ML Estimates for Fractionally-Differenced ARIMA (p,d,q) models
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).
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)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,
S-plus 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[t-1] - … - a[p]X[t-p] = e[t] - b[1]e[t-1] - … - b[q]e[t-q],
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_moving-average_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 fractional-differencing 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 least-squares optimization in the AR and MA parameters to
minimize white noise variance (uses the MINPACK subroutine lmDER).
written by Chris Fraley (March 1991).
The variance-covariance 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) Space-time Modelling with Long-memory Dependence: Assessing Ireland's Wind Power Resource (with Discussion); Applied Statistics 38, 1–50.
R. Brent (1973) Algorithms for Minimization without Derivatives, Prentice-Hall
J. J. More, B. S. Garbow, and K. E. Hillstrom (1980) Users Guide for MINPACK-1, Technical Report ANL-80-74, Applied Mathematics Division, Argonne National Laboratory.
coef.fracdiff and other methods for "fracdiff"
objects;
fracdiff.var() for re-estimation of variances or
standard errors;
fracdiff.sim
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.