Fractionally-Differenced ARIMA Modeling via Gaussian MLE

DESCRIPTION:

Calculates maximum likelihood parameter estimates for a fractionally-differenced ARIMA model.

USAGE:

arima.fracdiff(x, model, d.tol = <<see below>>, h = <<see below>>, M = 100, 
               d.range = c(0,.5)) 

REQUIRED ARGUMENTS:

x
a univariate time series or a vector. Missing values (NAs) are not allowed.

OPTIONAL ARGUMENTS:

model
a list specifying a fractionally-differenced ARIMA model. The model should be a list with component names from: d, ar, ma, where

d is the initial fractional differencing parameter (0 < d < .5). Default : NA

ar is the vector of initial values for the autoregressive parameters. Its length is the number of ar parameters that will be returned. An initial value of 0 is assumed for all components that are not assigned values. Default: NULL

ma is the vector of initial values for the moving average parameters. Its length is the number of ma parameters that will be returned. An initial value of 0 is assumed for all components that are not assigned values. Default: NULL
d.tol
interval of uncertainty for d. This should be a small positive quantity. If d.tol is initially nonpositive, too large in magnitude, or too small in magnitude for the achievable accuracy, it will be automatically altered by the program. Default: the fourth root of machine (double) precision.
h
finite-difference interval for approximating partial derivatives with respect to d in estimating var.coef. This should be a small positive quantity such that a perturbation of d by h produces a small but nonnegligeable perturbation in the likelihood. If h is initially not assigned, nonpositive, or unreasonably large or small in magnitude, it will be automatically altered by the program.
M
number of terms in the truncated series used to approximate the likelihood (see Haslett and Raftery, (1989)). Default: 100.
d.range
the range of values over which d is to be optimized. Default: c(0,.5).

VALUE:

a list containing the following elements :
model
same as input but containing the optimal values of d, ar, and ma
var.coef
covariance matrix of the parameter estimates (order : d, ar, ma)
loglik
a quantity proportional to the logarithm of the maximum likelihood (up to an additive constant).
d.tol
interval of uncertainty for d ultimately used by the method.
h
finite-difference interval ultimately used by the method to estimate var.coef .
M
number of terms in the truncated series used to approximate the likelihood.
aux
auxiliary information for recomputing var.coef with a different finite-difference interval.

NOTE:

Ordinarily the number of ar and ma parameters should not be too large (say < 10) to avoid degeneracy in the model.

DETAILS:

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 LMDER).

REFERENCES:

Haslett, J. and Raftery, A. E. (1989). Space-time modeling with long-memory dependence: assessing Ireland's wind power resource (with discussion). Applied Statistics 38, 1-50. Brent, R. (1973). Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ : Prentice-Hall. More, J. J., Garbow, B. S., and Hillstrom, K. E. (1980). Users Guide for MINPACK-1. Technical Report ANL-80-74, Applied Mathematics Division, Argonne National Laboratory.

The chapter "Analyzing Time Series" of the S-PLUS Guide to Statistical and Mathematical Analysis.

SEE ALSO:

, , ,

EXAMPLES:

# generate a fractionally-differenced ARIMA(1,d,1) model given initial values 
ts.sim <- arima.fracdiff.sim(model = list(d=.3, ar=.2, ma=.4), n = 3000) 
# estimate the parameters in an ARIMA(1,d,1) model for the simulated series 
arima.fracdiff(ts.sim, model = list(ar = NA, ma = NA))