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L29 Estimation under ARFIMA

Spectral Density of a TS

  • Spectral density = a way to understand how the variation in data is distributed across various frequencies.

    • Super-impose a frequency → A combination of a sine-wave or cos-wave. Frequency-based plot.
    • Rather than a time-based approach

  • Time Series: The actual data which is observed

  • Frequency: How often something offers
  • Spectral density: "How much of it" happens at different frequencies
    • Also telling us which cycles are present and how strong or weak they are.

Use?

  • Helps us identify repeating patterns
  • Helps us make predictions. By knowing the heights of the cycles/repetitions
  • Model the underlying frequencies and repetitions of TS

Spectral Density Formula

\[ S(f) = \lim_{ T \to \infty } \dfrac{1}{T} \left\lvert \sum_{t=0}^{T-1} y_{t}e^{ -i2\pi ft } \right\rvert ^{2} \]
  • \(S(f)\) = spectral density
  • \(f\) = frequency
  • \(T\) = total number of observations in the series

We evaluate the spectral density at the frequency, \(f\). The exponential term takes care of the cosine and sine functions.

  • The practical estimator of the spectral density is given by the periodogram with the form
\[ I(f) = \dfrac{1}{T} \left\lvert \sum_{t=0}^{T-1} y_{t}e^{ -i2\pi ft } \right\rvert ^{2} \]

### Estimation under the ARFIMA Model

Geweke and Porter-Hudak Estimation (GPH)

Use the log periodogram of the time series.

  1. Let \(\lambda = 2\pi f\). Compute and obtain \(\log(I(\lambda_{k}))\)
  2. Regress \(\log(I(\lambda_{k}))= a + b\log(\lambda_{k}) + \epsilon_{k}\)

We have to estimate \(a\) and \(b\). Interestingly, \(b\) is related to the fractional differencing parameter as follows:

\[ b = -2d \]

Thus, we can estimate, \(\hat{d} = -\dfrac{b}{2}\).

Then we can estimate AR and MA parameters as usual.

Advantages

  1. Robustness (to noise)
  2. Applicability (can be applied to wide range of TS (say, different \(H\) values))

Limitations

  1. Sensitivity to bandwidth → Biased estimates
  2. Assumption of Stationarity → Assumes stationary after differencing (which may not always hold)

Other Estimation Techniques

  • MLE
  • Yule Walker equations (method of moments)
    • \(d\) can be estimated using sample ACF
    • AR can be obtained by YW equations
  • Whittle estimation
    • Quasi-likelihood approach to estimate parameter
  • Local Whittle estimation
    • Focuses on local frequency range

The main focus is to estimate \(d\), the fractional difference parameter.