F. HESPELER

48

only imply real solutions, no further conditions beyond

the fact that all degrees of freedom must be restricted to

the real domain, are needed.

The paper is organized as follows. After this section’s

introduction section 2 presents the general method to be

employed. In sections 3 and 4 separate algebraic condi-

tions are developed which need to hold simultaneously in

order to guarantee a pure real-valued solutions. Section 5

concludes.

2. General Method

According to the solution algorithms for linear dynamic

models involving rational expectations, i.e.,

1

1

1

=

tt

tt

tt

vv

z

ww

GHCG

(1)

presented in [4,5,6-10], among others, any solution to

this type of models can be written in form of a VAR(1)-

process

1

122

=1

1

= .

tt

tqt

q

tt

vv

zE

ww

T

tq

z

(2)

Herein denotes the vector of endogenous

variables in period , while is an exogenous shock

vector realized in period . 1t

TT

tt

vw tt

z

t

is the vector of expec-

tational errors in the endogenous variables denoting the

difference between the unconditionally expected values

based on information available in period and the val-

ues actually realized in period . The shock vector

t might display some autocorrelation which explains

the appearance of expected future shock terms in Equa-

tion (2). In addition, the model in Equation (1) might be

required to fulfill the transversality condition

t

1t

z

>

=0.

lim tq

t

qtq

v

Ew

W (3)

Herein t denotes the operator for the unconditional

expectations based on information available in t. As

pointed out in [9] this requirement, if not guaranteed by

appropriate initial conditions, essentially restricts the

growth of the model’s unstable exogenous variables.

E

Depending on the characteristics of the used solution

method the coefficients 1 are defined as spe-

cific functions of the original coefficient matrices. The

central idea of asserting a real-valued solution starts with

the fact that often some of the coefficients are real by

construction. This claim will be discussed in the next

section for coefficients associated with the endogenous

variables and the expectational error terms. Afterwards it

will be analyzed with respect to the coefficient of the

exogenous shock term in the subsequent section. For any

coefficient for which this claim does not hold, we will

use any available degrees of freedom within that coeffi-

cient in order to force it into the real domain. Thus the

indeterminacy allows to guarantee a purely real-valued

solution for the model. Therefore such a coefficient

(,, )

will be decomposed into potentially complex,

, and

real, i.e.

, factors. The product of this factors has the

form

(() ())=()()i i

i1

(4)

where denotes the square root of , while

(

)

denotes the real (imaginary) part of its argument. When-

ever the last summand contains enough degrees of free-

dom in order to be forced down to zero, the entire coeffi-

cient will take on values from the set of real numbers.

Thus the entire solution of the macroeconomic model

will not include any complex numbers.

As already indicated the paper restricts the analysis on

the case of solution methods based on the generalized

Schur decomposition. Nevertheless, similar arguments

could be obtained for the case of solution methods based

on eigenvalue decompositions, Jordan decompositions

and ordinary Schur decompositions. This paper focuses

on the generalized Schur decomposition, because models

which can be solved by those methods nest all models

solvable by the mentioned alternative decompositions.

In addition the paper distinguishes two approaches to

balance the distorting influence of expectational errors

appearing within any rational expectations model. The

first approach has been presented in [8] and [10]. Herein

the mentioned distortion is eliminated by explaining the

expectational error’s influence on the model’s stable part

as a function of it’s influence on the model’s unstable

part, which itself is forced to be zero by the initial condi-

tions. On the other hand [5] explains the expectational

error directly as function of the exogenous shock term.

As shown in [2] this does not exclude sunspot solutions

because expectations might be driven by an additional

component which do not contribute to the model’s un-

stable part. For both approaches we also separate be-

tween the cases in which either a microfoundated trans-

versality condition is explicitly integrated into the solu-

tion as presented in [3] or the transversality condition is

only used in the traditional manner, e.g. [4], by forcing

the non-state variables to take on appropriate initial val-

ues.

3. Real-Valued Coefficients for Endogenous

Variables and Expectational Errors

In order to prove the claim that the coefficient of the en-

dogenous variables as well as those of the expectational

errors do not include complex numbers, some properties

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