Package 'fracdiff'

Confidence Intervals for Fracdiff Model Parameters

Description

Computes (Wald) confidence intervals for one or more parameters in a fitted fracdiff model, see fracdiff.

Usage

## S3 method for class 'fracdiff'
confint(object, parm, level = 0.95, ...)

Arguments

object	an object of class fracdiff, typically result of fracdiff(..).
parm	a specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.
level	the confidence level required.
...	additional argument(s) for methods.

Value

A matrix (or vector) with columns giving lower and upper confidence limits for each parameter. These will be labelled as (1-level)/2 and 1 - (1-level)/2 in % (by default 2.5% and 97.5%).

Warning

As these confidence intervals use the standard errors returned by fracdiff() (which are based on finite difference approximations to the Hessian) they may end up being much too narrow, see the example in fracdiff.var.

Author(s)

Spencer Graves posted the initial version to R-help.

See Also

the generic confint; fracdiff model fitting, notably fracdiff.var() for re-estimating the variance-covariance matrix on which confint() builds entirely.

Examples

set.seed(101)
ts2 <- fracdiff.sim(5000, ar = .2, ma = -.4, d = .3)
mFD <- fracdiff( ts2$series, nar = length(ts2$ar), nma = length(ts2$ma))
coef(mFD)
confint(mFD)

---

Fractionally Differenciate Data

Description

Differenciates the time series data using the approximated binomial expression of the long-memory filter and an estimate of the memory parameter in the ARFIMA(p,d,q) model.

Usage

diffseries(x, d)

Arguments

x	numeric vector or univariate time series.
d	number specifiying the fractional difference order.

Details

Since 2018, we are using (an important correction of) the fast algorithm based on the discrete Fourier transform (fft) by Jensen and Nielsen which is significantly faster for large n = length(x).

Value

the fractionally differenced series x.

Author(s)

Valderio A. Reisen [email protected] and Artur J. Lemonte (first slow version), now hidden as diffseries.0().

Current version: Jensen and Nielsen (2014); tweaks by Martin Maechler, 2018.

References

See those in fdSperio; additionally

Reisen, V. A. and Lopes, S. (1999) Some simulations and applications of forecasting long-memory time series models; Journal of Statistical Planning and Inference 80, 269–287.

Reisen, V. A. Cribari-Neto, F. and Jensen, M.J. (2003) Long Memory Inflationary Dynamics. The case of Brazil. Studies in Nonlinear Dynamics and Econometrics 7(3), 1–16.

Jensen, Andreas Noack and Nielsen, Morten Ørregaard (2014) A Fast Fractional Difference Algorithm. Journal of Time Series Analysis 35(5), 428–436; doi:10.1111/jtsa.12074.

See Also

fracdiff.sim

Examples

memory.long <- fracdiff.sim(80, d = 0.3)
str(mGPH <- fdGPH(memory.long$series))
r <- diffseries(memory.long$series, d = mGPH$d)
#acf(r) # shouldn't show structure - ideally

---

Geweke and Porter-Hudak Estimator for ARFIMA(p,d,q)

Description

Estimate the fractional (or “memory”) parameter $d$ in the ARFIMA(p,d,q) model by the method of Geweke and Porter-Hudak (GPH). The GPH estimator is based on the regression equation using the periodogram function as an estimate of the spectral density.

Usage

fdGPH(x, bandw.exp = 0.5)

Arguments

x	univariate time series
bandw.exp	the bandwidth used in the regression equation

Details

The function also provides the asymptotic standard deviation and the standard error deviation of the fractional estimator.

The bandwidth is bw = trunc(n ^ bandw.exp), where 0 < bandw.exp < 1 and n is the sample size. Default bandw.exp = 0.5.

Value

d	GPH estimate
sd.as	asymptotic standard deviation
sd.reg	standard error deviation

Author(s)

Valderio A. Reisen and Artur J. Lemonte

References

see those in fdSperio.

See Also

fdSperio, fracdiff

Examples

memory.long <- fracdiff.sim(1500, d = 0.3)
fdGPH(memory.long$series)

---

Sperio Estimate for 'd' in ARFIMA(p,d,q)

Description

This function makes use Reisen (1994) estimator to estimate the memory parameter d in the ARFIMA(p,d,q) model. It is based on the regression equation using the smoothed periodogram function as an estimate of the spectral density.

Usage

fdSperio(x, bandw.exp = 0.5, beta = 0.9)

Arguments

x	univariate time series data.
bandw.exp	numeric: exponent of the bandwidth used in the regression equation.
beta	numeric: exponent of the bandwidth used in the lag Parzen window.

