Theoretical Economics Letters, 2011, 1, 47-52
doi:10.4236/tel.[phone] Published Online November 2011 (http://www.SciRP.org/journal/tel)
Solution for Rational Expectation Models
Free of Complex Numbers
Frank Hespeler
Institute for Forecasting and Macroeconomic Research, Tashkent, Uzbekistan
Received July 2, 2011; revised August 25, 2011; accepted September 5, 2011
Abstract
This paper approaches the problem of the potential for complex-valued solutions within linear macroeco-
nomic models with rational expectations. It finds that these problems are associated with a specific solution
method for the underlying model. The paper establishes that the danger of complex-valued solutions always
can be eliminated by forcing those solutions to fulfill additional constraints. These constraints are essentially
restrictions on the degrees of freedoms in indeterminate solutions.
Keywords: Rational Expectation Models, Indeterminacy, Potentially Complex Coefficients
1. Introduction
Within the solution of dynamic economic models with
rational expectations regularly some kind of matrix de-
composition technique is employed in order to separate
the stable (generalized) eigenvalues from the unstable
(generalized) eigenvalues. Examples are the eigen de-
composition, the Schur decomposition or the generalized
Schur decomposition. All these decompositions share the
property that for real matrix polynomials it is sometimes
less costly to compute complex decompositions than real
decompositions. This raises the question whether we can
nevertheless assert real-valued solution paths. [1] shows
for the case of the eigen decomposition that this can be
achieved by choosing appropriate constants associated
with each eigenvalue. This paper discusses the same
question for the case of the generalized Schur decompo-
sition. It shows that while the structural properties of the
decomposition methods support real-valued solutions,
they do not always suffice to assert such solution paths.
In particular, the structural properes of the generalized
Schur decomposition can not assert real-valued solution
paths for the case that the model’s expectational error is
explained as a function of the exenous shocks to the
model as e.g. in [2]. Hence, in order to get a real-valued
solution of the model for this case, it is necessary to in-
troduce additional conditions to force the solution to the
real domain.
The paper develops such conditions. Of course this
requires some degrees of freedom within the solution of
the model, since a unique solution does not allow for the
development of additional conditions. Therefore the pa-
per applies to solutions of rational expectations models
which are characterized by indeterminacy. Within this
solution it attempts to use the available degrees of free-
dom to solve for a constellation, in which the coefficients
of endogenous and exogenous variables are forced into
the appropriately dimensioned real spaces . If
such a constellation exists, the entire solution of the
model is a real-valued one. To this purpose this paper
uses the fact that the product of two complex matrices
generates an imaginary part which consists in the sum of
the products of the real part of one matrix and the imagi-
nary one of the other matrix. Within this imaginary part
eventually complex-valued exogenous components can
be balanced by the imaginary part of the components
containing the degrees of freedom. Thus the product’s
imaginary part is set to zero. Applying this to the coeffi-
cient mentioned it can be guaranteed that the coefficient
is a real-valued matrix.
mn

The conditions obtained are sufficient conditions for
the existence of a real-valued solution. These conditions
built on the model’s transversality condition, which is
integrated into the model’s solution in a specific way that
does not allow any complex values for the model’s en-
dogenous variables. Thus, if necessary, the conditions
restrict the degrees of freedom available in the solution
further beyond the scope of restriction already obtained
by the mere integration of the transversality condition in
[3]. For the case that the transversality condition does

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
1t
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

F. HESPELER 49
of the generalized Schur decomposition of the matrix
pencil will be used. This decomposition takes
the form
(,)HG
=
H
RSPG (5)
=,
H
RTPH (6)
where and are unitary matrices and T is an
upper triangular matrix. Depending on the exact form of
the decomposition is either upper triangular or upper
Hessenberg. The details of the computation of this de-
composition are discussed in [11]. For the purpose of the
present paper just one characteristic of this algorithm is
needed. So long as is only transformed up to upper
Hessenberg form, i.e. the socalled real generalized Schur
decomposition, the operations involved will secure that
for any real-valued initial matrix-pencil both and T
are real-valued matrices, while the matrices and P
will be potentially complex-valued1. In order to exploit
these facts, the decomposition in Equation (6) is premul-
tiplied by the factor
R P
S
S
S
R

