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Estimation under ARFIMA

Estimating parameters for an ARFIMA model is more complex than standard ARIMA because of the fractional differencing parameter \(d\). Most techniques rely on understanding the process in the frequency domain rather than the time domain.

1. Spectral Density of a Time Series

Spectral density provides a way to understand how the variation in data is distributed across different frequencies. Instead of looking at a sequence of points in time, we look at the series as a combination of sine and cosine waves.

  • Time Series: The raw, observed sequence of data.
  • Frequency: How often a specific cycle repeats.
  • Spectral Density (\(S(f)\)): Measures "how much" of the variance happens at different frequencies, revealing which cycles are dominant.

The Spectral Density Formula

The theoretical spectral density is defined as:
$\(S(f) = \lim_{ T \to \infty } \dfrac{1}{T} \left\lvert \sum_{t=0}^{T-1} y_{t}e^{ -i2\pi ft } \right\rvert ^{2}\)$

In practice, we use the Periodogram (\(I(f)\)) as an estimator of the spectral density for a finite series of length \(T\).

2. Estimation under the ARFIMA Model

The primary goal in ARFIMA estimation is to isolate and estimate \(d\), the fractional differencing parameter, before calculating the standard AR (\(\phi\)) and MA (\(\theta\)) parameters.

Geweke and Porter-Hudak (GPH) Estimation

The GPH method is a popular semi-parametric approach that uses the log-periodogram of the time series.

  1. Calculate the periodogram \(I(\lambda_{k})\) at frequencies \(\lambda_{k} = 2\pi f\).
  2. Perform a simple linear regression:
    $\(\log(I(\lambda_{k})) = a + b\log(\lambda_{k}) + \epsilon_{k}\)$
  3. The slope \(b\) is directly related to the fractional parameter \(d\) by the relationship:
    $\(b = -2d \implies \hat{d} = -\dfrac{b}{2}\)$

Pros and Cons of GPH:
* Advantages: It is highly robust to noise and applicable to a wide range of long-memory processes.
* Limitations: It is sensitive to the choice of bandwidth (the range of frequencies used), which can lead to biased estimates. It also assumes the series is stationary after the fractional differencing.

3. Other Estimation Techniques

  • Maximum Likelihood Estimation (MLE): An exact but computationally intensive method that maximizes the likelihood function in either the time or frequency domain.
  • Whittle Estimation: A popular frequency-domain approach that uses a quasi-likelihood function. It is generally more computationally efficient than exact MLE.
  • Local Whittle Estimation: A variation that focuses only on a local range of frequencies near zero, making it more robust to short-run dynamics.
  • Yule-Walker (Method of Moments): Estimates \(d\) using the sample Autocorrelation Function (ACF) and then solves for AR parameters using the standard Yule-Walker equations.