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A time-independent state-space model is defined by the two equations
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In order to keep the state-space model reasonably simple we assume
the sequence
to be orthogonal.
The matrices
,
,
and
(sometimes they are referred to
as system matrices) were originally
(in engineering applications) assumed to be known but in econometric applications
some of them may be unknown (but we still assume they are nonstochastic).
One can easily check that the sequence has a Markovian property
if the vectors in the sequence
are independent.
This is satisfied for example in a case when
and
are Gaussian errors (the model is then said to be
Gaussian).
The state-space models include most of the well-known time series and linear models.
In the following examples state representations of Holt-Winters
method and ARMA() model are introduced.
The state-space equivalent of the Holt-Winters method has the
form
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ARMA(
) model
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More detailed information about state-space models may be found for example in Harvey (1990).
Time-invariant state-space models can be easily simulated using
the
XploRe
quantlet
kemitor
. One hundred observations of the state-space
model
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library("times") T = 100 randomize(0) ex = normal(T)~(vec(1:T).*0) ey = normal(T).*2 H = 1~0 F = #(0.5,1)~#(-0.3,0) x0 = #(0,0) ar2 = kemitor(T,x0,H,F,ey,ex)
The fact that the errors are pregenerated and supplied as parameters
of the quantlet allows us to model errors with distributions different from
the Gaussian one or error terms with some special interdependencies.
In this framework, it is for example possible to model easily time series
with different kinds of outliers.