H
RI ΦR yielding the expres-
sion

11 122212
=() ,
HH H
TTΦTPR ΦRH (7)
while the results and 22 2
are derived from Equations (5) and (6), whereby
1
11 11
=H SPGR

2
=H TRHP
H
i
P
stands for the rows of matrix
H
P
S
TR  X
X
associated with the
model’s stable (unstable) eigenvalues for the case of
(), while ij denotes essentially a block of
after partitioning the latter according to the criterion
of stable and unstable eigenvalues. Symmetric notations
hold for matrices and . denotes the pseudo-
inverse of any matrix , while
=1i
S
=2i
H
X symbolizes the
conjugate transpose of that matrix.
For the solution methods based on generalized Schur
decompositions [5] shows that the coefficients 1

and
take on the following general forms
2q


1
11
111122
=

2
H

 


S
PTTΦTP
0
 (8)

1
11 1222
2
111
22 22222
=1
=
().
q
qH
q








SSΦS
PI
TS TRC

(9)
in which the matrix depends on the specific solution
technique chosen ande question whether a transversal-
ity question is integrated or not2. Subs
(7) and the expressiobtained for and into
these coefficients anplifying yields:
Φ
th
ons
d sim
tituting Equation
1
11

S22 T
11111 1112
=
HH HH
PPGRHPPGR ΦRH (10)
211221222
1
2222
=1
= ( ( ))
().
R
H
HH
q
HqH
q





GRS PGRΦRG IPP
PPHGPP HC

(11)

Because the product of any set of columns of a
factor of the generalized Schur decomposition with its
pseudoinverse, i.e. with its conjugate transpose, is a
real-valued matrix by construction3 and the expre
unitary
ssion
112 2
H
RS P can be proven to be real-valued as well, the
two expressions in Equations (10) and (11) are real-val-
ued matrices iff the expression 12
H
RΦR is real-valued.
For the solution along the lines proposed in [10] the
expression 12
H
RΦR is a zero matrix,4 if there exists an
olution, while for any indeterminate solution
12
unique s
H
RΦR depends on the question whether an explicit
transversality condition is intego the solution or rated int
not. For the latter case there are no additional restrictions
on the matriich implies that for real-valued
12 1122
12 22
=
()
HH H
HH



RΦRRRGGRR
RZRIGG RR


the expression 12
x Φ, wh
H
RZR need to be real-valued as well.
Without presenting details, it should be only mentioned
that the only psible solution for this case would coios
case of a
ich, as
n-
cide with the unique solution for the whole
model, for wh already stated, will be zero
riate dim
Φ
matrix of appropensions. For the case that the
solution includes an explicit transversality condition
12 1122112222
22 11 11
11 112222
=
(ker())
((ker()))
HHHH H
HH
H
HH


 


RΦRRRGGRRRTPHRR
BBRRRR GPWP
WPPGR TRBBRR

needs to hold. Herein B denotes the matrix =

BI

GG
 . Thus real-valoefficients require ued c112 2
H
RT R
pres-to be real
sion can
-valued. Accog to Equation (6
be rewritten as
rdin) this ex
11 22
H
H
RR HPP
that both 
, which is
truction. This pro and
real by
2q
consves 1


are
purely real-valued coefficients.
e solution for Φ in the method proposed]
again depends on the question whether it satisfies an ex-
plicit transversality condition or not. [3] reveals that in
the second case there are no additiona conditions on Φ
except for the fact that it does need toe of apriate
dimensions. Hence any matrix
1If the result of the generalized Schur decomposition is presented as a
p
air of triangular matrices and the appropriate pair of unitary matrices,
the real-valued upper Hessenberg form can be reconciled by simple
unitary transformations.
2Details on this topic can be found in [5], [8] and [10].
3This fact stems from the possibility presented in [11] to express both
unitary matrices as a product of symmetric orthogonal factors.
4[10] finds a solution for Φ by solving 21
. If no
consistent solution exists, the model is supposed to have no solution at
all.
=
HH
ΦRG RG
Th by [5
l
bprop
which produces a real
12
H
RΦR, qualifies as a solution. Due to the fact that 1 R

