Discussion Papers In Economics
And Business

Is There a Direct Effect of Money?:
Money’s Role in an Estimated Monetary Business
Cycle Model of the Japanese Economy
Ippei Fujiwara
Discussion Paper 03-15

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIs There a Direct Effect of Money?:
Money’s Role in an Estimated Monetary Business
Cycle Model of the Japanese Economy
Ippei Fujiwara
Discussion Paper 03-15
December 2003
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIs There a Direct Effect of Money?:
Money’s Role in an Estimated Monetary Business
Cycle Model of the Japanese Economy
Ippei Fujiwara∗†
Research and Statistics Department, Bank of Japan, andOsaka University
This Version, October 2003

Abstract
In this paper, I estimate the monetary business cycle model of theJapanese economy by the method advocated by Ireland (2002a), the maximum likelihood estimation of the dynamic stochastic general equilibrium
model in a state-space representation.
The model estimated here includes the direct role of money on output
and inflation so that we could study the alternative transmission mechanism of monetary policy to traditional interest rate channel, which may
even work under the zero nominal interest rate as in Japan now.
However, estimation results report that the direct effect of money isextremely small even if there could be. This finding is consistent with theones obtained for US data in Ireland (2002a) and Euro area in Andres,
Lopez-Salido and Valles (2001).

JEL Classification: C31; E32; E52.
Key words: Direct Role of Money; Cross-Restriction;
Maximum Likelihood Estimation;
Dynamic Stochastic General Equilibrium Model∗ Author’s correspondence: ippei.fujiwara@boj.or.jp† The author would like to thank Peter Ireland (Boston College) for sharing the MATLAB
programme and techinical advices. Further I appreciate the helpful comments from Kosuke
Aoki (Universitat Pompeu Fabra), Kanemi Ban (Osaka University) and semianr participants
at Osaka University. Importantly, the view in this paper should not be taken as those of the
Bank of Japan nor any of its respective monetary policy or other decision making bodies.
Further, any errors is my responsibilibily alone.

1

1

Introduction

Japanese economy has been stagnant since the burst of the bubble economy
around 1990. To respond to those economic downturns, the Bank of Japan
gradually lowered the overnight call rate until it hit a record low in 1996, 0.25%
per annum. However, because the negative shocks were more severe than had
been expected initially; magnified by the balance sheet problem due to the collapse of stockes and land prices, the Bank of Japan and the Japanese government
just by intensifying public investments were not able to settle the economy on
the steady growth path. Then, in 1997, the bankruptcy of the fourth largest
securities company in Japan ”Yamaichi Shoken” ignited financial crises andhit the Japanese economy. Moreover, deflationary pressure become manifest, apressure brought on mainly by cheaper imported goods from China. As a resultof the financial crises and the deflationary pressure, the GDP annual growthrate dropped to 0.7% and the deflation rate per annum measured by the GDP
deflator to 1.6% for the average of last five years. Such numbers were not reallyseen by the industrially advanced countries since the Great Depression.
Although there were almost no room left for stimulating the economy by
way of traditional interest rate channel literally after the introduction of ”zeronominal interest rate policy” in February 1999, the Bank of Japan resumed newmonetary policy scheme called ”quantitative easing” as an alternative deviceof monetary policy to the traditional short-term interest rate control in March
2001. With ”quantitative easing” scheme, the Bank of Japan started to target
not the overnight call rate but the outstanding balance of the current accounts
at the Bank.1
It has been only two years since the introduction of the ”quantitative easing” monetary policy. Therefore, it may not be appropriate to judge the effectof the ”quantitative easing” monetary as the policy may effect with long lags.
However, recent researches with VAR (Vector AutoRegression) models statethat the effect of monetary expansion without traditional interest rate channel
is limited. Kimura, Kobayashi, Muranaga and Ugai (2002) claim that according to the Bayesian VAR with time varying parameters, although there canbe found some positive effect on output and inflation from quantitative easing,it is minuscule. Fujiwara (2003a) concludes with Markov switching VAR thatsince the traditional interest rate channel becomes almost incompetent in the
middle of 1990s, the effects of monetary policy either through lowering nominalinterest rate or monetary expansion become significantly weaker. Furthermore,
the Bank of Japan’s Outlook and Risk Assessment of the Economy and Prices
released in April 2003 assesses the ”quantitative easing” policy as effective but
not satisfactory such that ”As seen so far, the Bank’s ”quantitative easing”
1 In Q&A: New Procedures for Money Market Operations, the Bank of Japan declares”The Bank has conducted market operations based on the guidelines set by the Monetary
Policy Meeting of the Policy Board in terms of the uncollateralized overnight call rate. In
the new procedures, the Monetary Policy Meeting will decide the guidelines in terms of the”outstanding balance” of the current accounts at the BOJ and operations will be conducted
to meet the target balance.”2

