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L42 Spectral Representation of a series

Spectral Representation Theorem (SRT)

Theorem

Any stationary time series can be expressed as a combination of sinusoidal functions of different frequencies, each with its own amplitude and phase

\[ y_{t} = \int_{-\infty}^{\infty} e^{ i \omega t } Z(d\omega) \]
  • \(y_{t}\) → stationary time series
  • \(\omega\) → angular frequency
  • \(Z(d\omega)\) → complex valued stochastic process which determines the contribution of frequency \(\omega\) to \(y_{t}\)

Definition of SDF

\(S(\omega)\) describes how the variance of the TS is distributed across different frequencies.

  • SDF provides insight into periodic components
  • widely used in signal processing, finance and other fields

For a weakly stationary time series \(y_{t}\), SDF is the Fourier transform of the ACF \(\gamma(h)\)

\[ S(\omega) = \dfrac{1}{2\pi}\sum_{h=-\infty}^{\infty} \gamma(h) e^{ -i \omega h } \]
  • \(\gamma(h)= E(y_{t}y_{t+h})\)

Properties of SDF

  • For a real valued TS, \(S(\omega)\) is symmetric.
\[ S(\omega) = S(-\omega) \]
  • Non-negativity: \(S(\omega) \geq 0\) for all \(\omega\)
  • Total variance: Total variance of TS
\[ Var(y_{t}) = \sum_{h=-\infty}^{\infty} \gamma(h) = \int_{-\pi}^{\pi} S(\omega) d\omega \]
  • Periodicity is \(2\pi\)
    • \(S(\omega)\) repeats itself every \(2\pi\) length
  • Inverse relationship exists between ACF \(\gamma(h) \cap mkS(\omega)\)
\[ \gamma(h) = \int_{-\pi}^{\pi} S(\omega) e^{ i \omega h } d\omega \]

Interpretation

  • Low frequencies (\(\omega \approx 0\)) correspond to long-term trends (slow-moving components in the TS)
  • High frequency (\(\omega \approx \pi\)) → Noise (rapid fluctuations)
  • Peaks in \(S(\omega)\) indicate dominant period components at specific frequencies.1

Examples

White Noise

  • \(y_{t} \sim N(0, \sigma^{2})\)
  • ACF: \(\gamma(h) = \begin{cases} \sigma^2 & h=0 \\ 0 & \text{ow} \end{cases}\)
  • SDF: \(S(\omega)= \dfrac{\sigma^{2}}{2\pi}\) \(\implies\) constant for all \(\omega\). So, the white noise has equal power across all frequencies.

AR(1) process

  • \(y_{t} = \phi y_{t-1} + \epsilon_{t}\), where \(|\phi| \lt 1\), \(e_{t} \sim N(0, \sigma^{2})\)
  • ACF: \(\gamma(h) = \dfrac{\sigma^{2}}{1-\phi^{2}}\phi^{|h|}\)
  • SDF: \(S(\omega) = \dfrac{\sigma^{2}}{2\pi} \dfrac{1}{ | 1 - \phi e^{ -i\omega }|^{2}}\)
  • Low frequencies dominate for \(\phi\) close to 1 \(\implies\) slow moving behavior.

MA(q) process

  • limited to frequencies below a certain level, determined by \(q\)
  • Smoother than an AR process of similar order since it is based on past shocks.

Random Walk

  • SDF: \(S(\omega) = \dfrac{\sigma^{2}}{2\pi} \dfrac{1}{ | 1 - \phi e^{ -i\omega }|^{2}}\)
  • The SDF diverges as \(\omega\to 0\) \(\implies\) non-stationary process dominated by low frequencies.

What would happen to \(S(\omega)\) for different values of \(h\)

  1. We can curve out the path of an SDF. There will be peaks in specific points in the SDF. Those peaks represent the dominant periodic components ↩