Chari, Kehoe and McGrattan (2007) (CKM) show that a large class of dynamic stochastic general equilibrium (DSGE) models with various frictions and shocks is observationally equivalent to a benchmark real business cycle (RBC) model with correlated "wedges" in the RBC model's first-order conditions. The wedges in the static first-order conditions of the RBC model can be readily computed by evaluating the first-order conditions at the data and then solving for the wedges. In contrast, identification of the "investment wedge" in the RBC model's dynamic Euler equation requires the researcher to make assumptions about the expectation formation by agents in the RBC model. In particular, CKM assume that expectations are formed as if, from the perspective of the model's agents, wedges followed a vector autoregressive process of order one (VAR(1)). We show that wedges generally do not have a VAR(1) representation, implying that CKM's procedure is based on model-inconsistent expectations. We also provide an alternative, model-consistent approach to modeling expectation formation. On the former issue, we present a necessary and sufficient ``rank condition'' under which a detailed economy can be mapped into a benchmark model where wedges follow a VAR(1) process. On the latter issue, we suggest that the information set underlying the expectation formation should not only contain current wedges, but also all predetermined variables.Business Cycle Accounting; Model Consistent Expectations

Daniel Burren

Gregor Baeurle

English

Research Papers in Economics

Faculty of Economics and 
Social Sciences 
 
Department of Economics 
A Note on Business Cycle Accounting 
 
 
 
Gregor Bäurle 
Daniel Burren 
 
07-05 
 
 
October 2007 
 
DISCUSSION PAPERS 
Schanzeneckstrasse 1 
Postfach 8573 
CH-3001 Bern, Switzerland
http://www.vwi.unibe.ch A Note on Business Cycle Accounting∗
Gregor B¨ aurle†and Daniel Burren‡
October 17, 2007
Abstract
Chari, Kehoe, and McGrattan (2007) (CKM) show that a large class of dynamic
stochastic general equilibrium (DSGE) models with various frictions and shocks is
observationally equivalent to a benchmark real business cycle (RBC) model with
correlated “wedges” in the RBC model’s ﬁrst-order conditions. The wedges in the
static ﬁrst-order conditions of the RBC model can be readily computed by evaluating
the ﬁrst-order conditions at the data and then solving for the wedges. In contrast,
identiﬁcation of the “investment wedge” in the RBC model’s dynamic Euler equa-
tion requires the researcher to make assumptions about the expectation formation
by agents in the RBC model. In particular, CKM assume that expectations are
formed as if, from the perspective of the model’s agents, wedges followed a vector
autoregressive process of order one (VAR(1)). We show that wedges generally do not
have a VAR(1) representation, implying that CKM’s procedure is based on model-
inconsistent expectations. We also provide an alternative, model-consistent approach
to modeling expectation formation. On the former issue, we present a necessary and
suﬃcient “rank condition” under which a detailed economy can be mapped into a
benchmark model where wedges follow a VAR(1) process. On the latter issue, we
suggest that the information set underlying the expectation formation should not
only contain current wedges, but also all predetermined variables.
JEL-Classiﬁcation: C50, E10
Keywords: Business Cycle Accounting, Model Consistent Expectations
∗We thank Klaus Neusser, Dirk Niepelt and seminar participants at the Study Center Gerzensee for
valuable discussions and comments. The usual disclaimer applies.
†University of Bern, gregor.baeurle@vwi.unibe.ch
‡University of Bern and Study Center Gerzensee, daniel.burren@szgerzensee.ch
11 Introduction
Chari, Kehoe, and McGrattan (2007) (CKM) show that a large class of dynamic
stochastic general equilibrium (DSGE) models with various frictions and shocks is
observationally equivalent to a benchmark real business cycle (RBC) model with cor-
related “wedges” in the RBC model’s ﬁrst-order conditions. Since diﬀerent DSGE
models - CKM refer to these models as “detailed economies” - have diﬀerent implica-
tions for the dynamic properties of the wedges, the wedges reveal information about
the structure of the unkown data generating economy.
The wedges in the static ﬁrst-order conditions of the RBC model can be read-
ily computed by evaluating the ﬁrst-order conditions at the data and then solving
for the wedges. In contrast, identiﬁcation of the “investment wedge” in the RBC
model’s dynamic Euler equation requires the researcher to make assumptions about
the expectation formation by agents in the RBC model. In particular, CKM as-
sume that expectations are formed as if, from the perspective of the model’s agents,
wedges followed a vector autoregressive process of order one (VAR(1)). We argue that
this assumption is inappropriate for some interesting and widely discussed detailed
economies.1
We argue further that the VAR(1) assumption is not only critical for computing
the investment wedge but also for implementing the accounting as proposed by CKM.
Indeed, the impact on equilibrium quantities might be wrongly assessed even for
correctly measured wedges.
This can be seen as follows. In order to assess the contribution of diﬀerent wedges
to business cycle movements, CKM suggest to set the values of the other wedges to
constants, leaving the distribution of the wedges of interest unchanged. They then
calculate the decision rules as functions of the operating wedges, their expected future
values and the predetermined variables in the RBC model. Finally, they plug the
measured wedges and their expected future values, as obtained from the VAR(1),
into the decision rules in order to get simulated equilibrium quantities which they
compare to the data.
We show that wedges generally do not have a VAR(1) representation, imply-
ing that CKM’s procedure is based on model-inconsistent expectations. This result
holds independently of whether the wedges were correctly measured in the ﬁrst place.
We also provide an alternative, model-consistent approach to modeling expectation
formation. On the former issue, we present a necessary and suﬃcient “rank condi-
tion” under which a detailed economy can be mapped into a benchmark model where
1One example is mentioned in the critique of the accounting procedure by Christiano and Davies (2006).
2wedges follow a VAR(1) process. On the latter issue, we suggest that the information
set underlying the expectation formation should not only contain current wedges, but
also all predetermined variables.
We illustrate our results for the sticky wage model discussed in CKM. For that
model, the rank condition is not satisﬁed, implying that the accounting procedure
proposed by CKM is inconsistent with the assumption of rational expectations. We
also show that a simple application of our proposal - augmenting the VAR(1) of the
wedges by capital - resolves these problems.
2 Rank Condition
Suppose that the solution of the detailed rational expectations model can be written
in the following state-space form
ct = Mcppt + Mceet (1)
 
