On equivalence results in business cycle accounting∗ Kengo Nutahara† The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan. Masaru Inaba‡ Research Institute of Economy, Trade, and Industry (RIETI), Daidoseimei-Kasumigaseki Building 20th Floor, 1-4-2 Kasumigaseki, Chiyoda-Ku, Tokyo 100-0013, Japan. Revised: November 2008

∗

We would like to thank Richard Anton Braun, Fumio Hayashi, Julen Esteban-Pretel, Keiichiro Kobayashi, Koji Miyawaki, Kensuke Miyazawa, Tomoyuki Nakajima, Keisuke Otsu, Hikaru Saijo, Etsuro Shioji, Fumihiko Suga, Harald Uhlig and seminar participants at the 2008 Autumn Meeting of Japanese Economic Association, the RIETI for their helpful comments. The first author is particularly grateful to Kensuke Miyazawa for valuable discussions on this topic. Of course, the remaining errors are our own. † Corresponding author. Tel.: +81-3-3812-2111. E-mail: [email protected] ‡ E-mail: [email protected]

1

1

Abstract

2

Business cycle accounting is premised upon the insight that the prototype neo-

3

classical growth model with time-varying wedges can achieve the same allocation

4

generated by a large class of frictional detailed models: equivalence results. Equiv-

5

alence results are shown under unrestricted general class of the process of wedges

6

while it is often specified to be the first order vector autoregressive when business

7

cycle accounting is applied to actual data. In this paper, we characterize the class

8

of detailed models that are covered by the prototype model under the conventional

9

first order vector autoregressive specification of wedges and find that the class is

10

much smaller than that believed in previous literature. We also apply business cycle

11

accounting to an artificial economy where the equivalence does not hold and find

12

that business cycle accounting works well even in such an economy.

13 14

Keywords: business cycle accounting; equivalence results

15

JEL classification: C68, E10, E32

2

1

1

Introduction

2

Business cycle accounting (hereafter BCA) is a method that is used to (i) measure dis-

3

tortions using a prototype model with time-varying wedges – which resemble aggregate

4

productivity, labor and investment taxes, and government consumption – such that the

5

prototype model perfectly accounts for observed data and (ii) investigate the importance

6

of each wedge in business cycles through counterfactual simulations. BCA is a popular

7

method of business cycle analysis and it is applied to many countries.1

8

In order to justify the prototype model, Chari, Kehoe, and McGrattan (2007a) (here-

9

after CKM) claim equivalence results, that is, many detailed models are equivalent to (or

10

covered by) the prototype model in the situation where any realized sequences of con-

11

sumption, investment, labor, and output generated by the detailed model are achieved by

12

the prototype model through the adjustment of wedges. CKM show equivalence results

13

under unrestricted general class of the stochastic process of wedges.

14

However, in practice, they assert that wedges evolve according to the first order

15

vector autoregressive (hereafter VAR(1)) process. It is not clear whether CKM’s VAR(1)

16

specification of wedges is consistent with the conditions in terms of equivalence results.

17

Many papers that apply BCA also employ CKM’s VAR(1) specification of wedges.

18

Therefore, it is important to investigate the class of models that are covered by

19

the prototype model with the VAR(1) specification of wedges. In addition, if the class

20

of models covered by the prototype model is found to be small, it is also important

21

to examine whether or not BCA can measure true wedges in an economy where the

22

equivalence does not hold. Non-equivalence might result in a mismeasurement with

23

respect to wedges and model predictions in counterfactual simulations have bias. In

24

the BCA literature, two contradicting results have been reported in the importance

25

of investment wedge in business cycles These results depend on minor differences in the

26

procedure of BCA. The diversion of results might be accounted for by the non-equivalence

27

between the prototype model and the true model of the real world. 1

For example, Chari, Kehoe, and McGrattan (2002, 2007); Ahearne, Kydland, and Wynne (2006); Bridji (2007); Kersting (2008); Kobayashi and Inaba (2006); Otsu (2007); and Saijo (2008).

3

1

In this paper, we examine the equivalence results by focusing on the VAR(1) rep-

2

resentation of wedges. We characterize the class of frictional detailed models that are

3

covered by the prototype model with the conventional VAR(1) specification of wedges.

4

We find that the prototype model covers a detailed model if and only if wedges have

5

sufficient information about the endogenous and exogenous states of the detailed model.

6

Intuitively, the number of independent wedges should be larger than the number of en-

7

dogenous and exogenous states variables in the detailed model. We also find that, in the

8

case of the VAR(1) specification of wedges, the class of detailed models covered by the

9

prototype model is much smaller than that shown by CKM under unrestricted general

10

class of stochastic process of wedges. Some examples of the equivalence shown by CKM

11

are not covered by the prototype model with the VAR(1) specification of wedges. More-

12

over, a medium-scale dynamic stochastic general equilibrium (hereafter DSGE) model

13

`a la Christiano, Eichenbaum, and Evans (2005) (hereafter CEE) is also not covered al-

14

though it is frequently used model that can capture many aspects of business cycles. We

15

extend our analysis to an alternative specification that has the VAR(1) specification as

16

a special case and we find that the class of models covered by the prototype model is

17

not so large even in this case. Furthermore, we apply BCA to an artificial medium-scale

18

DSGE model, which is not covered by the prototype model but is “realistic,” in order

19

to assess the empirical usefulness of BCA. We find that measured wedges capture the

20

properties of the true wedges with a great deal of accuracy even in such an economy.

21

Our theoretical results highlight a problem pertaining to the theoretical foundation of

22

BCA. However, we also find that, in practice, BCA works well since the prototype model

23

is a good approximation of our artificial medium-scale DSGE economy. According to our

24

numerical experiments, the variation in empirical results with respect to the importance

25

of investment wedge is not due to the problem of non-equivalence.

26

The following is a description of the related literature. CKM propose BCA and

27

claim that their prototype model covers a large class of frictional business cycle models.

28

CKM also conclude that investment wedge is not promising for business cycle research.

29

Christiano and Davis (2006) critique the BCA procedures and claim that the results of

4

1

BCA are very sensitive to changes in procedures.2 This paper is closely related to B¨aeurle

2

and Burren (2007). They investigate the class of frictional models that are covered by

3

the prototype model, and their study was conducted at the same time as the one in

4

this paper. However, they consider the class of models where there exists the VAR(1)

5

specification of wedges, which is a necessary condition for the equivalence.

6

The remainder of this paper is organized as follows. Section 2 introduces the basic

7

framework for the analysis: the prototype model, a detailed model, and the definition of

8

the equivalence. Section 3 presents our main results. We characterize the class of models

9

covered by the prototype model with the VAR(1) specification of wedges and discuss the

10

implications of our results. We extend our analysis to a more general class of the process

11

of wedges in Section 4. In Section 5, we apply BCA to an artificial economy where the

12

equivalence result does not hold and provide a numerical example that BCA works well

13

even in such an economy. Section 6 provides some concluding remarks.

14

2

15

2.1

16

The prototype model is as follows.3 The representative household problem is given by

17

the following:

Basic setting Prototype model

max

{ct ,ℓt ,it }∞ t=0

E0

∞ ∑

[ β

t

] log(ct ) + ν log(1 − ℓt ) ,

(1)

t=0

s.t. ct + (1 + τx,t )it ≤ (1 − τℓ,t )wt ℓt + rt kt−1 − Tt , kt = (1 − δ)kt−1 + it ,

2

(2) (3)

Chari, Kehoe, and McGrattan (2007b) critique the alternative procedure of Christiano and Davis (2006). 3 While CKM consider a general model with history dependence, we restrict a model which has the recursive structure in the present paper.

