ECONOMETRIC METHODS II TA session 1 MATLAB Intro ...

ECONOMETRIC METHODS II TA Session 1
2 Preliminaries
2.1 VAR(p) process
Consider a VAR(p) process (L,2.1.1),
yt = v + A1yt 1 + + Apyt p + ut; t = 0; 1; 2; : : : ; (2.1)
where yt; ut and v are (K 1) vectors and Ai are (K K) matrices for each i = 1; : : : ; p.
In addition, the error term ut is a white noise random vector such that E (ut) = 0,
E (utu 0 t) = u and E (utu 0 s) = 0 for s 6= t, where u is a (K K) positive de…nite matrix.
It is useful to re-write (2:1) in its companion form (L,2.1.8),
where Yt = y 0 t; y 0 t 1; : : : ; y 0 t p+1
vectors and
6
A = 6
4
Yt = v + AYt 1 + Ut (2.2)
0 , v = [v 0 ; 0 0 ; : : : ; 0 0 ] 0 and Ut = [u 0 t; 0 0 ; : : : ; 0 0 ] 0 are (Kp 1)
2
A1 A2 Ap 1 Ap
IK 0 0 0
0 IK 0 0
.. . . . .
0 0 IK 0
is a (Kp Kp) matrix. The model (2:2) is said to be stable i¤ max (j Aj) < 1, where
A denotes the vector of eigenvalues of A. The stability condition implies that the model
can be re-written as a Moving-Average (MA) representation (L,2.1.13):
yt = JYt = J + J
3
7
5
1X
A i Ut i (2.3)
where =E (Yt) = (IKp A) 1 h
i
v is the unconditional expectation and J = IK 0 0
is a (K Kp) selection matrix. Notice also that Ut = J 0 JUt and ut = JUt, thus
yt = JYt = J +
1X
i=0
i=0
JA i J 0 JUt i
where i = JA i J 0 denotes the i th matrix lag on the MA representation (L,2.1.17)
yt = JYt = J +
1X
i=0
JA i J 0 ut i (2.4)
Fernando Pérez Forero 2