policy and ample liquidity provision have contributed to (1) dispelling liquidity
concerns, (2) reducing interest rates including those of longer-term maturities,and (3) shrinking credit spreads. Judging from such developments, the Bank’s
monetary easing seems to have effectively shielded channels through which various shocks lead to liquidity concerns, thereby securing financial market stability,and have contributed to preventing the economy from stumbling into a deflationary spiral. On the other hand, the growth rate of commercial lending has
been negative and bank’s financial intermediary function is still weak. Realeconomic activity has yet to be stimulated.” Thus, the ”quantitating easing”
policy seems to work as a bulwark against further deterioration, but it can beconcluded that its direct effect on output and inflation has been very limited sofar.
Can we expect any direct effects from it in the near future? Several answers
have been reported to this question. Such researches as Orphanides and Wieland(2000), and Coenen and Wieland (2002) conclude that there should be changes
in the risk premium caused by ”quantitative easing” policy as a result of thechanges in the composition of financial assets owned by the country as a whole
and this may bring the depreciation of the yen. These researches insist that
this is an important channel of monetary transmission especially when the zeronominal interest rate bound is binding. This is called the ”portfolio rebalance
effect” of ”quantitative easing” policy. However, the portfolio rebalance effectwill not be considered in this paper as it has already been neatly tested by above
two papers and its effect should be minuscule as the actual data has been telling
us so far.2
Instead, I will examine whether we can find any evidence for the direct effectof money on real economic activities as no major effect cannot be expectedwithout this channel. If the Bank’s expansion of base money results in thegrowing money supply, which has not materialised, the ”quantitative easing”
policy should have some positive influence on the real economy without thetraditional interest channel under the situation where money has some direct
effect on output and inflation. In other words, if we cannot find any evidenceof the direct effect even from money supply, there should almost be no effectfrom the current ”quantitative easing” monetary policy for the present nor in
the future.
Concerning this direct effect of money, Nelson (2002) shows that if the utility
is non-separable in consumption and real money holdings such that




Mt
∂ut
≈ a1 ct + a2 log
,
log
∂ct
Pt
then the optimised IS curve as follows is obtained after log linearisation of
the first order condition.
2 Indeed, Coenen and Wieland (2002) comment that the effect through this portfolio rebalance effect is ”small enough not to be noticeable in times of non-zero interest rates.”

3


 


Mt+1
Mt
a2
yt = b1 rt +log
− Et log+ Et yt+1
a1
Pt
Pt+1
u: utility from consumption and real money holdings, c: consumption,M
P : real money balances, y: output, r: real interest rate, a, b: parameters

The aim of this paper is to examine whether the utility function which therepresentative Japanese consumer owns is well represented by the utility form
with non-separability between consumption and real money holdings as above
and therefore the direct effect of money can be found in the macroeconomic
dynamics of the Japanese economy.
This paper consists as follows: In the next section, the direct effect of money
is tested by the estimation on a reduced form single equation such that the output is estimated by lagged real interest rate and real money balances. Howevereven if the existence of the direct money channel is established in a single estimation framework, it is inappropriate to conclude this is the direct channel on
output and inflation since it may just cover the effect through the traditional LM
curve relationship not fully captured by real interest rate. If this is just covering
the effect through the traditional channel additionally, it can be concluded thatthe monetary expansion has only limited impact on the macroeconomy when
the nominal interest rate is constrained at zero as is happening in Japan. Under this circumstance, we naturally cannot expect any effect from the current”quantitative easing” monetary policy. Therefore, we need a method to identify
the direct effect of money and to estimate it. The seminal research examinedin Ireland (2002a) proposes a method to identify the money’s effect on dynamic
macroeconomy. It constructs an optimised general equilibrium model based on
non-separable utility in consumption and real money holdings for the direct effect of money to be derived from microfoundations, then shows that ”real money
balances enter into a correctly-specified, forward-looking IS curve if and only ifthey enter into a correctly-specified, forward-looking Phillips curve,” and evaluate the direct effect of money balances on output and inflation by maximum
likelihood estimation on the dynamic stochastic general equilibrium (DSGE)
model in state space form.3 In section three and four, the model and its estimation procedure of Ireland (2002a) are summarised. In section five, I firstshow that the direct effect of money has not been found in Japanese data. Further, it is shown that the impulse responses of this estimated DSGE model has
reasonable simulated properties and then simulated autocorrelations, variancedecompositions obtained from an estimated DSGE model of Japanese economy
are similar to those of identified VAR which mimics the actual data tendencies.
Finally, section six concludes with comments on the possible future extension.
3 Recently, constructing the more data-oriented dynamic stochastic general equilbrium
model has become popular. These are well-summarised in Ruge-Murcia (2002).