5

1

where ct denotes consumption; ℓt , labor supply; 1/(1 + τx,t ), investment wedge; (1 − τℓ,t ),

2

labor wedge; it , investment; wt , wage rate; rt , rental rate of capital; kt−1 , capital stock

3

at the end of period t; and Tt , the lump-sum tax. The production function of competitive firms is

4

α yt = At kt−1 ℓ1−α , t

(4)

5

where yt denotes output, and At denotes efficiency wedge.4 Finally, the market clearing

6

condition is

ct + it + gt = yt ,

7

(5)

where gt denotes government wedge. The equilibrium system of the prototype model is summarized as

8

ct yt = (1 − τℓ,t )(1 − α) · , 1 − ℓt ℓt [ { }] 1 + τx,t 1 yt+1 = βEt (1 + τx,t+1 )(1 − δ) + α , ct ct+1 kt

ν

(6) (7)

α yt = At kt−1 ℓ1−α , t

(8)

kt = (1 − δ)kt−1 + it ,

(9)

ct + it + gt = yt ,

(10)

9

where (6) denotes the consumption-leisure-choice optimization condition; (7), the Euler

10

equation; (8), the aggregate production function; (9), the evolution of aggregate capital

11

stock; (10), the resource constraint. To close the model, we have to specify the evolution

12

of wedges. CKM specify that the vector of wedges st ≡ [log(At ), τℓ,t , τx,t , log(gt )]′ evolves 4

We consider a stationary economy for simplicity.

6

1

according to the first order vector autoregressive, VAR(1), process as

st+1 = P 0 + P st + εt+1 ,

2

(11)

where εt is i.i.d. over time and is normally distributed with mean zero.

3

CKM also consider the prototype model with capital wedge, which resembles to capital

4

income tax, instead of investment wedge. In this case, the budget constraint (2) becomes

ct + it ≤ (1 − τℓ,t )wt ℓt + (1 − τk,t )rt kt−1 − Tt ,

5

(12)

where (1 − τk,t ) denotes capital wedge. The analogue of (7) is

[ { }] 1 1 yt+1 = βEt (1 − δ) + (1 − τk,t+1 )α , ct ct+1 kt

(13)

6

and st ≡ [log(At ), τℓ,t , τk,t , log(gt )]′ .5 We consider the case of the prototype model with

7

investment wedge in this paper, but the analogues of results in Section 3 are applicable

8

to the prototype model with capital wedge.

9

2.2

Detailed and extended detailed models

10

Here, we define two models: a detailed model and an extended detailed model for the

11

analysis.

12

Let xt be a vector of the endogenous state variables; y t , a vector of endogenous jump

13

variables; and z t , a vector of exogenous state variables. let n and q is the numbers of

14

variables contained in [x′t−1 , z ′t ]′ and y t , respectively. We consider these variables to be

15

defined as deviations from the deterministic steady-state.

5

CKM employ some types of capital wedge. This version is employed for equivalence result of the input-financing friction model as we will show in Section 3.

7

1

2

As described by Uhlig (1999), generally, a linearized DSGE detailed model is described as

Axt + Bxt−1 + Cy t + Dz t = 0, ] [ Et F xt+1 + Gxt + Hxt−1 + J y t+1 + Ky t + Lz t+1 + M z t = 0, ] [ z t+1 = N z t + ut+1 , Et ut+1 = 0,

(14) (15) (16)

3

where ut+1 is i.i.d. over time and is normally distributed with mean zero. We assume

4

that consumption ct , investment it , labor ℓt , capital stock kt , and output yt are definable

5

in this detailed model and that capital stock at the beginning of period kt−1 is included

6

in xt−1 . The state-space form solution of this detailed model is  

 xt



 = Ψ

z t+1

 yt = Ω 

 xt−1 zt xt−1



 0

+ 

,

(17)

ut+1

,

(18)

zt

7

where Ψ is an n × n matrix and Ω is a q × n matrix. The aggregate decision rule of

8

consumption, investment, labor, output, and capital stock at the end of period is



          

9

cˆt

   ˆit    x t−1  , ℓˆt  = Θ   zt  yˆt   kˆt

(19)

where Θ is a 5 × n matrix and variables with hat ˆ denote (log) deviations from the

8

1

deterministic steady-state. Here, we introduce the extended detailed model in order to consider wedges which are

2

3

consistent with the detailed model as follows.

4

Definition 1. An extended detailed model consists of (i) an equilibrium system of a

5

detailed model (14), (15), and (16), and (ii) linearized equations of the equilibrium con-

6

ditions of the prototype model (6), (7), (8), (9), and (10).

7

The extended detailed model implies that any realized sequences of consumption,

8

investment, labor, and output generated by the detailed model are consistent with the

9

equilibrium conditions of the prototype model (6), (7), (8), (9), and (10). Since wedges sˆt

10

are endogenous variables in the extended detailed model, we can calculate the aggregate

11

decision rule of sˆt as  sˆt = Φ 

 xt−1

,

(20)

zt 12

where Φ is an m×n matrix; m, the number of wedges sˆt ; and n, the number of endogenous

13

and exogenous state variables [x′t−1 , z ′t ]′ .6 We are also able to calculate the aggregate

14

decision rule of endogenous and exogenous state variables as 

 

xt z t+1



 = Ψ

 xt−1 zt

15

where Ψ is an n × n matrix.

16

2.3

17

The definition of the equivalence is as follows.



+

 0

,

(21)

ut+1

Equivalence results

6

We assume that this extended detailed model satisfies the suitable conditions in order to solve the aggregate decision rule, that is, the Blanchard-Kahn condition.

9

1

Definition 2. A detailed model is equivalent to (or covered by) the prototype model if the

2

prototype model can achieve any realized sequences of consumption, investment, labor,

3

output, and capital stock generated in the detailed model.

4

In other words, a detailed model is equivalent to the prototype model if the intratem-

5

poral condition (6), the Euler equation (7), the aggregate production function (8), and

6

the resource constraint (10) are satisfied given any realized sequences of {ct , it , ℓt , yt , kt }

7

generated by the detailed model. These four equations are satisfied by suitable adjust-

8

ments of wedges. CKM claim some examples of equivalence results by comparing the

9

equilibrium conditions of two models. Their equivalence results are true under unre-

10

stricted general class of stochastic process of wedges. However, in practice, CKM assert

11

that wedges evolve according to VAR(1) and it is not clear whether or not VAR(1)

12

specification of wedges is consistent with such conditions of the equivalence. It is also

13

unclear that even if such VAR(1) specification exists, a detailed model is equivalent to

14

the prototype model in general.

15

Generally, the necessary and sufficient condition of the equivalence is summarized as

16

follows.

17

Proposition 1. A detailed model is equivalent to the prototype model if and only if (i)

18

a process of wedges to close the prototype model is consistent with the detailed model,

19

and (ii) there exists, in the extended detailed model, a mapping from the states of the

20

prototype model given the process of wedges to variables of the prototype model.

