Section \[sec:5\] on numerical examples, first simulating some ARFIMA process with a time-varying $d$ and comparing the performance of the two-sided (rectangular) kernel with the recursive weight scheme. Second, we apply the kernel estimator to a series of realized log volatilities (see also [@bauwens:wang:2008]), namely of the exchange rate of the YEN versus USD, from June 1986 to September 2004. We conclude in Section \[sec:6\] before an appendix section presents all technical details of our derivations including our proofs.

Model set-up and examples {#sec:2}
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Define the difference operator $[\diffop X]_k = X_k - X_{k-1}$ and $\diffop^p$ recursively. This will allow $d(u)$ to take values up to $p+1/2$ in the following model.

We adapt the approach of [@dahlhaus96] to the case where the spectral density is allowed to have a singularity at the zero frequency. Let us fix $p=0,1,2,\dots$ and $A^0_{t,T}(\lambda)$ be an array of $L^2([-\pi,\pi])$ functions with real-valued Fourier coefficients. Let $\{X_{t,T}\}$ be an array of real-valued random variables such that $$\label{eq:LocStatProcSpectralExt}
\diffop^p X_{t,T} = 
\int_{-\pi}^\pi A^0_{t,T}(\lambda)\ \rme^{\rmi\lambda t}\;\rmd Z(\lambda)\ , \qquad t=1,\ldots T, \ T \geq 1\ ,$$ where $\rmd Z(\lambda)$ is the spectral representation of a centered weak white noise with unit variance, $$\label{eq:epsZ}
  \varepsilon_t= \int_{-\pi}^\pi\rme^{\rmi\lambda t}\;\rmd 