SSMSPECTRUM Procedure

@SSMSPECTRUM returns the estimated multivariate spectrum from a state space model given by the A, F and SW options. It returns in its argument a series of complex matrices.

@SSMSpectrum(A=A matrix,SW=SW matrix,other options) spectrum

Parameters

spectrum

SERIES[CMATRIX] with the estimated spectral density. 0 frequency will be in entry 1; in general, frequency $2\pi(t-1)/\text{ORDINATES}$ will be in entry $t$. In order to avoid problems with non-stationary models, the 0 frequency isn't calculated.

Options

A=transition matrix for the state space model

F=mapping from fundamental shocks to shocks to the states [not used]

SW=covariance matrix of the fundamental shocks.

ORDINATES=# of frequency ordinates on which calculations are based [512].

This will return values only out to frequency $\pi$, thus to entries ORDINATES/2+1.

COMPONENTS=VECT[INT] of components (numbered between 1,...,# of states)

which are actually wanted. For instance, COMPONENTS=||1|| will return the estimate just for the first component.

Example

This compute the implied spectrum from a state-space model with a 2 element state vector, where only one is of direct interest (the other is an augmenting state). It's computed over 256 ordinates, and graphed over the first half of those (since it will be symmetrical).

@SSMSpectrum(a=af(1),sw=sigsqkappa*%identity(2),ordinates=256,$

components=||1||) ssmspec

set ssmreal 1 128 = %real(ssmspec(t)(1,1))

set frequencies 1 128 = (t-1.0)/128

scatter(style=lines,header="Model Spectrum")

# frequencies ssmreal 2 128