F. HESPELER
50
has full column rank and 2
H
R has full row rank, any
solution involving nullspaces reduces to the trivial solu-
tion =Φ0. Additional solutions are given by any m
trix of the form 12
H
RΨR where Ψ is an arbitrary
quadratic matrix in nn . On the other hand an explicit
a-
trans
2
vers ality condition implies
12112222 1111
22 22
=(ker())
ker((ker()))
HHHH
HH




RΦRRTPHRRRRGP WP
WPJPHR R
.
In order to guarantee-valued entries for this expres-
sion, 11
(ker( ))ker((ker( )))
real
2
H
PWP WPJP needs to con-
tain exclusively real numbers as welm
hf
ker
H
Presult yields
t of row
l. So
of a unitary m
ore we
e properties of
atrices help
notice that
the nullspace
to establish t
) =WP
s the se
of any subm
is property
( )W. Usi
s of
atrix
. There
ng this
ker(
ker((ker())) = ker(ker())
H
ii
WPPW where =1i (=2i)
denote
H
P associated with the stable
(unstable) eigenvalues. For =2i, this expression is de-
noted as Θ below. Thus the result for 12
H
RΦR sim-
plifies to
11 122 2
er()
H
12 11222
1k
H
2
=
H
H
H
RR .
HH



RR
RR GPPWΘJPHR R
This matrix’s elements belonhe set of real numbers,
iff 2
RΦTPH
g to t
H
ΘJP s real-valued. A sufficient con
is
i
()
s. The
t-hand (le
within Equation
ght-ha
dition for this
2)
ybuild on the
righft-hace of tst (third factor)
w-spof the fi
tor’s rind cludes the row-space of the

()
)()
potential alterna
nd) nu
(12). If the
nullspace i
2
2
() ()
( =.
() ()
H
H


 



ΘΘ 0
P




P
utions
he fir
ace


JJ
JJ
tive sol
llspa
ro
n
(1
Potentially this equation has several solutions. The most
obvious is the trivial solution =J0, which holds al-
wa
rst fac-
third factor, i.e.
 
rankker( )()ΘΘ
 



122
rankker()()()
T
TTT
HH


 




ΘΘ PP,
there is a set of additional solutions of the form

HH




PP
r
y m
2
2
1
H
H






P
P
d m
p
trix. A si
1
()
ker ()
()



 



I
22
22
,
() ()
HHi







 

DI I
PP
( )(
()

ΘΘ
rbitrary
with at
and D
rwise arbitrar
)
)




J
in which Σ1
dime
( )
d, but ot
(13
is an aeal-valueatrix of appro-
priate nsions least the same rank as
is an apropriately dimen-
amilar solution
exists if

ker( )ΘΘ
sione he





 22
2
2
2
rank ker(())(())
(( ))
rank()()ker
(( ))
HT HT
HT
HT
 







 

 







PP
P
ΘΘ
P
T
T

holds. This solution turns out to be

2
()
= (()()ker
T
T
H
T






2
2
1
()
()()()()).
H
i

 





 

P
I
IΘΘ ΘΘEI

(14)
Herein



 
P
JΘΘ
2
T

is a arbitrary matrix of appropriate dimen-
sions with at least the same rank as the factor with which
it is postmplied, while is arbitrary, but of appro-
lutions discussed the
coefficients
ulti E
priate dimensions. For all three so
1

and 2q


are purely real-valued coef-
ficients.
Hence, we can conclude this section by emphasizing
that the structural properties of the generalized Schu
composition guarantee the existence of real-valued coef-
ficients for any solution obtained by the method of [10],
r solut
lim line
tio both solution
s
r de-
while foions derived along the arguments of [3,5]
those properties are not sufficient, because the solution’s
degrees of indeterminacy allow for complex-vaued coef-
ficients as well. In order to avoid those, the degrees of
freedoms need to be restricted in the sense that the inde-
terminacy is ited to ar subspaces, which render the
mentioned coefficients to real-valued matrices. Never-
theless, the existence of a real-valued solution is in any
case asserted by the possibility of the trivial solution for
J, which holds under any circumstances.
4. Conditions for a Real-Valued Coefficient
of the Contemporary Shock Variables
In the process of establishing the conditions for a real-
valued coefficient of the contemporaneous shock vari-
ables the paper follows the same classification as used in
the last section. Thus again the solution are decided into
one balancing the expectational errors within the unsta-
ble part of the model and one explaining those as func-
ns of the shock terms. In addition in
methods the cases in which the transversality condition i
or is not used explicitly are distinguished as well.
The first method produces the coefficient
211 112
().
HHH
PPG RRΦRC (15)
This coefficient appears to be real-valued whenever

