Download Program GAP∗- Version 2.2 User Manual May 2003

where Z·t is a vector of (N ex1 × 1) exogenous variables, with N ex1 less or equal to
3. The trend can be specified as a second order random walk:
N
(1 − L) X1t
= µ1t + aN t .
(2)
where L is the lag operator. The slope is such that
(1 − L) µ1t = aµt .
(3)
Both aN t , and aµt , are niid-innovations with variances VN and Vµ . Notice that if
V ar (aµt ) = 0, then (2) reduces to a random walk plus drift. The short-term component is such that:
³
´
S
1 − φ1 L − φ2 L2 X1t
= aSt .
(4)
The variables aSt are niid-innovations with variance VS .
Equation 2: in agreement with the Phillips curve framework, series 1 is related
to a second series according to:
X2t = µ2 +
NX
ex2
α2i Yit + γ (1 − L) X1t−1 +
i=1
+φ∗1 X2t−1
+
φ∗2 X2t−2
³
4
X
i=0
S
βi X1t−i
´
+ 1 + θ1 L + θ2 L2 + θ3 L3 aπt
(5)
where Y·t is a vector of (N ex2×1) exogenous variables, with N ex2 less or equal to 10,
and aπt is a nid-innovation with variance Vπ that can be correlated with aSt . That
correlation is denoted by ρ.
Warnings:
1. the second series and the regressors on the second equation should be covariance
stationary;
S
2. if X1t
is included in the Phillips curve equation (2nd equation), the innovations
aπt and aSt must be uncorrelated, so the program imposes ρ = 0 when β0 6= 0.
S
in the second
Without that restriction, the innovation aπt and the regressor X1t
equation would not be orthogonal and the corresponding estimator would not
be consistent.
4