21

The necessity of (i) is obvious since it is needed to close the prototype model. The

22

necessity of (ii) is as follow. There exists, in the prototype model, a mapping from

23

the states to variables and if the condition (ii) does not hold, realized sequences of

24

variables generated by the detailed model are not achieved in the prototype model. The

25

sufficiency is also easily shown as follows. The mapping (ii) generated in the extended

26

detailed model must satisfy the equilibrium conditions of the prototype model since they

27

are also included in the extended detailed model. Note that the condition (ii) requires

28

that the state-space form solution of the prototype model must exists in the extended

29

detailed model. 10

1

We investigate these problems through a formal discussion of the linearized economy

2

in the following two sections.7 In Section 3, we consider the case of the VAR(1) spec-

3

ification of wedges. We extend our analysis to an alternative specification which has

4

the VAR(1) specification as a special case and provide an example the condition (ii) in

5

Proposition 1 does not hold even if the condition (i) holds in Section 4.

6

3

7

3.1

8

In the case of CKM’s VAR(1) specification (11), the condition (i) is the existence of

VAR(1) specification of wedges Conditions for equivalence

sˆt+1 = P sˆt + εt+1 ,

(22)

9

in the extended detailed model. Since, in the case of VAR(1) specification of wedges,

10

[kˆt−1 , sˆ′t ]′ are the state variables in the prototype model, the condition (ii) is the existence

11

of the mapping from [kˆt−1 , sˆ′t ]′ to consumption, investment, labor, output and capital

12

stock at the end of period:

          

 cˆt

   ˆit    ˆt−1 k  , ℓˆt  = Λ   s ˆ t  yˆt   kˆt

(23)

13

where Λ is a 5 × (n + 1) matrix, in the extended detailed model. Note that we do not

14

need to consider the existence of the mapping from [kˆt−1 , sˆ′t ]′ to sˆt+1 if the VAR(1)

15

specification of wedges (22) exists. 7

We only consider a linearized economy. However, results in this paper hold in a non-linear economy in the neighborhood of a steady-state.

11

1

First, we consider the existence of the VAR(1) specification of wedges. The necessary

2

and sufficient condition for the existence of the VAR(1) representation is as follows.

3

Lemma 1. Assume that (20) and (21) hold in the extended detailed model. There exists

4

P that satisfies CKM’s specification (22) if and only if

([ rank(Φ) = rank

5

6

. Φ .. Ψ′ Φ′ ′

]) .

(24)

Proof. See Appendix A. If rank(Φ) = n, the following Lemma 1 holds and provides us with some intuitions

7

with regard to the existence of P .

8

Lemma 2. Assume that (20) and (21) hold in the extended detailed model. If the rank

9

of Φ equals the number of endogenous and exogenous states in the detailed model, n,

10

there exists P that satisfies CKM’s specification (22).

11

Proof. Since rank(Φ) = n, there exists (Φ′ Φ)−1 . Then, (20) becomes 

 (Φ′ Φ)−1 Φ′ sˆt = 

xt−1

.

(25)

zt 12

By (25) and (21), we obtain  sˆt+1 = ΦΨ(Φ′ Φ)−1 Φ′ sˆt + Φ 

 0

.

(26)

ut+1 13

14

Therefore, there exists a VAR(1) representation of wedges (22) where P = ΦΨ(Φ′ Φ)−1 Φ′ and εt+1 = Φ[0, u′t+1 ]′ .

15

Lemma 2 can be interpreted as follows. As seen in (25), rank(Φ) = n implies that

16

[x′t−1 z ′t ]′ are identified by sˆt if we know Φ. In other words, wedges sˆt has sufficient 12

1

information with regard to the endogenous and exogenous states [x′t−1 z ′t ]′ , and such

2

wedges can be written as a VAR(1) process. Intuitively, the number of (linearly indepen-

3

dent) wedges should be that of (linearly independent) endogenous and exogenous state

4

variables in the detailed model at least in order to let the prototype model cover the de-

5

tailed model. It is easily shown that, in general, P does not exist if rank(Φ) < n. If the

6

number of wedges is strictly smaller than that of the endogenous and exogenous states in

7

the detailed model or n > m, wedges cannot contain adequate information regarding the

8

endogenous and exogenous states in general. Then, there is no VAR(1) representation

9

of wedges. Even if the number of wedges are larger than that of the endogenous and

10

exogenous states in the detailed model or m ≥ n, and if rank(Φ) < n, it implies that

11

wedges do not have sufficient information about the endogenous and exogenous states in

12

the detailed model and there is no VAR(1) representation in general.

13

˜ to be Next, we consider the existence of (23). Let a (m + 1) × n matrix Φ 

 e

˜ = Φ

,

(27)

Φ 14

where e = [1, 0, 0, · · · , 0] is a 1 × n vector and it satisfies  

kˆt−1





˜ =Φ

sˆt

 xt−1

.

(28)

zt

15

The necessary and sufficient condition for the existence of (23) is as follows.

16

Lemma 3. Assume that (20) and (28) hold in the extended detailed model. There exists

17

Λ that satisfies (23) if and only if

([ ˜ = rank rank(Φ)

13

˜ ′ ... Θ′ Φ

]) .

(29)

1

2

Proof. See Appendix B. ˜ = n, the following Lemma 4 holds and provides us with some intuitions If rank(Φ)

3

with regard to the existence of Λ.

4

Lemma 4. Assume that (19) and (20) hold in the extended detailed model. If the rank

5

˜ equals the number of endogenous and exogenous states in the detailed model, n, of Φ

6

there exists Λ that satisfies (23).

7

˜ −1 and ˜ = n, there exists (Φ ˜ ′ Φ) Proof. Since rank(Φ)  

 xt−1 zt

8

  ) −1 k ˜ ′Φ ˜ ˜ ′  t−1  . = Φ Φ st (

(30)

( ′ )−1 ′ ˜Φ ˜ ˜ is a solution. By (19) and (30), it is easily shown that Λ = Θ Φ Φ

9

˜ < n as in the case of It is easily shown that, in general, Λ does not exist if rank(Φ)

10

P . Roughly speaking, if the number of wedges plus one is strictly smaller than that of

11

the endogenous and exogenous states in the detailed model or n > m + 1, wedges cannot

12

contain adequate information regarding the endogenous and exogenous states in general.

13

Then, there is no mapping from [kˆt−1 , sˆ′t ]′ to consumption, investment, labor, output,

14

and capital stock.

15

Finally, the necessary and sufficient condition for the equivalence is summarized as

16

follow.

17

Theorem 1. Assume that (19), (20), (21), and (28) hold in the extended detailed model.

18

The detailed model is equivalent to the prototype model with the VAR(1) specification of

19

wedges if and only if

([

]) . Φ′ .. Ψ′ Φ′ , ([ ]) ′ .. ′ ˜ ˜ rank(Φ) = rank Φ .Θ . rank(Φ) = rank

14

1

2

Proof. It is obvious by Lemma 1 and 3. The following sufficient condition is useful to understand the intuition.

3

Theorem 2. Assume that (19), (20), (21), and (28) hold in the extended detailed model.

4

The detailed model is equivalent to the prototype model with the VAR(1) specification of

5

wedges if the rank of Φ equals the number of endogenous and exogenous states in the

6

detailed model, n.

