fracdiff: ML Estimates for Fractionally-Differenced ARIMA (p,d,q)...

In fracdiff: Fractionally Differenced ARIMA aka ARFIMA(P,d,q) Models

ML Estimates for Fractionally-Differenced ARIMA (p,d,q) models

Description

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).

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))), where clff := log. max.likelihood (as returned) and eps.5 := sqrt(.Machine$double.neg.eps) (typically 1.05e-8). This is used to compute a finite difference approximation to the Hessian, and hence only influences the cov, cor, and std.error computations; use fracdiff.var() to change this after the estimation process.
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, 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_1X_{t-1} - \cdots - a_pX_{t-p} = e_t - b_1e_{t-1} - \dots - b_qe_{t-q},

where e_i are mean zero i.i.d., for fracdiff()'s estimation, e_i \sim \mathcal{N}(0,\sigma^2). This model indeed has the signs of the MA coefficients b_j inverted, compared to other parametrizations, including Wikipedia's https://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.

Value

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 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}.

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 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).

Warning

The variance-covariance matrix and consequently the standard errors may be quite inaccurate, see the example in fracdiff.var.

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.

References

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.

See Also

coef.fracdiff and other methods for "fracdiff" objects; fracdiff.var() for re-estimation of variances or standard errors; 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)

fracdiff documentation built on May 29, 2024, 3:37 a.m.