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

fracdiff {fracdiff} | R Documentation

## 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; 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, 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.

### 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 `lm`DER). written by Chris Fraley (March 1991).

### 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.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)

[Package fracdiff version 1.4-2 Index]