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).

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

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 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.

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.

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$ma))
fd.
#> 
#> Call:
#>   fracdiff(x = ts.test$series, nar = length(ts.test$ar), nma = length(ts.test$ma)) 
#> 
#> Coefficients:
#>          d         ar         ma 
#>  0.3111219  0.2376722 -0.3873700 
#> sigma[eps] = 1.015861 
#> a list with components:
#>  [1] "log.likelihood"  "n"               "msg"             "d"              
#>  [5] "ar"              "ma"              "covariance.dpq"  "fnormMin"       
#>  [9] "sigma"           "stderror.dpq"    "correlation.dpq" "h"              
#> [13] "d.tol"           "M"               "hessian.dpq"     "length.w"       
#> [17] "residuals"       "fitted"          "call"           
## Confidence intervals
confint(fd.)
#>         2.5 %     97.5 %
#> d   0.2788698  0.3433740
#> ar  0.1726554  0.3026889
#> ma -0.4335520 -0.3411879

## with iteration output
fd2 <- fracdiff(ts.test$series, nar = 1, nma = 1, trace = 1)
all.equal(fd., fd2)
#> [1] "Component “call”: target, current do not match when deparsed"