QUEST II. A Multi-Country Business Cycle and Growth Model

OhWKrG 3: 7huplqdO &rqGLWlrq Lq 'LIIhuhqfhv (7&') - Define jumping variables y ¹ as
transformations of the initial jumping variables yⁱ_t with the property that yⁱ_++τ+1 is
invariant to exogenous interventions.

This method consists of exploiting properties of the steady state solution for the definition of
the jumping variables. Knowing that the model reaches a steady state implies a certain
knowledge about the change of variables between two successive periods. If the system is
formulated in efficiency units, for example, then we know that in the steady state, the
percentage change of y'! is equal to zero for any shock and any steady state reached by the
model solution¹⁴. If we define a new vector of jumping variables y¹ = yj - y¹_-1, then
y+_+τ+1 = 0 if we choose T large enough such that the model reaches a steady state in period
t+T. In terms of the example given above, this amounts to the following specification.

У- = $ У --ι + %yⁱ_f + Yι[_r

^y = Iy tι + I^y'ⁱ_t

(^y L = &y L ⁺ ( ' ^{- i} )y^⁺ γ 2 [_f

Notice that in this system of equations, the variable y ⁱ_j+1 is the jumping variable, while yⁱ_r
has now become a predetermined state variable. Thus the model can be reformulated such
that the terminal conditions are invariant to the policy shock. This seems to be the most
elegant solution and it is the solution we have implemented in QUEST II. There is only a
small cost associated with it, namely the model must be extended by adding P equations
defining the vector y ,' .

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In Roeger and in ’t Veld (1997c) we have compared these three methods by applying three
types of permanent shocks: a fiscal shock, a monetary shock and a technology shock. We
found that in practice the first method can give satisfactory results if a long enough horizon
can be applied in the simulation. As this is often not practical, it fails as a general model
solution procedure. The third method gives almost identical results to that obtained under the
(theoretically correct) second method, but has the additional advantage of easy
implementation. This is the method used in our standard simulations.

Chart 1 below shows the results of three methods with a permanent fiscal shock under money
targeting. In these simulations governement purchases are reduced by 1 per cent of GDP
permanently and the debt to GDP ratio is reduced by 10 per cent. The three methods yield
fairly similar results. It is however visible that it takes the model longer to reach the long run
solution and there seems to be some small endpoint problems also associated with the
solution methods TCL and TCD. It is interesting to notice that TCD does best in this case.
We attribute this to the fact that the discrepancy between the steady state solution and the
solution value from the dynamic simulation is larger if the terminal conditions are formulated
in levels instead of first differences. It can also be seen from Chart 2 that the solution under
NTC deviates from TCL and TCD already after 10 years. Thus it is not generally true that
NTC replicates the correct solution over one half of the simulation horizon.

¹⁴ This holds for QUEST II which is formulated in efficiency units. However, this method also works if
all variables are specified in levels. In that case the vector y j_{+ +1} contains the values of the steady
state growth rate of the jumping variables.

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