7

˜ = n. Proof. It is obvious by Lemma 2, 4 and that rank(Φ) = n implies rank(Φ)

8

Theorem 2 implies the sufficient condition of the existence of the VAR(1) representa-

9

tion of wedges rank(Φ) = n is the sufficient conditions of the existence of the mapping

10

from the states of the prototype model to consumption, investment, labor, capital, and

11

output in the extended detailed model. A detailed model is equivalent to the prototype

12

model if the VAR(1) representation of wedges exists in many cases. This tells us that,

13

in the case of VAR(1) of wedges, the condition (ii) in Proposition 1 is less restrictive.

14

3.2

15

We showed that the VAR(1) representation of wedges exists if and only if wedges have

16

sufficient information about the endogenous and exogenous states in the detailed model

17

and that equivalence holds in many cases if the VAR(1) representation exists. We can

18

roughly verify the condition for the existence of P by comparing the number of wedges

19

m and that of endogenous and exogenous states n of the detailed model. If rank(Φ) < n,

20

there is no P in general.

Implications of our results

21

If the detailed model is a medium-scale DSGE model `a la CEE, which has many

22

exogenous shocks and endogenous states, the equivalence results do not hold in many

23

cases since the maximum number of wedges is at most four. Such a model is often

24

used for business cycle analysis since it can capture many aspects of business cycle facts.

25

However, such a “realistic” model cannot be covered since there are many endogenous

26

and exogenous variables and this is a serious problem when we apply business cycle

27

accounting to the actual data. 15

1

Even if the number of endogenous and exogenous states is small, a VAR(1) represen-

2

tation might not exist. Here, we show that the class of frictional models covered by the

3

prototype model with the VAR(1) specification of wedges is much smaller than that is

4

shown by CKM under unrestricted general class of stochastic process of wedges.

5

The sticky wage model of CKM is not covered by the associated prototype model. In

6

their sticky wage model, there are monopolistically competitive households (indexed on

7

the unit interval by h), each of which supplies a differentiated labor service ℓt (h) to the

8

production sector. Households set nominal wage rate Wt (h) in the previous period and

9

the labor demand function for ℓt (h) is

[

Wt−1 ℓt (h) = Wt−1 (h)

]

1 1−ν

ℓt ,

(31)

10

where ℓt denotes aggregate labor demand and Wt denotes aggregate nominal wage rate.

11

Monetary authority follows money growth rule:

Mt = µt Mt−1 ,

(32)

µt = ρµ µt−1 + εµt ,

(33)

12

where Mt denotes nominal money supply; µt , growth rate of money supply; and εµt , i.i.d.

13

money supply shock. As shown in Appendix C, in this model, the endogenous states are

14

aggregate capital kt−1 , real money balance mt−1 ≡ Mt−1 /Pt−1 , and real wage rate wt−1 ≡

15

Wt−1 /Pt−1 and the exogenous state is money growth rate µt . In the associated prototype

16

model, there is only one wedge – labor wedge, or rank(Φ) = 1. Then, according to

17

Lemma 1, a VAR(1) representation does not exist in general since the number of wedges

18

is strict smaller than that of endogenous and exogenous variables: m < n.8 The input-financing friction model of CKM is also not covered by the associated

19

8

Rigorously speaking, the rank of Ψ in the case of sticky wage model is three, not four, since real wage rate and real money are linearly dependent. This is easily verified by numerical simulations.

16

1

prototype model. The endogenous state is aggregate capital kt−1 , and the exogenous

2

states are the interest rate spreads of sector 1 and 2: τ1,t and τ2,t in the model with

3

input-financing friction.9 In the associated prototype model, there are three wedges –

4

efficiency, labor, and capital wedges. Subsequently, in this case, n = m in this case;

5

however CKM’s Proposition 1 shows that τℓ,t equals τk,t . This implies that 

 st = Φ   ⇐⇒

   

 log(At ) τℓ,t τk,t

xt−1



zt



ϕ1,1 ϕ1,2 ϕ1,3

     =  ϕ2,1 ϕ2,2 ϕ2,3   ϕ3,1 ϕ3,2 ϕ3,3



 kt−1

    τ1,t  τ2,t

  , 

(34)

6

where ϕ2,i = ϕ3,i for i = 1, 2, and 3. It is obvious that rank(Φ) = 2 < 3 = n in general,

7

and then there is no VAR(1) representation.10 Finally, these results imply that equivalence result is highly restrictive if we employ

8

9

the prototype model with the VAR(1) specification of wedges.

10

4

Alternative specification of wedges

11

In Section 3, we consider the case of the VAR(1) specification of wedges. Here, we

12

consider an alternative specification of wedges, which lets the prototype model cover

13

larger class of frictional models than that in the case of the VAR(1) representation. The

14

main message of this section is that the prototype model with our alternative specification

15

can cover larger class of models but such class is not so large. We also provide an example

16

where the condition (ii) in Proposition 1 does not hold even if the condition (i) holds in

17

this section. Contrary to this, in the case of VAR(1) specification of wedge, the condition

18

(i) in Proposition 1 implies the condition (ii) in many cases. 9

CKM do not specify what are exogenous shocks. Here, we consider two interest rate spreads are exogenous shocks to close the model. 10 This non equivalence arises since the prototype model is described by capital wedge as in (13). The prototype model with investment wedge is equivalent to this input-financing friction model.

17

1

2

Let π t is a r × 1 vector of variables included in both the prototype model and the detailed model such that  πt = Ξ 

 xt−1

,

(35)

zt 3

where Ξ is an r ×n matrix. Then, a candidate of variables included in π t is jump or state

4

variables at period t in the extended detailed model. Consider the following alternative

5

specification:

st+1 = R0 + Rπ t + η t+1 ,

(36)

6

where R0 is an m × 1 vector, R is an m × r matrix, and η t+1 is i.i.d. over time and is

7

normally distributed with mean zero. Note that the VAR(1) specification of wedges (11)

8

is a special case of (36) where π t = st . The linearized version of (36) is

sˆt+1 = Rˆ π t + η t+1 .

9

(37)

Even if we do not specify variables in π t , we can discuss about the existence of (37) by

10

(35). The necessary and sufficient condition is as follows.

11

Lemma 5. Assume that (20), (21) and (35) hold in the extended detailed model. There

12

exists R that satisfies our specification (37) if and only if

([ rank(Ξ) = rank

. Ξ .. Ψ′ Φ′ ′

]) .

(38)

13

Proof. It is easily shown that there exists R that satisfies RΞ = ΨΦ if and only if (38)

14

holds. The rest of the proof is almost similar to that in the proof of Lemma 1. Necessity 18

1

2

and sufficiency are easily shown. Lemma 5 is the analogue of Lemma 1. The analogue of Lemma 2 is as follows.

3

Lemma 6. Assume that (20), (21), and (35) hold in the extended detailed model. If the

4

rank of Ξ equals the number of endogenous and exogenous states in the detailed model,

5

n, there exists a unique R that satisfies the alternative specification (37).

6

Proof. It is easily shown by the same logic in the proof of Lemma 2.

7

These lemmas imply that our specification (36) holds if and only if π t has sufficient

8

information about the endogenous and exogenous states in the detailed model. Roughly

9

speaking, if the number of πt is larger than that of the endogenous and exogenous states

10

11

in the detailed model, there exists (36). The rest is the condition (ii) in Proposition 2. To consider it, we have to specify π t .