Details

The function also provides the asymptotic standard deviation and the standard error deviation of the fractional estimator.

The bandwidths are bw = trunc(n ^ bandw.exp), where 0 < bandw.exp < 1 and n is the sample size. Default bandw.exp= 0.5;
and bw2 = trunc(n ^ beta), where 0 < beta < 1 and n is the sample size. Default beta = 0.9.

Value

a list with components

d	Sperio estimate
sd.as	asymptotic standard deviation
sd.reg	standard error deviation

Author(s)

Valderio A. Reisen [email protected] and Artur J. Lemonte

References

Geweke, J. and Porter-Hudak, S. (1983) The estimation and application of long memory time series models. Journal of Time Series Analysis 4(4), 221–238.

Reisen, V. A. (1994) Estimation of the fractional difference parameter in the ARFIMA(p,d,q) model using the smoothed periodogram. Journal Time Series Analysis, 15(1), 335–350.

Reisen, V. A., B. Abraham, and E. M. M. Toscano (2001) Parametric and semiparametric estimations of stationary univariate ARFIMA model. Brazilian Journal of Probability and Statistics 14, 185–206.

See Also

fdGPH, fracdiff

Examples

memory.long <- fracdiff.sim(1500, d = 0.3)
spm <- fdSperio(memory.long$series)
str(spm, digits=6)

---

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)

---

Many Methods for "fracdiff" Objects

Description

Many “accessor” methods for fracdiff objects, notably summary, coef, vcov, and logLik; further print() methods were needed.

Usage

## S3 method for class 'fracdiff'
coef(object, ...)
## S3 method for class 'fracdiff'
logLik(object, ...)
## S3 method for class 'fracdiff'
print(x, digits = getOption("digits"), ...)
## S3 method for class 'fracdiff'
summary(object, symbolic.cor = FALSE, ...)
## S3 method for class 'summary.fracdiff'
print(x, digits = max(3, getOption("digits") - 3),
        correlation = FALSE, symbolic.cor = x$symbolic.cor,
        signif.stars = getOption("show.signif.stars"), ...)
## S3 method for class 'fracdiff'
fitted(object, ...)
## S3 method for class 'fracdiff'
residuals(object, ...)
## S3 method for class 'fracdiff'
vcov(object, ...)

Arguments

x, object	object of class fracdiff.
digits	the number of significant digits to use when printing.
...	further arguments passed from and to methods.
correlation	logical; if TRUE, the correlation matrix of the estimated parameters is returned and printed.
symbolic.cor	logical. If TRUE, print the correlations in a symbolic form (see symnum) rather than as numbers.
signif.stars	logical. If TRUE, “significance stars” are printed for each coefficient.

Author(s)

Martin Maechler; Rob Hyndman contributed the residuals() and fitted() methods.

See Also

fracdiff to get "fracdiff" objects, confint.fracdiff for the confint method; further, fracdiff.var.

Examples

set.seed(7)
ts4 <- fracdiff.sim(10000, ar = c(0.6, -.05, -0.2), ma = -0.4, d = 0.2)
modFD <- fracdiff( ts4$series, nar = length(ts4$ar), nma = length(ts4$ma))
## -> warning (singular Hessian) %% FIXME ???
coef(modFD) # the estimated parameters
vcov(modFD)
smFD <- summary(modFD)
smFD
coef(smFD) # gives the whole table
AIC(modFD) # AIC works because of the logLik() method
stopifnot(exprs = {

})

---

Simulate fractional ARIMA Time Series

Description

Generates simulated long-memory time series data from the fractional ARIMA(p,d,q) model. This is a test problem generator for fracdiff.

Note that the MA coefficients have inverted signs compared to other parametrizations, see the details in fracdiff.

