# fracdiff

##### 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).

- 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)))`

, 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
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:

logarithm of the maximum likelihood

optimal fractional-differencing parameter

vector of optimal autoregressive parameters

vector of optimal moving average parameters

covariance matrix of the parameter estimates (order : d, ar, ma).

standard errors of the parameter estimates
`c(d, ar, ma)`

.

correlation matrix of the parameter estimates (order : d, ar, ma).

interval used for numerical derivatives, see `h`

argument.

interval of uncertainty for d; possibly altered from input
`dtol`

.

as input.

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
\(C = (-H)^{-1}\).

##### 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 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).

##### Warning

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

.

##### 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

```
# NOT RUN {
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)
# }
```

*Documentation reproduced from package fracdiff, version 1.5-1, License: GPL (>= 2)*