12

Specification of B¨ aurle and Burren (2007):

13

is as follows.  πt ≡ 

A candidate of the specification of π t

 k t−1

.

(39)

st 14

where k t−1 is the “aggregate” capital stock and k t−1 = kt−1 in equilibrium. This “aggre-

15

gate” trick is employed in order to let the first order conditions of the prototype model to

16

be same (6), (7), (8), (9), and (10). This specification of π t is the same as that proposed

17

in B¨aurle and Burren (2007).

18

Since, under the specification (37), the state variables of the prototype model is

19

[kˆt−1 , sˆ′t ]′ and the condition (ii) in Proposition 1 is the same as (23). Therefore, Lemma

20

3 and 4 is applicable. In the case of this alternative specification, the necessary and

21

sufficient condition for the equivalence is summarized as follow.

22

Theorem 3. Assume that (19), (20), (21), and (28) hold in the extended detailed model.

23

The detailed model is equivalent to the prototype model with the specification of wedges: 19

st+1 = R0 + Rπ t + η t+1 ,

1

where 

 πt ≡ 

k t−1



st 2

if and only if

([

]) . Ξ′ .. Ψ′ Φ′ , ([ ]) ′ .. ′ ˜ ˜ rank(Φ) = rank Φ .Θ .

rank(Ξ) = rank

3

(40) (41)

Proof. It is obvious by Lemma 3 and 5.

4

˜ and the sufficient condition for the existence In this case of the specification, Ξ = Φ

5

of the alternative specification (37), Ξ, implies the condition (23): the mapping from the

6

states of the prototype model to variables. Thus, the equivalence holds if the existence

7

of the alternative specification (37) exists in many cases.

8

Since there are m + 1 variables in π t , this alternative specification exists in the case

9

where the number of endogenous and exogenous states is smaller than m+1 in the detailed

10

model. It is easily verified that the prototype model with this specification covers the

11

input-financing friction model shown in Section 3. However, CKM’s sticky wage model

12

is not still covered even if we employ this specification. Note that the prototype model

13

with this specification, generally, can only cover a detailed model in which the number

14

of the state variables is less than five.

20

1

Even if we augment π t ...

2

For example,

Consider the case where there are more variables in π t .



 k t−1

  π t ≡  st  ct

  , 

(42)

3

where ct is aggregate consumption and ct = ct in equilibrium. We augment the previous

4

specification of π t by ct .

5

This alternative specification exists even if the number of the endogenous and exoge-

6

nous wedges is six, which is the specification of B¨aurle and Burren (2007) does not exist.

7

However, the class of detailed models covered by the prototype model with this specifi-

8

cation is not larger than that covered by the prototype model with the specification of

9

B¨aurle and Burren (2007) since the condition (ii) in Proposition 1 is restrictive. In this

10

specification, the condition (ii) in Proposition 1 is the same as (23). It is described as in

11

Proposition 2.

12

Proposition 2. If a detailed model is covered by the prototype model with the alternative

13

specification (36) and π t ≡ [k t−1 , s′t , ct ]′ , it is also covered by the prototype model with

14

(36) and π t ≡ [k t−1 , s′t ]′ .

15

Proof. Assume that a detailed model is covered by the prototype model with (37) and

16

(42). Then, (23) and 

kˆt−1

  sˆt+1 = R  sˆt  cˆt 17

    + ηt 

hold. Since there is a mapping from [kˆt−1 , sˆ′t ]′ to cˆt , (43) is rewritten as

21

(43)

 ˜ sˆt+1 = R

kˆt−1 sˆt

  + ηt.

(44)

1

(23) and (44) imply that this detailed model is covered by the prototype model with (37)

2

and (42).

3

The analogue of Proposition 2 holds if we augment π t by other variables in the

4

prototype model. Therefore, generally, the number linearly independent endogenous and

5

exogenous variables in detailed models covered by the prototype model with (36) is less

6

than six and the class of detailed models is not so large.11

7

5

Business cycle accounting in an economy where the equivalence does not hold

8

9

As shown in previous two sections, the class of models covered by the prototype model

10

with the VAR(1) specification is small. However, there is a possibility that the proto-

11

type model works well as an approximation of the detailed model. In this section, we

12

investigate what happens if we apply BCA to an economy where the equivalence result

13

does not hold in order to assess the usefulness of BCA.12

14

5.1

15

We employ a kind of medium-scale DSGE model `a la CEE.13 The reason why we employ

16

this medium-scale DSGE model is that such model has rich enough dynamics to produce

17

the actual tendency found in the data and we already know the empirically plausible

An artificial economy

11

It is well known that a vector autoregressive moving-average (VARMA) representation exists under more general conditions than VAR(1) by the VAR literature as shown by Ravenna (2007) and it would be a solution of this theoretical problem. 12 In the literature of the structural VARs, Ravenna (2007) and Christiano, Eichenbaum, and Vigfusson (2006) investigate the empirical usefulness of VARs by applying VARs to artificial economies. 13 The details about our model in Appendix D.

22

1

range of parameters in this sort of models. So, using this model, we investigate the

2

empirical usefulness of BCA under plausible parameter values in a realistic economy.

3

Our model has the following properties: (i) sticky price with backward price indexa-

4

tion, (ii) sticky wage with backward price indexation, (iii) habit persistence in preference,

5

(iv) flow adjustment costs of investment, and (v) forward-looking Taylor rule. In our

6

model, there are eight linearly independent endogenous and exogenous state variables

7

and both prototype models with VAR(1) specification and alternative specification with

8

π t ≡ [kˆt−1 , sˆ′t ]′ cannot cover this model. The parameter values of our medium-scale

9

DSGE model are described in Table 1.

10

[Insert Table 1]

11

We specify the model to be quarterly and most values are taken from Christiano, Ilut,

12

Motto, and Rostagno (2007). We employ the log-linear approximation to calculate the

13

aggregate decision rule and generate 10,000 periods artificial data in order to avoid the

14

small-sample bias.

15

5.2

16

Our procedure of BCA follows the standard way. We apply BCA to our medium-scale

17

DSGE economy using the prototype model with the VAR(1) specification of wedges.

18

We set the parameter values of the prototype model as follows: discount factor β, de-

19

preciation rate of capital δ, and share of capital in production α to be the same as in

20

our detailed model, and the weight of leisure in utility ν to be 2.14 Note that there is

21

no adjustment costs of investment in our prototype model.15 We estimate the process

22

of wedge P by maximum likelihood based on the Kalman filter. Table 2 summarizes

23

estimation results.

Business cycle accounting

[Insert Table 2]

24

14

We do so since the utility function in our detailed model is different from that of the prototype model. 15 The adjustment costs of investment do not affect the evolution of capital (9) in the linearized economy because of quadratic costs. The adjustment costs only affect investment wedge.

23

1

Given the evolution of wedges P , wedges are measured by 



    ˆt−1 k  ˆd  ˜  ,  ℓt  = Λ   sˆt yˆtd ˆidt

(45)

2

d , and g ˆt = gˆtd , where variables with superscript d denote kˆt = (1 − δ)kˆt−1 + δˆidt , kˆ−1 = kˆ−1

3

˜ is a 3×5 matrix. There are three equations and three unknown wedges actual data and Λ

4

– At , τℓ,t , and τx,t , in (45) and wedges are measured as solution. One possible problem of non equivalence is mismeasurement of wedges. Figure 1

5

6

shows the true and measured wedges.16

7

[Insert Figure 1]

8

The true wedges are generated by the extended detailed model. The estimated process of

9

wedge P affects only the measurement of the investment wedge and others are measured

10

correctly by the intratemporal conditions of the prototype model (6), (8), and (10).

