Kernels

The SPECTRA Procedure

Kernels

Kernels are used to smooth the periodogram by using a weighted moving average of nearby points. A smoothed periodogram is defined by the following equation.

$\hat{J}_{i}(\rm{l}(q)) = \sum_{{\tau} = -\rm{l}(q)}^{\rm{l}(q)}{\rm{w}( \frac{{\tau}}{\rm{l}(q)} ) \tilde{J}_{i+{\tau}}}$

where w(x) is the kernel or weight function. At the endpoints, the moving average is computed cyclically; that is,

$\tilde{J}_{i+{\tau}} = \cases{ J_{i+{\tau}} & 0 \lt= i+{\tau}\space \lt= q \c... ...)} & i+{\tau}\space \lt 0 \cr J_{q-(i+{\tau})} & i+{\tau}\space \gt q \cr }$

The SPECTRA procedure supports the following kernels. They are listed with their default bandwidth functions.

Bartlett: KERNEL BART

$\rm{w}(x) &=& \cases{ 1-{| x|} & {| x|}{\le}1\space \cr 0 & \rm{otherwise} \cr } \cr \rm{l}(q) &=& \frac{1}2 q^{1 / 3}$

Parzen: KERNEL PARZEN

$\rm{w}(x) &=& \cases{ 1-6{| x|}^2 + 6{| x|}^3 & {0{\le}{| x|}{\le}\frac{... ...{\le}1}\space \cr 0 & \rm{otherwise} } \cr \rm{l}(q) &=& q^{1 / 5}$

Quadratic Spectral: KERNEL QS

$\rm{w}(x) &=& \frac{25}{12{\pi}^2 x^2} ( \frac{{sin}(6{\pi}x/5)}{6{\pi}x/5} - {cos}(6{\pi}x/5) ) \cr \rm{l}(q) &=& \frac{1}2 q^{1 / 5}$

Tukey-Hanning: KERNEL TUKEY

$\rm{w}(x) &=& \cases{ (1+{cos}( {\pi} x))/2 & {| x|}{\le}1\space \cr 0 & \rm{otherwise} \cr } \cr \rm{l}(q) &=& \frac{2}3 q^{1 / 5}$

Truncated: KERNEL TRUNCAT

$\rm{w}(x) &=& \cases{ 1 & {| x|}{\le}1\space \cr 0 & \rm{otherwise} \cr } \cr \rm{l}(q) &=& \frac{1}4 q^{1 / 5}$

Figure 17.1: Kernels for Smoothing

Refer to Andrews (1991) for details on the properties of these kernels.

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