pt
et
!
| {z }
:=St
=
 
Npp Npe
0 ρ
!
| {z }
:=Θ
 
pt−1
et−1
!
+
 
0
σ
!
|{z}
:=Σ
εt (2)
where ct is a vector of non-predetermined, pt a vector of predetermined variables2 and
et an exogenous vector autoregressive process of order 1 (with serially uncorrelated
innovations εt).
Let Wt be the vector of the wedges needed for mapping this model into the
benchmark RBC model of CKM. In order to understand how the wedges are related to
the state variables St of the detailed economy, one has to plug the solved equilibrium
processes (in closed form) of the detailed economy into the linearized ﬁrst order
conditions of the benchmark RBC model. The wedges which distort the static ﬁrst
order conditions can then directly be written as a linear combination of the states pt
and et. This is not so clear for the investment wedge which distorts the Euler equation
because this equation involves both, the current and the expected future investment
wedge. However, this equation can be solved forward and also the investment wedge
turns up to be a linear combination of pt and et. Hence, the closed form solution of
the wedges is given by
Wt = Zppt + Zeet.
Note that when solving the expectational equation for the investment wedge forward,
2In the sense of Blanchard and Kahn (1980), i.e. Et−1pt = pt.
3we implicitely use the correct speciﬁcation of the expectation. This is no longer the
case when the expactations are infered from a VAR(1) in the wedges when these do
not have a VAR(1) representation.
We now discuss when the wedges have a VAR(1) representation. Using equation
(2), the process of the wedges can be written as
Wt = Z
 
pt
et
!
= ZSt = Z(ΘSt−1 + Σεt) (3)
where
Z =

Zp
. . . Ze

Theorem 1. Assuming that the detailed economy, described by (1) and (2), maps
into a benchmark RBC model with wedges Wt = ZSt, then the process of the wedges
has a VAR(1) representation, i.e. Et−1[Wt−ΦWt−1] = 0 where Et−1 is the expectation
conditional on all information up to time t − 1, if and only if
rank

Z0 . . . Θ0Z0

= rank(Z) (4)
Proof. We show that the condition is necessary (step i) and suﬃcient (step ii).
i) Plugging in (3) into Et−1[Wt − ΦWt−1] = 0 yields
Et−1[ZΘSt−1 − ΦZSt−1] = 0.
Almost surely, it follows that
ZΘ = ΦZ. (5)
Equation (5) states that each row of ZΘ lies in the row space of Z. Since the
dimension of the row space of Z is equal to rank(Z), (4) follows.
ii) Given that the rank condition (4) is veriﬁed, it follows that the product Θ0Z0
lies in the column space of Z0. Hence, there exists a matrix Φ such that
ZΘ = ΦZ.
Since, by assumption,
Wt = ZΘSt−1 + ZΣεt,
it follows that
Wt = ΦWt−1 + ZΣεt.
4Theorem 2. Assuming that the detailed economy, described by (1) and (2), maps
into a benchmark RBC model with wedges Wt = ZSt and that the vector of predeter-
mined variables in the benchmark model, kt, is the same as in the detailed economy,
then Wk
t :=

k0
t W0
t
0
has a VAR(1) representation if Ze is invertible.
Proof. By assumption
 
kt
Wt
!
=
 
I 0
Zp Ze
! 
pt
et
!
Since it is assumed that the inverse of Ze exists, it follows
 