11

The measured investment wedge looks to be close to the true one. Table 3 reports

12

the cyclical behavior of the true and measured investment wedges: means, standard

13

deviations, autocorrelations, correlations with current output, and correlations the true

14

current investment wedge.

15

[Insert Table 3]

16

The true investment wedge is slightly more volatile and persistent than the measured

17

one and the correlation between the true investment wedge and output is slightly larger

18

than that between the measured and output. However, differences are quite small and

19

BCA works well in measurement of wedges.

20

The other possible problem is about predictions of counterfactual simulations: wedge

21

decompositions. In wedge decompositions, counterfactual sequences of wedges are con-

22

structed as follows. For example, in order to investigate the contribution of efficiency 16

We only show the data of first 200 periods for the viewability. The complete outputs will be available from authors upon request.

24

17

1

wedge, efficiency wedge is the same as measured and others are constants over time.

2

Mismeasurement of P might affect wedge decompositions of all wedges, not only invest-

3

ment wedge, since the aggregate decision rule depends on estimated value of P . Figure

4

2 shows output decomposition by each wedge.

5

[Insert Figure 2]

6

The dashed-dotted lines are actual data of output. The solid line are output predictions

7

by true wedge, and the bold solid line are output predictions by measured wedges. It is

8

easily verified that both output predictions by the true and measured wedges are very

9

close. Then, biases in wedge decompositions are also quite small.

10

Finally, BCA works well even in an economy where the equivalence does not hold.

11

Of course, this is just an example. However, we think that it is important that BCA

12

works well in a “realistic” model like CEE.

13

6

14

Business cycle accounting is premised upon he insight that the prototype neoclassical

15

growth model with time-varying wedges can achieve the same allocation generated by a

16

large class of frictional models: equivalence results. Equivalence results are shown under

17

unrestricted general class of stochastic process of wedges while it is often specified to be

18

the first order vector autoregressive when business cycle accounting is applied to actual

19

data.

Conclusion

20

In this paper, we characterized the class of frictional detailed models that are covered

21

by the prototype model under the conventional specification and found that the class of

22

models is much smaller than that believed in previous literature. Sticky wage model and

23

input-financing friction model are not covered by the prototype model and moreover,

24

models that have many state variables are not covered. We also show that even if

25

we employ alternative specification of the process of wedges, the class covered by the 17

There are some decomposition methods and we employ the same method of CKM, here. The details are described by Chari, Kehoe, and McGrattan (2007b).

25

1

prototype model is not so large. In order to investigate the empirical usefulness of

2

BCA, we applied BCA to an artificial medium-scale DSGE economy `a la CEE where

3

the equivalence does not hold and find that BCA works well even in such an economy.

4

This result tells us that equivalence results is highly restrictive from the theoretical view.

5

However, BCA works well empirically since the prototype model is a good approximation

6

of the detailed model.

7

Appendix A: Proof of Lemma 1

8

Proof. There exist P that satisfies ΦΨ = P Φ if and only if (24) holds according to the

9

standard knowledge of linear algebra. In the rest of the proof, we elaborate on sufficiency

10

and then subsequently, on necessity.

11

(Sufficiency) Assume that there exists P that satisfies ΦΨ = P Φ. By (20), at period

12

t + 1,  st+1 = Φ 

 xt

 z t+1    xt−1 0  + Φ = ΦΨ  zt ut+1    xt−1 0  + Φ = PΦ ut+1 zt = P st + εt+1 ,

13

    (46)

where  εt+1 ≡ Φ 

 0 ut+1

26

.

(47)

1

(Necessity) Assume that there exists P that satisfies CKM’s specification (22). (22) and

2

(20) imply that 

 xt

Φ



 = PΦ

z t+1 3

 + εt+1 .

(48)

zt

(21) and (48) imply that  ΦΨ 

 xt−1 zt

4

 xt−1



 + Φ

 0



 = PΦ

ut+1

 xt−1

 + εt+1 .

(49)

zt

Since Et [εt+1 ] = 0 and (49) must hold for any xt−1 , z t and ut+1 , (47) and

ΦΨ = P Φ

(50)

5

hold.

6

Appendix B: Proof of Lemma 3

7

˜ = Θ if and only if (38) holds. In the rest of the Proof. There exists Λ that satisfies ΛΦ

8

proof, we elaborate on sufficiency and then subsequently, on necessity.

9

˜ = Θ. (19) becomes (Sufficiency) Assume that there exists Λ that satisfies ΛΦ

27



          

1

cˆt

   ˆit    x  ˜  t−1  ℓˆt  = ΛΦ  zt  yˆt   kˆt   kˆt−1 . = Λ sˆt

(51)

(52)

(Necessity) Assume that (23). By (19) and (28),

(

)



˜ −Θ  ΛΦ

 xt−1

 = 0.

(53)

zt 2

˜ = Θ must hold. Therefore, ΛΦ

3

Appendix C: Sticky wage model

4

Here, we briefly explain a version of the sticky wage model of CKM. In CKM’s sticky

5

wage economy, households belong to monopolistically competitive unions that decide

6

nominal wage in the one-period advance. Firms are perfectly competitive and monetary

7

authority follows money growth rule.

8

Households in union j solve

max

{ct (j),Mt (j),Wt (j)}∞ t=0

Et

∞ ∑

[ β

t

(

) ( ) log ct (j) + γℓ log 1 − ℓt (j) + γM log

t=0

Mt (j) Wt−1 (j) Mt−1 (j) ≤ ℓt (j) + + Tt , Pt Pt Pt [ ]1−ν Wt ℓt (j) = ℓt , Wt−1 (j)

s.t. ct (j) +

28

(

Mt (j) Pt

)] , (54) (55) (56)

1

where ct (j) denotes consumption; Mt (j), nominal money of j at the end of period; Wt (j),

2

nominal wage rate offered by j; Wt , aggregate nominal wage; Wt (j), labor demand for j;

3

ℓt , aggregate labor demand; Pt , aggregate price level; and Tt , lump-sum transfer. For the

4

simplicity, the functional form of the utility function is specified and we ignore nominal

5

bond.

6

Firms problem is

max

{kt ,ℓt ,xt }∞ t=0

∞ ∑

[ Λt

t=0

] Wt ℓt − xt , yt − Pt

s.t. kt = (1 − δ)kt−1 + xt , α yt = kt−1 ℓ1−α . t

7

where kt is capital stock at the end of period; xt , investment; yt , output; and Λt ≡

8

where λt β t is the Lagrange multiplier of household’s budget constraint.

9

(57) (58) (59) λt t λ0 β

The monetary authority follows the money growth rule:

Mt = µt Mt−1 ,

(60)

µt = ρµ µt−1 + (1 − ρµ )¯ µ + εµt ,

(61)

10

where µt is the money growth rate; µ ¯, steady-state money growth; εµt , money growth

11

shock. The stochastic process of growth rate is specified to AR(1) following to the

12

standard literature.

