11-FinEcon-Time Series Page 307 Wednesday, February 4, 2004 12:58 PM
Financial Econometrics: Time Series Concepts, Representations, and Models 307Xt + 1 = α 0 Xt + α 1 Yt + εt + 1Yt + 1 = XtThis transformation can be applied to systems of any order and with
any number of equations. Recall from Chapter 9 that a similar proce-
dures is applied to systems of differential equations.
Note that this state-space representation is not restricted to white
noise inputs. A state-space representation is a mapping of inputs into
outputs. Given a realization of the inputs ut and an initial state z 0 , the
realization of the outputs xt is fixed. The state-space representation can
be seen as a black-box, characterized by A, B, C, D, and z 0 that maps
any m-dimensional input sequence into an n-dimensional output
sequence. The mapping S = S(A,B,C,D,z 0 ) of u →x is called a black-box
representation in system theory.
State-space representations are not unique. Given a state-space rep-
resentation, there are infinite other state-space representations that
implement the same mapping u →x. In fact, given any nonsingular
(invertible) matrix Q, it can be easily verified thatS(ABCDz,,, , 0 ) = S(QAQ , DQz 0 )- 1
,QB CQ- 1
, ,
- 1
Any two representations that satisfy the above condition are called
equivalent.
The minimal size of a system that admits a state-space representa-
tion is the minimum possible size k of the state vector. A representation
is called minimal if its state vector has size k.
We can now establish the connection between state-space and infi-
nite moving-average representations and the equivalence of ARMA and
state-space representations. Consider a n-dimensional process xt, which
admits an infinite moving-average representation∞xt = ∑Hiεεεεti–
i = 0where εεεεt is an n-dimensional, zero-mean, white noise process with non-
singular variance-covariance matrix ΩΩΩΩand H 0 = I, or a linear moving
average model