4

2

Simple Estimates of Real Balance Effect

Nelson (2002) estimates several forms of reduced form IS curve following Koenig(1990) and Rudebusch and Svensson (2000). Here I estimate three equationsof reduced form the IS curve. In the first estimated equation, output gap isestimated by its own lags and lagged real interest rate. Then, lagged real money
balances and real exchange rate are added in the second and third estimatedequation. Number of lags are always set to three.
These three estimated reduced form IS equations take forms as follows, where
ye denotes the output gap measured in Hirose and Kamada (2002), rr real interest
rate measured by LIBOR three month rate4 minus CPI inflation rate,5 rmb realmoney gap6 expressed as its log deviation from its HP filtered value, q realexchange rate7 and a, b, and c are parameters.
IS equation (1)yet = cons tan t+3X

ai yet−i +

i=1

3
X

bi rrt−ii=1

IS equation (2)yet = cons tan t+3X

ai yet−i +

i=1

3
X

bi rrt−i +

i=1

3
X

ci rmbt−ii=1

IS equation (3)yet = cons tan t+

3
X
i=1

ai yet−i +3Xbi rrt−i +

i=1

3
X

ci rmbt−i +

i=1

3
X

di qt−ii=1

The estimated period is set from 1983/Q2 to 1995/Q1. The starting date is
fixed due to the availability of output gap series defined in Hirose and Kamada(2002).8 Estimation is ended at 1995/Q1 as the estimation of DSGE model insection five.9 The estimated results from these three IS equations are as follows.
4 This series is downloaded from IFS, International Financial Statistics published by IMF.
5 Perishables and utility prices are excluded.

6 M2+CD is deflated by CPI excluding perishables and utility prices.

7 Dollar/Yen exchange rate is deflated by Japanese CPI and US CPI.

8 Results do not change significantly if the output gap estimated by other methods is
employed.
9 As the DSGE estimated includes monetary policy rule which determines the nominal interest rate, it is not appropriate to include the periods where Japanese economy is constrainedby zero bound and short-term nominal interest rate is not following the specified policy rule
which should be valid for the whole sample.

5

Table 1: Estimated results of IS equation (1)Estimate Standard Error
constant
0.003
0.004
0.947
0.162
yet−1
yet−2
0.265
0.204
-0.2670.167
yet−3-0.548
0.353
rrt−1
1.012
0.565
rrt−2
-0.559
0.357
rrt−3
2
Adjusted R : 0.87, S.E: 0.01
Table 2: Estimated results of IS equation (2)Estimate Standard Error
constant
0.003
0.001
0.415
0.173
yet−1
0.426
0.166
yet−2
0.279
0.179
yet−3
-0.736
0.308
rrt−1
0.685
0.533
rrt−2
-0.261
0.331
rrt−3
0.502
0.193
rmbt−10.086
0.266
rmbt−20.012
0.172
rmbt−3Adjusted R2 : 0.92, S.E: 0.01

Table 3: Estimated results of IS equation (3)Estimate Standard Error
constant-0.012
0.022
0.383
0.176
yet−1
0.427
0.168
yet−2
0.293
0.184
yet−3
-0.761
0.346
rrt−1
0.671
0.537
rrt−2
-0.261
0.333
rrt−3
0.448
0.200
rmbt−1
0.092
0.268
rmbt−20.051
0.176
rmbt−3
0.010
0.017
qt−1
0.010
0.026
qt−2
-0.0160.019
qt−3
Adjusted R2 : 0.92, S.E: 0.01
6

