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model (V ECM(p
1)) is:
y t = 1 y t 1 + : : : + p 1 y t p+1 + 0 y t p + t ; (3.1)
where the variance-covariance matrix of the error terms E ( t 0 t) is not necessarily diagonal, and
independence among shocks to consumption is not imposed.
We turn now to the discussion of how to extract trends and cycles using (3.1).
To simplify
notation, we jump straight to our empirical results, where we found that the system (3.1) is well
described by a V ECM(1). It can be put in state-space form, as discussed in Proietti (1997):
where,
f t+1 =
2
6
4
with the associated VECM being,
y t+1 = Zf t+1 (3.2)
f t+1 = T f t + Z 0 t+1 ;
3
y t+1
y t
0 y t 1
7
5 ; T =
2
6
4
1 0
I 2 0 0
0 0 1
y t = 1 y t 1 + 0 y t 2 + t ; and,
h i
Z = I 2 0 0 :
From the work of Beveridge and Nelson (1981), and Stock and Watson (1988), and ignoring
initial conditions and deterministic components, the series in y t can be decomposed into a trend
( t ) and a cyclical component (' t ), as follows:
where,
t = y t + lim
' t =
y t = t + ' t ;
lX
l!1
i=1
lX
lim
l!1
i=1
E t [y t+i ] ; and,
E t [y t+i ] :
It is straightforward to show that t is a martingale. Using the state-space representation (3.2),
we can compute the limits above. The cyclical and trend components will be, respectively:
' t = Z [I m T ]
1 T f t ;
t = y t ' t : (3.3)
3
7
5
16