L44 Numerical Examples and Further

Smoothing the Periodogram
- To reduce variance and obtain a consistent estimator
- Average the periodogram by averaging across frequencies
Kernel Smoothing
- Apply weighted moving average using (kernel function $W$)
- $\hat{S}(\omega) = \sum_{k=-K}^{K} W(k)I(\omega+k \Delta \omega)$, its a moving average before and after $\omega$
- Common Kernels
  - Barlett kernel: Triangular weights
  - Dianiell kernel: Simple moving average
  - Parzen kernel: Smoother weights for gradual tapering
Averaging over bands
- Divide frequency range into bands (baskets) and average periodogram within each band
Welch's method
- Divide TS into overlapping segments
- Apply a windowing function (e.g. Hamming, Hann) to each segment to reduce edge effects
- Compute periodogram for each segment.
- Average them $\hat{S}(\omega) = \dfrac{1}{N}\sum_{k=1}^{N}I_{k}(\omega)$

Method	Advantages	Disadvantages
Parametric	Smooth, interpretable. Efficient for stationary models	Requires correct specification
Periodogram	Simple, high resolution	Noisy, inconsistent
Smoothed Periodogram	Reduces noise, improves consistency	May lose frequency resolution
Welch's Method	Robust to noise, reduces variance	Reduced frequency resolution due to segmentation

Choice of windowing/kernel
- Good window = reduce spectral leakage, & maintain frequency resolution
- Popular choices
  - Rectangular: High res, but leakage
  - Hamming/Hann: Reduce leakage with moderate resolution loss
Choice of bandwidth
- Smoothing bandwidth (trade-off between variance and resolution)
- should not be very wide or narrow
- Wide → less variance, more smoothing
- Narrow → more variance, higher resolution
Handling non-stationary
- For such situations, use time -frequency methods
  - Wavelet Transform
  - Short-Time Fourier Transform (STFT)
Sampling Rage
- Sampling frequency $f_{s}$ satisfies Nyquist criterion:
  - If $f_{s} > 2 f_{\max}$ then we are good to go (maximum frequency $f_{\max}$)

\[ Y_{t} = \sin(2\pi f_{1}t) + 0.5 \sin(2\pi f_{2}t) + \epsilon_{t} \]

Task

Generate the time series… let sampling rate be $\Delta t = 1$ and total time series length $T = 20$. $t = 0,1,2,\dots,19$

The TS is,

\[ Y_{t} = \sin(2\pi \times 0.1)t + 0.5 \sin(2\pi \times 0.3t) + \epsilon_{t} \]

\[ Y(\omega_{k}) = \sum_{t=0}^{T-1} Y_{t}e^{ -i\omega_{k}t }, \quad \omega_{k} = \dfrac{2\pi k}{T},\ k=0,1,\dots,T-1 \]

\[ I(\omega_{k}) = \dfrac{1}{T}|Y(\omega_{k})|^{2} \]

Frequencies of interest.
- For $T=20$ are $f_{k} = \dfrac{k}{T}$
- Corresponding to $f_{k} =[0,0.05,0.1,0.15,\dots,0.5] Hz$
Compute the periodogram $I(f_{k})$ for each $f_{k}$

Computing periodogram for $f_{1}=0.1 Hz$ and $f_{2}=0.3Hz$
At $f_{1}$, $Y(0.1) = \sum_{t=0}^{19}Y_{t}e^{ -i\times 2\pi \times 0.1t }$
At $f_{2}$, $Y(0.3) = \sum_{t=0}^{19}Y_{t}e^{ -i\times 2\pi \times 0.3t }$
Power spectrum: $I(0.1) = \dfrac{1}{20}|Y(0.1)|^{2}$, $I(0.3) = \dfrac{1}{20}|Y(0.3)|^{2}$

Let there be two stationary TS with mean 0
Ask two questions
- Are periodicities related to each other?
- If so, what's the phase relationship between them?

Let $\gamma_{k}^{xy} = Cov(x_{t},y_{t-k})$ be the cross covariance

$$
f_{xy}(\omega) = \sum_{k=-\infty}^{\infty} e^{ -ik\omega } \gamma_{k}^{xy}
$$
- Compute the spectrum of a sum, $z_{t} = z_{t} + y_{t}$

\[ f_{z}(\omega) = \sum_{k=-\infty}^{\infty} e^{ -ik\omega }E(z_{t}z_{t-k}) \]

Thus,

\[ f_{z}(\omega) = f_{x}(\omega) + f_{xy}(\omega) + f_{yx}(\omega) + f_{y}(\omega) \]

and if $x_{t}$ and $y_{t}$ are uncorrelated

\[ f_{z}(\omega) = f_{x}(\omega) + f_{y}(\omega) \]

Identifying leading and lagging relationships between GDP growth and stock market returns
Portfolio diversification
Forex and commodities (hedging strategies)