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

There are various ways of solving forward looking models with rational expectations. Most
of them are based on linearisations of the model around the steady state and then applying
closed form solution algorithms to the linearised system, like for example the method
suggested by Blanchard and Kahn (1980). TROLL uses a method developed by Laffarque
(1990), Boucekkine (1995) and Juillard (1996) to solve the nonlinear model by Newton-
Raphson directly. This certainly has the advantage of increased accuracy and applicability of
the method to economies which are not operating close to a steady state initially, but it has
the drawback that terminal conditions must be explicitly specified. This appendix compares
three alternative methods of imposing terminal conditions.

The stacked-time solution algorithm in TROLL essentially works as follows. Let \t (Q[) and
[t (N[) be vectors of endogenous and exogenous variables respectively. The model can be
expressed compactly as

I, (\t-1,\,^, (, \ +1 ^, [, ) = 0                                                          ⁽¹⁾

where It is a vector of Q nonlinear dynamic equations. The presence of predetermined state
variables \t-1 and forward looking expectations (jumping variables) Et\t+1 introduces
simultaneity across time periods. One way of solving that model (with starting date W), is to
stack the system for the T+1 periods
where ]t+j=(\t+j-1, \t+j, Et\t+j+1). This stacked system of equations is then solved by Newton-
Raphson¹³ subject to the predetermined variable \t-1 and the terminal condition \t+T+1.

7KUHH $OWHUQDWLYH 7HUPLQDO &RQGLWLRQV

We consider three methods of dealing with the problem of selecting terminal conditions. This
analysis is conducted under the assumption that the model can be formulated in efficiency
units and the model reaches a steady state growth path in the long run. Of course, the
methods discussed can also be applied to the level of the variables, but it would slightly
complicate the notation. The three methods can briefly be described as follows

0HWKRG 1R 7HUPLQDO &RQGLWLRQ 17& - LH don’t worry about the problem and use the
same terminal condition as for the baseline.

¹² This appendix draws on Roeger and in’t Veld (1997c)

¹³ The Jacobian matrix, which has to be inverted, can become very large, but the algorithm makes use
of the repetitive structure of the stacked system, which is triangular by blocks corresponding to the
different time periods. The computational costs can be minimised by inverting the matrix by blocks and
taking advantage of the sparsity within the single period blocks. Details on this method can be found in
Hollinger and Spivakovsky (1996).

33

<<   <   31   32   33   34   35   36   37   >   >>