Above results demonstrate that the coefficients on lagged real money balances are always positive and the first lag of them enters the equation significantly. From these reduced form IS equation, real money balances seem to havesome additional direct effect on output to the traditional interest rate channel,
which effects should be captured by real nominal interest rate terms. Howevereven if the existence of the direct money channel is established in a single estimation on reduced form equation to avoid simultaneity bias, it cannot be an
evidence for the existence of direct effect of money in actual economy due toendogeneity bias etc. Kato (2002) shows the defect of single estimation withsimple but explicit example. It first constructs DSGE model based on Kiyotaki
and Moore (1998), where the debt outstanding of the firm affects the investment decision and therefore aggregate demand, and then collects artificial data
series from stochastic simulation. Simple single estimation of debt outstanding
on investment leads to insignificant coefficient although it should be strongly
significant as the model is set to have a direct effect of debt on investment. This
simple example tells us that a system estimation of the DSGE model where
direct effect of money is explicitly derived from rigid micro-foundations is indispensable for testing the direct effect of money.10 This will be examined in thefollowing sections.

3

Dynamic Stochastic General Equilibrium Modelwith the Direct Effect of Money

I estimate the DSGE model examined in Ireland (2002a) for Japanese data inorder to test if the direct effect of money really exists. In this section, this modelis briefly summarised to understand the money’s role in the estimated monetarybusiness cycle model of the Japanese economy.11
In Ireland (2002a), the economy consists of a representative household, a representative finished goods-producing firm, a continuum of intermediate goodsproducing firms indexed by i ∈ [0, 1], and a monetary authority. A distinct,
perishable intermediate good also indexed by i ∈ [0, 1] are produced by eachintermediate goods-producing firm at t = 0, 1, 2, ....

3.1

Households

The representative household maximises the following expected stream of utility:
E0∞X

β t at {u[ct , (Mt /Pt )/et ] − ηht }

(1)

t=0

subject to the budget constraint as below.
10 Of course, there is risk of mis-specification. However, the DSGE model here estimated isfairlly small and has very standard form.
11 For the detailed derivation of the model and estimation procedure, see Ireland (2002b).

7

Bt /rt + Mt
Mt−1 + Tt + Bt−1 + Wt ht + Dt
= ct +
.
Pt
Pt

(2)

β: subjective discount rate, u [·]: instantaneous utility function, Mt : money,
Pt : price level, Tt : lump-sum nominal transfer from monetary authority,
Bt : bonds, Wt : nominal wage,
ct : finished goods consumption purchased at Pt , rt : nominal interest rate, anda and e are preference shocks which are assumed to follow the stationary
AR processes.
ln(at ) = ρa ln(at−1 ) + εat(3)ln(et ) = (1 − ρe ) ln(e) + ρe ln(et−1 ) + εet ,(4)

1 > ρa > −1, 1 > ρe > −1, εat ˜NID(0, σ a ) and εet ˜NID(0, σ e )h denotes units ofRlabor supplied to each intermediate goods-producing firm
1
for a total of ht = 0 ht (i)di during period t, and D is the nominal profits
received by representative household from each intermediate goods-producing
R1
firm for a total of Dt = 0 Dt (i)di.
As a result of household’s choosing ct , ht , Bt , and Mt to maximise the utilityin (1) subject to the budget constraint in (2), the first order conditions as follows
are obtained.
at∂u= λt ,∂ct(5)

ηat = λt wt ,


λt+1
,
λt = βrt Et
π t+1


λt+1
at ∂ut= λt − βEt,et ∂mt
π t+1

(6)
(7)
(8)

t
and mt = M
λ: Lagrange multiplier on budget constraint, π t+1 = PPt+1
Pt
t

3.2

Firms

3.2.1

Finished Goods

The finished goods-producing firm maximises its profits, given by
Z 1
Pt

0

yt (i)(θ−1)/θ diθ/(θ−1)
−

8

Z 1
0

Pt (i)yt (i)di,(9)

subject to the constant-returns-to-scale technology described by
Z 1
yt =

(θ−1)/θ

yt (i)

0θ/(θ−1)
di

.

