The TIMESERIES Procedure

SPECTRA Statement

Subsections:

Kernel Option Details

SPECTRA statistics < / options> ;

A SPECTRA statement can be used with the TIMESERIES procedure to specify which statistics appear in the OUTSPECTRA= data set. The SPECTRA statement options are used in performing a spectral analysis on the variables listed in the VAR statement. These options affect values that are produced in the PROC TIMESERIES statement’s OUTSPECTRA= data set, and in the periodogram and spectral density estimate. Only one SPECTRA statement is allowed.

The following univariate frequency domain statistics are available:

FREQ: frequency in radians from 0 to $\pi$
PERIOD: period or wavelength
COS: cosine transform
SIN: sine transform
P: periodogram
S: spectral density estimates

If none of the frequency domain statistics are specified, the default is as follows:

   spectra period p;

The following options can be specified in the SPECTRA statement following the slash (/):

C=coefficient

specifies the scale coefficient for the kernel function. See the section Kernel Option Details for more information.

E=exponent EXP=exponent EXPON=exponent

specifies the exponent for the kernel function. See the section Kernel Option Details for more information.

ADJUSTMEAN=NO | YES CENTER=NO | YES

specifies whether or not the series should be adjusted by its mean prior to performing the Fourier decomposition. This sets the first periodogram ordinate to 0 rather than to $2 n$ times the squared mean. This option is commonly used when the periodograms are to be plotted to prevent a large first periodogram ordinate from distorting the scale of the plot. ADJUSTMEAN=YES specifies that the series be transformed by subtracting its mean. ADJUSTMEAN=NO specifies that no adjustment of the series be performed. The default is ADJUSTMEAN=NO.

ALPHA=num

specifies the width of a window that is drawn around the spectral density estimate in a spectral density versus frequency plot. Based on approximations proposed by Brockwell and Davis (1991), periodogram ordinates fall within this window with a confidence level of $1-\Argument{num}$ . The value $\Argument{num}$ must be between 0 and 1; the default is 0.5.

DOMAIN=domain

specifies how the smoothing function is interpreted. You can specify the following domain values:

FREQUENCY: smooths the periodogram ordinates.
TIME: applies the kernel as a filter to the time series autocovariance function.

By default DOMAIN=FREQUENCY, and smoothing is applied in the same manner as weights are applied when you specify the WEIGHTS= option.

kernel

specifies the smoothing function to use to calculate a spectral density estimate as the moving average of periodogram ordinates. The kernel function is an alternative smoothing method to using the WEIGHTS= option. You can specify the following kernel values:

PARZEN: Parzen kernel
BARTLETT: Bartlett kernel
TUKEY: Tukey-Hanning kernel
TRUNC | TRUNCAT: truncated kernel
QS | QUADR: quadratic spectral kernel

If neither a WEIGHTS= option nor a kernel function is specified, the spectral density estimate is identical to the unmodified periodogram.

WEIGHTS=numlist

specifies the relative weights to use to compute a spectral density estimate as the moving average smoothing of periodogram ordinates. If neither a WEIGHTS= option nor a kernel function is specified, the spectral density estimate is identical to the unmodified periodogram. The following SPECTRA statement uses the WEIGHTS= option to specify equal weighting for each of the three adjacent periodogram ordinates centered on each spectral density estimate:

   spectra / weights 1 1 1;

For information about how the weights are applied, see the section Using Specification of Weight Constants.

Kernel Option Details

You can further parameterize each of the kernel functions with a kernel scale factor by using the C= and E= options. The default values of the kernel scale parameters, $c$ and $e$ , that are associated with each of the kernel functions together with their kernel scale factor values, $M$ , for a series with 100 periodogram ordinates are listed in Table 32.2. The formula that is used to generate the table entries is $M=c{K}^{e}$ , where $K$ is the number of Fourier component frequencies.

Table 32.2: Default Kernel Scale Factor Parameters

Kernel	$c$	$e$	$M$
Bartlett	$1/2$	$1/3$	2.32
Parzen	$1$	$1/5$	2.51
Quadratic	$1/2$	$1/5$	1.26
Tukey-Hanning	$2/3$	$1/5$	1.67
Truncated	$1/4$	$1/5$	0.63

For example, to apply the truncated kernel by using default scale factor parameters in the frequency domain, you could use the following SPECTRA statement:

   spectra / truncat;

For more information about the kernel function parameterization and the DOMAIN= option, see the section Using Kernel Specifications.