I 0
−Z−1
e Zp Z−1
e
! 
kt
Wt
!
=
 
pt
et
!
and the VAR(1) representation follows from equation (2).
3 Example: Sticky Wages
CKM present a sticky wage economy which is observationally equivalent to a bench-
mark model with a labor wedge given by
Wt = 1 − τL,t = −
ULt
UCt
1
FLt
where ULt (resp. UCt) is the marginal utility of labor (resp. consumption) and FLt
is the marginal productivity of labor. The labor wedge captures the distortions
between the marginal product of labor and the marginal rate of substitution between
leisure and consumption. The detailed economy is driven by a stochastic money
supply, which is called et, and the only predetermined variable is capital kt. Since
the solution to the detailed economy is a linear combination of capital and the money
supply shock, the ﬁrst order accurate dynamics of the labor wedge is determined by
the underneath state space system.
1 − τL,t =

z1 z2
 
kt
et
!
 
kt
et
!
=
 
nkk nke
0 ρ
! 
kt−1
et−1
!
+
 
0
1
!
εt
5where Z =

z1 z2

. Since rank(Z) = 1, the rank condition of Theorem 1 is not
satisﬁed for meaningful calibrations:
rank

Z0 . . . Θ0Z0

= rank
  
z1 nkkz1
z2 nkez1 + ρz2
!!
≤ 2.
The misspeciﬁcation by imposing a VAR(1) in the wedges may lead to wrong ac-
counting results. This is the case even if the realized wedges are correctly measured.
The reason is that the solution of the benchmark model depends on the process of
the wedges. If the process is wrong, also the (rational expectation) solution to the
model is wrong.3
Following Theorem 2, there is a simple solution to this potential misspeciﬁcation:
Writing
 
kt
1 − τL,t
!
=
 
1 0
z1 z2
! 
kt
et
!
it can be seen that the assumptions of Theorem 2 are satisﬁed if z2 6= 0, which is
the case for most calibrations of the parameters in the detailed economy. Hence, by
not restricting the correlations between the labor wedge and capital to be zero in the
estimation, we mitigate the need for estimating an inﬁnite order VAR or VARMA
process for the wedges.
4 Conclusion
We have derived a necessary and suﬃcient condition for the existence of a VAR(1)
representation of the wedges. We have then shown that for the sticky wage model
of CKM, this condition is not satisﬁed. Hence, we conclude that the model is not
representable in the form that CKM estimate.
We suggest an extended econometric model that allows to accurately estimate
the dynamics of the wedges. The solution is based on the fact that there is a VAR(1)
representation in the vector of the wedges augmented by the capital stock.
Obviousely, this extension does not provide a solution for all DSGE models pro-
posed in the literature. For example, if the stochastic money supply is replaced by
an interest rate rule with interest rate smoothing, then the lagged interest rate is an
3In the language of CKM, p. 797, the decision rules are not correctly computed. Hence, the real-
ized sequences of output, labor and investment and therefore also of the capital stock are not correctly
computed.
6additional predetermined variable which does not have a counterpart in the bench-
mark RBC. In this case, the vector with the wedges and capital does not have a
VAR(1) representation. However, one could generalize the benchmark RBC model
by adding predetermined variables such that there is a VAR(1) representation in
the wedge vector augmented with the predetermined variables for a larger class of
detailed economies.
References
Blanchard, O. J., and C. M. Kahn (1980): “The solution of linear diﬀerence
models under rational expectations,” Econometrica, 48(5), 1305–1311.
Chari, V., P. J. Kehoe, and E. R. McGrattan (2007): “Business Cycle Ac-
counting,” Econometrica, 75, 782–836.
Christiano, L. J., and J. M. Davies (2006): “Two Flaws in Business Cycle
Accounting,” Federal Reserve Bank of Chicago Working Paper, 2006-10.
7

The solution of linear diﬀerence models under rational expectations,”

Two Flaws in Business Cycle Accounting,” Federal Reserve Bank of Chicago Working Paper,

A Note on Business Cycle Accounting

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A Note on Business Cycle Accounting

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