13

The symmetric equilibrium system of this economy is as follows:

29

[ ] 1 1 Pt = γM + βEt , ct Mt /Pt Pt+1 ct+1 [ ] ℓt+1 γℓ Et 1−ℓ t [ ], Wt = ℓt+1 νEt Pt+1 ct+1 [ ( )] 1 yt+1 1 = Et 1−δ+α , ct ct+1 kt+1 yt Wt−1 = (1 − α) , Pt ℓt

(62)

(63)

(64) (65)

α yt = kt−1 ℓ1−α , t

(66)

ct + kt = (1 − δ)kt−1 + yt ,

(67)

Mt = µt Mt−1 ,

(68)

µt = ρµ µt−1 + (1 − ρµ )¯ µ + εµt .

(69)

1

Since nominal variables, Wt , Mt and Pt , are growing in this economy, we detrend the

2

system as follows:

[ ] 1 1 1 = γM + βEt , ct mt πt+1 ct+1 [ ] ℓt+1 γℓ Et 1−ℓ t [ ], wt = ℓt+1 νEt πt+1 ct+1 [ ( )] 1 1 yt+1 = Et 1−δ+α , ct ct+1 kt+1 wt−1 yt = (1 − α) , πt ℓt

(70)

(71)

(72) (73)

α yt = kt−1 ℓ1−α , t

(74)

ct + kt = (1 − δ)kt−1 + yt ,

(75)

mt πt = µt mt−1 ,

(76)

µt = ρµ µt−1 + (1 − ρµ )¯ µ + εµt ,

(77)

30

1

where wt ≡

Wt Pt ,

mt ≡

Mt Pt ,

and πt ≡

Pt Pt−1 .

2

By the detrended equilibrium system, it is obvious that the endogenous state variables

3

are real wage wt , real money mt , and capital stock kt . (since wt−1 , mt−1 , and kt−1 show

4

up.) The exogenous variable is only money growth µt .

5

Appendix D: Our medium-scale DSGE model

6

In this appendix, we present our medium-scale DSGE model used in the investigation of

7

BCA. Our model has the following properties as in CEE: (i) sticky price with backward

8

price indexation, (ii) sticky wage with backward price indexation, (iii) habit persistence,

9

(iv) flow adjustment costs of investment, and (iv) forward-looking Taylor rule. We employ

10

the Rotemberg adjustment costs for sticky price and wage, not Calvo-pricing. However,

11

this does not matter since both specification is approximately equivalent in a linearized

12

economy.18 There are eight independent endogenous and exogenous state variables.19

13

Therefore, Theorem 1 does not hold and the prototype model cannot cover it.

14

D.1 Firm

15

There are competitive final-goods firms and monopolistic competitive intermediate-goods

16

firms. The production function of final-goods firms is as follows:

[∫ yt =

θp

1

Yt (z)

1−θp θp

] 1−θp dz

.

(78)

0

17

The profit maximization of final-goods firms implies demand function of intermediate

18

goods indexed by z:

18

We employ the log-linear approximation to calculate the aggregate decision rule. The six endogenous states are consumption c, capital k, nominal interest rate R, investment i, inflation rate π, and wage rate w and the three exogenous states are technology A, government consumption g, and monetary policy shock εR t . However, nominal interest rate is not independent from monetary policy shock because of the Taylor rule. 19

31

( Yt (z) =

1

Pt (z) Pt

)−θp yt .

The production function of intermediate-goods firm indexed by z is

Yt (z) = At Kt (z)α Lt (z)1−α ,

2

(79)

(80)

and the productivity At evolves according to

log(At+1 ) = ρA log(At ) + (1 − ρA ) log(A) + σ A εA t+1 .

(81)

3

where A denotes the steady-state of At ; σ A , the standard deviation of technology shocks;

4

and εA t+1 , i.i.d. shock with standard normal distribution. The cost minimization of

5

intermediate-goods firms implies

Yt (z) , Kt (z) Yt (z) wt = (1 − α) · mct · . Lt (z)

rt = α · mct ·

(82) (83)

6

The Rotemberg adjustment costs which is approximately equivalent to the Calvo-pricing

7

with price indexation is

[ /( ) ]2 ϕp Pt (z) 1−ξp ξp π πt−1 − 1 yt , 2 Pt−1 (z)

(84)

8

where πt ≡ Pt /Pt−1 is the gross inflation rate. Finally, intermediate-goods firm indexed

9

by z sets price such that it is a solution to

32

max Pt (z)

∞ ∑ t=0

β

t λt

{(

λ0

Pt (z) Pt

)1−θp

( − mct

Pt (z) Pt

)−θp

[ /( ) ]2 } ϕp Pt (z) 1−ξp ξp − π πt−1 − 1 yt . 2 Pt−1 (z) (85)

1

D.2 Households

2

The utility function with habit persistence of household indexed by i is

U (i) =

∞ ∑

[ β

t

t=0

] ) ℓt (i)σℓ +1 , log ct (i) − bct−1 (i) − ψℓ σℓ + 1 (

(86)

3

where b > 0 means habit persistence and σℓ is the Frisch elasticity. The evolution of

4

capital stock is

σG kt (i) = (1 − δ)kt−1 (i) + xt (i) − 2

(

xt (i) −1 xt−1 (i)

)2 xt (i).

(87)

5

Households have differentiated labor as endowment and they have a power to offer nom-

6

inal wage. The labor demand is

( ℓt (i) =

Wt (i) Wt

)−θw Lt ,

(88)

7

where Wt denotes nominal wage rate. The Rotemberg adjustment costs of nominal wage

8

which is approximately equivalent to the Calvo-pricing with price indexation is

[ /( ) ]2 ϕw Wt (i) Wt 1−ξw ξw π πt−1 − 1 ℓt . 2 Wt−1 (i) Pt 9

Finally, the budget constraint is

33

(89)

Bt (i) Bt−1 (i) = Rt−1 + rt kt−1 (i) Pt Pt { [ /( ) ]2 } Wt (i) ϕw Wt (i) 1−ξw ξw + ℓt (i) 1 − π πt−1 − 1 − Tt + Ωt , Pt 2 Wt−1 (i)

ct (i) + xt (i) +

1

where Tt is transfer from the government and Ωt is profit of firms.

2

D.3 Policy

3

The government consumption gt is AR(1):

log(gt+1 ) = ρg log(gt ) + (1 − ρg ) log(g) + σ g εgt+1 ,

(90)

(91)

4

where g denotes the steady-state of gt ; σ g , standard deviation of fiscal policy shock; and

5

εgt+1 , i.i.d. shock with standard normal distribution. The monetary authority follows the

6

forward looking Taylor rule:

[ { } ] ˆ ˆ Rt = ρR Rt−1 + (1 − ρR ) ρπ Et π ˆt+1 + ρy yˆt + σ R εR t ,

(92)

7

where variables with the hatˆdenotes the log-deviations from the steady-state and ϵR t is

8

monetary policy shock.

9

D.4 Resource constraint

10

The resource constraint is

{ [ /( ) ]2 } ϕp 1−ξp ξp ct + xt + gt = yt 1 − πt π πt−1 − 1 2 [ /( ) ]2 ϕw Wt 1−ξw ξw − πw,t π πt−1 − 1 ℓt , 2 Pt 34

(93)

1

where πw,t = Wt /Wt−1 .

