots  a_pX_{tp} = e_t  b_1e_{t1}  \dots  b_qe_{tq},$$
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.
Value

an object of S3
 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 c(d, ar, ma).
 correlation.dpq
 correlation matrix of the parameter estimates (order : d, ar, ma).
 h
 interval used for numerical derivatives, see
h
argument.  dtol
 interval of uncertainty for d; possibly altered from input
dtol
.  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
covariance.dpq
, the approximate asymptotic covariance matrix as $C = (H)^{1}$.
class
"fracdiff"
, which is
a list with components:
Note
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.
Method
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).
References
J. Haslett and A. E. Raftery (1989) Spacetime Modelling with Longmemory Dependence: Assessing Ireland's Wind Power Resource (with Discussion); Applied Statistics 38, 150.
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.
See Also
coef.fracdiff
and other methods for "fracdiff"
objects;
fracdiff.sim
Examples
ts.test < fracdiff.sim( 5000, ar = .2, ma = .4, d = .3)
fd. < fracdiff( ts.test$series,
nar = length(ts.test$ar), nma = length(ts.test$ma))
fd.
## Confidence intervals
confint(fd.)
## with iteration output
fd2 < fracdiff(ts.test$series, nar = 1, nma = 1, trace = 1)
all.equal(fd., fd2)