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.

     

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

     

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

Bartlett: KERNEL BART

     
     

Parzen: KERNEL PARZEN

     
     

Quadratic spectral: KERNEL QS

     
     

Tukey-Hanning: KERNEL TUKEY

     
     

Truncated: KERNEL TRUNCAT

     
     

A summary of the default values of the bandwidth parameters, c and e, associated with the kernel smoothers in PROC SPECTRA are listed below in Table 26.2:

Table 26.2 Bandwidth Parameters

Kernel

c

e

Bartlett

Parzen

quadratic

Tukey-Hanning

truncated

Figure 26.1 Kernels for Smoothing
Kernels for Smoothing

See Andrews (1991) for details about the properties of these kernels.