2

References

3

[1] Ahearne, A., Kydland, F., Wynne, M.A., 2006. Ireland’s Great Depression, The

4

5

6

7

8

9

10

11

12

Economic and Social Review 37, 215–243. [2] B¨aurle, G., Burren, D., 2007. A note on business cycle accounting, University of Bern, Discussion Paper 07–05. [3] Bridji S., 2007. Accounting for the French Great Depression, Paper presented The 2007 Meeting of the Society for Economic Dynamics. [4] Chari, V.V., Kehoe, P.J., McGrattan, E.R., 2002. Accounting for the Great Depression, American Economic Review 92, 22–27. [5] Chari, V.V., Kehoe, P.J., McGrattan, E.R., 2007a. Business cycle accounting, Econometrica 75, 781–836.

13

[6] Chari, V.V., Kehoe, P.J., McGrattan, E.R., 2007b. Comparing alternative represen-

14

tations and alternative methodologies in business cycle accounting, Federal Reserve

15

Bank of Minneapolis Staff Report 384.

16

17

[7] Christiano, L.J., Davis, J.M., 2006. Two flaws in business cycle accounting, Federal Reserve Bank of Chicago Working Paper, 2006–10.

18

[8] Christiano, L.J., Eichembaum, M., Evans, C., 2005. Nominal rigidities and the dy-

19

namic effects of a shock to monetary policy, Journal of Political Economy 113, 1–45.

20

[9] Christiano, L.J., Eichembaum, M., Vigfusson, R., 2006. Assessing structural VARs,

21

In: Acemoglu, D., Rogoff, K., Woodford, M. (Eds.), NBER Macroeconomics Annual

22

2006. MIT Press., Cambridge, MA. pp. 1–72.

35

1

[10] Christiano, L.J., Ilut, C., Motto, R., Rostagno, M., 2007. Monetary policy and stock

2

market boom-bust cycles,

3

http://faculty.wcas.northwestern.edu/%7Elchrist/research/ECB/mirage.htm.

4

5

6

7

[11] Kersting, E.K., 2008. The 1980s recession in the UK: A business cycle accounting perspective, Review of Economic Dynamics 11, 179–191. [12] Kobayashi, K., Inaba, M., 2006. Business cycle accounting for the Japanese economy, Japan and the World Economy 18, 418–440.

8

[13] Otsu, K., 2007. A neoclassical analysis of the Asian Crisis: Business cycle accounting

9

of a small open economy, IMES Discussion Paper Series 07-E-16, Bank of Japan.

10

[14] Ravenna, F., 2007. Vector autoregressions and reduced form representations of

11

12

13

DSGE models, Journal of Monetary Economics 54 , 2048–2064. [15] Saijo, H., 2008. The Japanese depression in the interwar period: A general equilibrium analysis, The B.E. Journal of Macroeconomics 8, (Topics), Article 25.

14

[16] Uhlig, H., 1999. A toolkit for analyzing nonlinear dynamic stochastic models easily,

15

In: Marimon, R., Scott, A., (Eds.), Computational Methods for the Study of Dynamic

16

Economies. Oxford University Press, Oxford. pp. 30–61.

36

Figure 1: True and measured wedges Efficiency Wedge

Labor Wedge

1.1

1.05 True BCA

True BCA

1

1.05 0.95 1

0.9 0.85

0.95 0.8 0.9

0.75 0

50

100

150

200

0

Investment Wedge

50

100

150

200

Government Wedge

0.9

0.17 True BCA

0.88

True BCA

0.165

0.86 0.16 0.84 0.155 0.82 0.15

0.8 0.78

0.145 0

50

100

150

200

0

50

100

150

200

Notes: The solid lines are true wedges generated by the extended detailed model. The crosses are measured wedges by the prototype model with the VAR(1) specification of wedges.

37

Figure 2: Output decomposition by true and measured wedges Efficiency Wedge Only Economy

Labor Wedge Only Economy

1.7

1.7

1.65

1.65

1.6

1.6

1.55

1.55

1.5

1.5

1.45

1.45 Data

BCA

True

Data

1.4

BCA

True

1.4 0

50

100

150

200

0

Investment Wedge Only Economy

50

100

150

200

Government Wedge Only Economy

1.7

1.7

1.65

1.65

1.6

1.6

1.55

1.55

1.5

1.5

1.45

1.45 Data

BCA

True

Data

1.4

BCA

True

1.4 0

50

100

150

200

0

50

100

150

200

Notes: The dashed-dotted lines are actual data of output. The bold solid line are output predictions by measured wedges. The solid line are output predictions by true wedge.

38

Table 1: Parameter values of our medium-scale DSGE model parameter discount factor of households Frisch elasticity of labor substitution habit persistence steady-state labor supply capital share in production depreciation rate of capital adjustment costs of investment Rotemberg adjustment cost of price Rotemberg adjustment cost of wage indexation of price indexation of wage persistence of technology level steady-state technology level standard deviation of technology shock persistence of government consumption standard deviation of g shock steady-state ratio of government consumption steady-state gross inflation steady-state markup of price steady-state markup of wage persistence of nominal interest rate weight of inflation in Taylor rule weight of output in Taylor rule standard deviation of monetary policy shock

39

symbol

value

β σℓ b ℓ α δ σG ϕp ϕw ξp ξw ρA A σA ρg σg g/y π

1.01358−.25 1 .63 .3 .4 .025 15.1 27.454 199.0819 .84 .13 .83 1 .01 .83 .01 .1 1 1.2 1.05 .81 1.95 .18 .01

θp θp −1 θw θw −1

ρR ρπ ρy σR

Table 2: Estimation results of the VAR(1) specification of wedges name parameters P (1, 1) P (2, 2) P (3, 3) P (4, 4) P (1, 2) P (1, 3) P (1, 4) P (2, 1) P (2, 3) P (2, 4) P (3, 1) P (3, 2) P (3, 4) P (4, 1) P (4, 2) P (4, 3) log(A) log(1 ( − τℓ))

mean

standard error

1.0256 .4218 1.2414 .9082 .0342 -.0127 .0079 -.7726 .8660 -.0793 -.2524 -.1847 -.0257 -.0492 -.0332 .1348 .0041 -.1483

.1060 .0424 .0359 .0229 .0035 .0120 .0004 .0018 .0035 .0004 .0001 1.8664e-5 .0001 .0002 2.3197e-5 2.1816e-5 1.1136e-6 1.0450e-6

.1818

1.2238e-6

log(g)

-1.8677

2.7913e-7

standard deviations of shocks εA εℓ εx εg

.0092 .0203 .0058 .0101

.1268 .2903 .0831 .0957

correlations of shocks (εA , εℓ ) (εA , εx ) (εA , εg ) (εℓ , εx ) (εℓ , εg ) (εx , εg )

-.7464 .1364 -.0355 .5572 -.0170 -.0974

.0005 .0001 .0018 .0001 .0003 .0001

log

1 1−τx

Notes: P (i, j) is the (i, j) component of P and ε ≡ [εA , εℓ , εx , εg ]′ denotes the error term. The maximum number of the absolute value of eigenvalue of P is .9821.

40

Table 3: Cyclical behavior of true and measured investment wedges

true VAR(1)

mean .8334 .8358

std .0143 .0132

autocorr. .8898 .8774

corr w/ yt -0.9225 -0.8958

corr w/ true 1 .9699

Notes: Means, standard deviations, autocorrelations, correlations with current output, and correlations with the true investment wedge are reported.

41