3. Semiparametric Estimators in the Spectral Domain

These estimators are based on the behaviour of the spectrum of a long-memory time series near the zero frequency, and are estimated in the frequency band $(0,m]$, where $m$ is a bandwidth parameter less than or equal to $[n/2]$, where $[.]$ denotes the integer part operator. The idea is that the range of frequencies between zero and $m$ captures the long term component, whilst the remainder of the frequencies capture the local variations which could be linear or nonlinear. These estimators are denoted semiparametric in the sense that they depend on a bandwidth parameter $m$.

3.1 Log-periodogram Regression

Under the assumption of normality, Geweke, and Porter-Hudak (1983) assumed that the spectrum $f(\lambda)$ near the zero frequency can be approximated by

$$f(\lambda) = C\{(4\sin^2(\lambda_j/2)\}^{-d}$$ (9)

and then propose to estimate the long-memory parameter $d$ with the following spectral regression:

$$\log\{I(\lambda_j)\} = c - d \log\{4\sin^2(\lambda_j/2)\}+ \varepsilon_j$$ (10)

where $n$ is the sample size. We consider for this estimator only harmonic frequencies $\lambda_j$, with $j \in (l,m]$, where $l$ is a trimming parameter discarding the lowest frequencies and $m$ is a bandwidth parameter.

  library("times")

  y = read("dmus58.dat")

  ar = abs(tdiff(y[1:2000]))

  d = gph(ar)

  d

longmem07.xpl

We obtain the following output:

  Contents of d



  [1,]  0.088369

3.1.1 Average periodogram estimator

The Robinson (1994b) averaged periodogram estimator is defined by:

$$\hat{d} = \frac{1}{2} - \frac{\ln \left( \hat{F}(q\lambda_m)/\hat{F}(\lambda_m) \right )}{2 \ln(q)},$$ (11)

where $\hat{F}(\lambda)$ is the average periodogram

$$\hat{F}(\lambda) = \frac{2 \pi}{n} \sum_{j=1}^{[n \lambda/2\pi]} I(\lambda_j).$$ (12)

By construction, the estimated parameter $\hat{d}$ is $< 1/2$, i.e., is in the stationarity range. This estimator has the following asymptotic distribution if $d < 1/4$

$$\sqrt{m} (\hat{d}-d) \sim \textrm{N}\left(0,\frac{\pi^2}{24}\right)$$ (13)

We evaluate the degree of long-memory with this estimator as follows:

  library("times")

  y = read("dmus58.dat")

  ar = abs(tdiff(y[1:2000]))

  d = roblm(ar)

  d

longmem08.xpl

We obtain the following output:

  Contents of d



  [1,] "     d     Bandwidth     q   "

  [2,] "______________________________"

  [3,] ""

  [4,] "  0.0927      500       0.5"

  [5,] "  0.1019      250       0.5"

  [6,] "  0.1199      125       0.5"

3.2 Semiparametric Gaussian Estimator

The Robinson (1995a) semiparametric estimator, suggested by Künsch (1987), is based on the approximation (2) of the spectrum of a long-memory process in the Whittle approximate maximum likelihood estimator. An estimator of the fractional degree of integration $d$ is obtained by solving the minimization problem:

$$\{\hat{C},\hat{d} \} = \textrm{arg}\min_{C,d} L(C,d) =\frac{1}{m} \sum_{j=1}^m... ...eft(C \lambda_j^{-2d}\right) +\frac{I(\lambda_j)}{C \lambda_j^{-2d}} \right \},$$ (14)

where $I(\lambda_j)$ is evaluated for a range of harmonic frequencies $\lambda_j = 2 \pi j/n$, $j=1,\ldots,m \ll [n/2]$ bounded by the bandwidth parameter $m$, which increases with the sample size $n$ but more slowly: the bandwidth $m$ must satisfy

$$\frac{1}{m} + \frac{m}{n} \rightarrow 0 \quad \textrm{as} \quad n\rightarrow \infty.$$ (15)

If $m=n/2$, this estimator is a Gaussian estimator for the parametric model $f(\lambda)= C \lambda^{-2d}$. After eliminating $C$, the estimator $\hat{d}$ is equal to:

$$\hat{d} = \textrm{arg} \min_{d} \left\{ \ln \left( \frac{1}{m}\sum_{j=1}^m \fr... ...a_j)}{\lambda_j^{-2d}} \right) -\frac{2d}{m} \sum_{j=1} \ln(\lambda_j)\right\}.$$ (16)

Although this Gaussian estimator has no closed form, it is more efficient than the averaged periodogram estimator since

$$\sqrt{m} (\hat{d}-d) \sim \textrm{N}\left(0,\frac{1}{4}\right).$$ (17)

Furthermore, Velasco (1998) has considered the nonstationary case, i.e., where $d \ge 0.5$, and has shown that, with tapered data, this estimator is consistent for $d \in (-1/2,1)$ and asymptotically normal for $d \in (-1/2,3/4)$, i.e., the statistical properties are robust to nonstationary but nonexplosive alternatives.

The quantlet robwhittle computes this local Whittle estimator. Its syntax is:

  d = robwhittle(ary{, m})

where

ary

is the series

m

is a vector of bandwidth parameters. If this optional argument is not provided, the default bandwith vector m = $[T/4]$, $[T/8]$, $[T/16]$, where $T$ denotes the sample size.

The results are displayed in the form of a table, the first column contains the value of the bandwidth parameter, while the second column contains the estimated value of $d$.

The instructions

  library("times")

  y = read("dmus58.dat")

  ar = abs(tdiff(y[1:2000]))

  d = robwhittle(ar)

  d

longmem09.xpl

yield the following table

  Contents of d



  [1,] "     d     Bandwidth"

  [2,] "_____________________"

  [3,] ""

  [4,] "  0.0948      500"

  [5,] "  0.1078      250"

  [6,] "  0.1188      125"

	MD*TECH Method and Data Technologies
	http://www.mdtech.de  mdtech@mdtech.de