waveslim (version 1.12)

fdp.mle: Wavelet-based Maximum Likelihood Estimation for a Fractional Difference Process

Description

Parameter estimation for a fractional difference (long-memory, self-similar) process is performed via maximum likelihood on the wavelet coefficients.

Usage

fdp.mle(y, wf, J=log(length(y),2))

Arguments

y

Dyadic length time series.

wf

Name of the wavelet filter to use in the decomposition. See wave.filter for those wavelet filters available.

J

Depth of the discrete wavelet transform.

Value

• List containing the maximum likelihood estimates (MLEs) of $d$ and $\sigma^2$, along with the value of the likelihood for those estimates.

Details

The variance-covariance matrix of the original time series is approximated by its wavelet-based equivalent. A Whittle-type likelihood is then constructed where the sums of squared wavelet coefficients are compared to bandpass filtered version of the true spectrum. Minimization occurs only for the fractional difference parameter $d$, while variance is estimated afterwards.

References

M. J. Jensen (2000) An alternative maximum likelihood estimator of long-memory processes using compactly supported wavelets, Journal of Economic Dynamics and Control, 24, No. 3, 361-387.

McCoy, E. J., and A. T. Walden (1996) Wavelet analysis and synthesis of stationary long-memory processes, Journal for Computational and Graphical Statistics, 5, No. 1, 26-56.

Percival, D. B. and A. T. Walden (2000) Wavelet Methods for Time Series Analysis, Cambridge University Press.

Examples

## Figure 5.5 in Gencay, Selcuk and Whitcher (2001)
fdp.sdf <- function(freq, d, sigma2=1)
  sigma2 / ((2*sin(pi * freq))^2)^d
dB <- function(x) 10 * log10(x)
per <- function(z) {
  n <- length(z)
  (Mod(fft(z))**2/(2*pi*n))[1:(n %/% 2 + 1)]
}
data(ibm)     
ibm.returns <- diff(log(ibm))
ibm.volatility <- abs(ibm.returns)
ibm.vol.mle <- fdp.mle(ibm.volatility, "d4", 4)
freq <- 0:184/368
ibm.vol.per <- 2 * pi * per(ibm.volatility)
ibm.vol.resid <- ibm.vol.per/ fdp.sdf(freq, ibm.vol.mle$parameters[1])
par(mfrow=c(1,1), las=0, pty="m")
plot(freq, dB(ibm.vol.per), type="l", xlab="Frequency", ylab="Spectrum")
lines(freq, dB(fdp.sdf(freq, ibm.vol.mle$parameters[1],
                       ibm.vol.mle$parameters[2]/2)), col=2)