F. HESPELER 51
12
H
RΦR is real. According to the results of the last sec-
tion this condition holds always. Hence, the model’s so-
lution is real-valued by construction and no further con-
ditions are required.
For the second method the coefficient 2  has the
general form
H

 CGGRRC

22
211 11()
HH
H






PP GR R
GIGRRGZ
 

22
.

(16)
If the shock term explains the expectational error com
to invers
e soltion. In this situation the ma-
tri ing
tran integrated into
the model’s solution or For the case that no specific
condition applies, there is no
onbecauseother terms in Equation
7)
l onas the null-
space of this factor yields
(18)
ven by
Inserting any of these solutions into delivers
in il
o
a real-valued coefficient
can be asserted y circumce
the
existence of a purely real-valued solution
ic
po
ra
ntire solution path is a real-valued one. But for solutions
ctational error directly by the exoge-
properties are not sufficient to
l values, this paper
co
, pp. 273-285.
doi:10.1016/S0165-1889(02)00153-7

-
pletely, the pseudoinverses in Equation (16) are rendered
es and the matrix Z is a zeromatrix. Hence the
coefficient is real-valued. But if the shock term explains
the expectational error only partially, there remain inde-
terminacies within thu
sversality condition is
not.
all
x Z takes on two different values depend on whe-
ther a specific
transversality
restriction
additional
Z
(16) are real-valued by construction. Thus any matrix Z
belonging to mn  generates purely real-valued solu-
tions for the economic model. For the opposite case the
transversality condition implies
2211 11
1122
=( )
(()ker())
HHH
HH




ZIGRRGGRR GPP
GRRI GGRRCWΘF


where F is left undetermined, but appropriately dimen-
sioned. Inserting this expression into Equation (16) dem-
onstrates that a purely real solution requires
111 122
2211 11
(()
() ker())=.
HH H
HHH




PPG RRGIGRRG
IGRR GRR GPPWΘF0

 (1
Denoting the factor in front of F as Ξ and solving
Equation (17) for all terms invo

G

ving F

() =ker()( ), 
 

FΞΞE
()

F
where E is arbitrary, but real-valued and of appropriate
dimensions. Thus the solutions for F are gi
 
=iker()() .FIIΞΞE (19)
finally Z
stan
a real-valued coefficient 2 . Aga, simarly as in the
last section, it should be pointed out that Equation (17)
can be fulfilled independently of the rankf Ξ by the
trivial solution for F. Hence
r an
2
 undes.
5. Conclusions
In this paper we have established the conditions for
path to any
linear rational expectations model whh is based on the
generalized Schur decomsition. For solutions, which
explain the influence of the expectational error on the
model’s stable part as a function of its influence on the
model’s unstable part, the structural properties of the gen-
elized Schur decomposition suffice to guarantee that the
e
explaining the expe
ous shock term those n
exclude complex-valued solution paths. Nevertheless,
existing degrees of freedom allow to establish additional
constraints which force the solution paths into the real
domain. The paper shows also that the trivial solution for
the degrees of freedom generate a real-valued solution
path under any circumstances. Thus the existence of at
least one of those paths is asserted.
These results imply that a direct explanation of the
expectational error by the exogenous shock term gener-
ates the potential of complex-valued solutions. Certainly
this can be interpreted as a disadvantage compared to the
methods which force the unstable variables to balance
each other and use those to explain the expectational
error’s influence on the stable part. On the other hand the
first approach allows for a solution of models which have
been unsolvable with the second approach. Since it is
always possible to find solutions obtained by the first
approach which take one pure rea
ncludes that the mentioned disadvantages of this ap-
proach are balanced by its advantages. Hence this paper
supports a more wide-spread application of the method
presented first in [5] for the solution of macroeconomic
models.
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