Usage

fracdiff.sim(n, ar = NULL, ma = NULL, d,
             rand.gen = rnorm, innov = rand.gen(n+q, ...),
             n.start = NA, backComp = TRUE, allow.0.nstart = FALSE,
             start.innov = rand.gen(n.start, ...),
             ..., mu = 0)

Arguments

n	length of the time series.
ar	vector of autoregressive parameters; empty by default.
ma	vector of moving average parameters; empty by default.
d	fractional differencing parameter.
rand.gen	a function to generate the innovations; the default, rnorm generates white N(0,1) noise.
innov	an optional times series of innovations. If not provided, rand.gen() is used.
n.start	length of “burn-in” period. If NA, the default, the same value as in arima.sim is computed.
backComp	logical indicating if back compatibility with older versions of fracdiff.sim is desired. Otherwise, for d = 0, compatibility with R's arima.sim is achieved.
allow.0.nstart	logical indicating if n.start = 0 should be allowed even when $p + q > 0$. This not recommended unless for producing the same series as with older versions of fracdiff.sim.
start.innov	an optional vector of innovations to be used for the burn-in period. If supplied there must be at least n.start values.
...	additional arguments for rand.gen(). Most usefully, the standard deviation of the innovations generated by rnorm can be specified by sd.
mu	time series mean (added at the end).

Value

a list containing the following elements :

series	time series
ar, ma, d, mu, n.start	same as input

See Also

fracdiff, also for references; arima.sim

Examples

## Pretty (too) short to "see" the long memory
fracdiff.sim(100, ar = .2, ma = .4, d = .3)

## longer with "extreme" ar:
r <- fracdiff.sim(n=1500, ar=-0.9, d= 0.3)
plot(as.ts(r$series))

## Show that  MA  coefficients meaning is inverted
## compared to   stats :: arima :

AR <- 0.7
MA <- -0.5
n.st <- 2

AR <- c(0.7, -0.1)
MA <- c(-0.5, 0.4)
n <- 512 ; sd <- 0.1
n.st <- 10

set.seed(101)
Y1 <- arima.sim(list(ar = AR, ma = MA), n = n, n.start = n.st, sd = sd)
plot(Y1)

# For our fracdiff, reverse the MA sign:
set.seed(101)
Y2 <- fracdiff.sim(n = n, ar = AR, ma = - MA, d = 0,
                   n.start = n.st, sd = sd)$series
lines(Y2, col=adjustcolor("red", 0.5))
## .. no, you don't need glasses ;-)  Y2 is Y1 shifted slightly

##' rotate left by k (k < 0: rotate right)
rot <- function(x, k) {
  stopifnot(k == round(k))
  n <- length(x)
  if(k <- k %% n) x[c((k+1):n, 1:k)] else x
}
k <- n.st - 2
Y2.s <- rot(Y2, k)
head.matrix(cbind(Y1, Y2.s))
plot(Y1, Y2.s); i <- (n-k+1):n
text(Y1[i], Y2.s[i], i, adj = c(0,0)-.1, col=2)

## With  backComp = FALSE,  get *the same* as arima.sim():
set.seed(101)
Y2. <- fracdiff.sim(n = n, ar = AR, ma = - MA, d = 0,
                    n.start = n.st, sd = sd, backComp = FALSE)$series
stopifnot( all.equal( c(Y1), Y2., tolerance= 1e-15))

---

Recompute Covariance Estimate for fracdiff

Description

Allows the finite-difference interval to be altered for recomputation of the covariance estimate for fracdiff.

Usage

fracdiff.var(x, fracdiff.out, h)

Arguments

x	a univariate time series or a vector. Missing values (NAs) are not allowed.
fracdiff.out	output from fracdiff for time series x.
h	finite-difference interval length ($> 0$) for approximating partial derivatives with respect to the d parameter. Typically smaller than the one in fracdiff.out

Value

an object of S3 class "fracdiff", i.e., basically a list with the same elements as the result from fracdiff, but with possibly different values for the hessian, covariance, and correlation matrices and for standard error, as well as for h.

See Also

fracdiff, also for references.

Examples

## Generate a fractionally-differenced ARIMA(1,d,1) model :
set.seed(5) # reproducibility; x86_64 Lnx: get warning
tst <- fracdiff.sim(500, ar = .2, ma = .4, d = .3)$series
## estimate the parameters in an ARIMA(1,d,1) model for the simulated series
fd.out <- fracdiff(tst, nar= 1, nma = 1) # warning ... maybe change 'h'
summary(fd.out)## *** Warning ... {has been stored} --> h = 7.512e-6

## Modify the covariance estimate by changing the finite-difference interval
(fd.o2 <- fracdiff.var(tst, fd.out, h = 1e-3))
## looks identical as  print(fd.out),
## however these (e.g.) differ :
vcov(fd.out)
vcov(fd.o2)

## A case, were the default variance is *clearly* way too small:
set.seed(1); fdc <- fracdiff(X <- fracdiff.sim(n=100, d=0.25)$series)
fdc
# Confidence intervals just based on asymp.normal approx. and std.errors:
confint(fdc) # ridiculously too narrow

---