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.

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:

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