(10)

yt : finished goods, yt (·): intermediate goods, andθ: constant elasticity of demand for each intermediate goods
Finished goods-producing firm’s choice of yt (·) to maximise (9) subject tothe constraint (10) leads to the first order condition as follows.

yt (i) =3.2.2

Pt (i)
Pt

−θ
yt ,

(11)

Intermediate Goods

The representative intermediate goods-producing firm competing in monopolistically competitive market maximises its total market value expressed as2 )
Pt (i)
φ− 1 yt ,β λt−E
2 πPt−1 (i)t=0(12)
subject to the constant-returns-to-scale technology described by
∞
X

t

(

Pt (i)Pt1−θ

Pt (i)
yt −
Pt

−θ 

wt yt
zt



yt (i) = zt ht (i).

(13)

twt = W
Pt , π: steady state inflation rate

The aggregate technology shock z denotes the aggregate technology shock
following the stationary autoregressive process.
ln(zt ) = (1 − ρz ) ln(z) + ρz ln(zt−1 ) + εzt(14)

1 > ρz > −1, εzt ˜NID(0, σ z )β t λt in the components of total market value of intermediate goods-producing
firms in (12) measures the marginal utility value to the representative household of an additional dollar in profits received during period t. As the optimisation problem by finished goods-producing firm is expressed as in equation(10), terms within the parenthesis in (12) demonstrate profit defined as revenue
minus costs when the intermediate goods-producing firm must satisfy the representative finished goods-producing firm’s demand. In addition to the wage
cost, the intermediate goods-producing firm faces a quadratic cost of adjusting
i2
h
t (i)
−
1
yt , where φ > 0.
its nominal price given by φ2 πPPt−1(i)9The intermediate goods-producing firms’ optimal choice of Pt (i) is obtainedby maximising (12) subject to (13). The first order condition to this problem isexpressed as


0 =

3.3

−θ  
−θ−1 
Pt (i)yt
Pt (i)
yt wt+ θλt(1 − θ)λt
Pt
Pt
Pt
zt Pt



yt
Pt (i)−1−φλt
πPt−1 (i)πPt−1 (i)
yt+1 Pt+1 (i)Pt+1 (i)−1+βφEt λt+1
πPt (i)πPt (i)2(15)

Central Bank

The monetary authority conducts monetary policy according to an augmented
history dependent Taylor-type monetary policy rule, where lagged value of percentage deviation of nominal interest rate from its steady state is included.
Therefore, the nominal interest rate is adjusted gradually in response to deviations of output and inflation from their steady state values.
 

r
π 
r 
yt
t
t−1
t
= ρr ln+ ρy ln+ εrt ,
(16)+ ρπ lnlnrryπr: steady state gross nominal interest rate, y: steady state output,
εrt ˜NID(0, σ r ), ρr , ρy , ρπ : parameters

3.4

Equilibrium

In a symmetric equilibrium, all intermediate goods-producing firms make iden=
tical decisions, so that yt (i) = yt , ht (i) = ht , Pt (i) = Pt , and dt (i) = DPt (i)
t
Dt
=
d
.
In
addition,themarket-clearing
conditions
M
=
M
+
T
and
t
t
t−1
t
Pt
Bt = Bt−1 = 0 must hold for each period. After imposing these conditions,strait-forward manipulation and log-linear approximation around steady state,
equations (2), (3), (4), (5), (6), (7), (8), (11), (13), (14), (15) and (16) are
written in a compact linear system of equations consisted of eight equations
where variables
with hat denote the log difference from steady state, namely,
 





yt
rt
at
et
t
ŷt = ln y , π̂ t = ln ππt , m̂t = ln m
m , r̂t = ln r , ât = ln a , êt = ln e ,and ẑt = ln zzt .12ât = ρa ât−1 + εat(17)

12 It is worth noting that equation (22) shows that money gap should be negatively correlated

to inflation gap in this DSGE model with microfoundations, contrary to the behavioural
equation based on non-microfounded P* model estimated by Gerlach and Svensson (2001)
which find that real money stock terms enter very significantly as an explanatory variable inaggregate supply equation.

10

êt = ρe êt−1 + εet

ŷt

π̂ t =

π
r

=

(18)

Et ŷt+1 − ω1 (r̂t − Et π̂ t+1 ) + ω2 (m̂t − Et m̂t+1 )−ω 2 (1 − ρe )êt + ω1 (1 − ρa )ât

(19)m̂t = γ 1 ŷt − γ 2 r̂t + γ 3 êt(20)ẑt = ρz ẑt−1 + εzt

(21)


Et π̂ t+1 + ψ

1
ω1




ŷt −

ω2
ω1




m̂t +

ω2
ω1




êt − ẑt ,r̂t = ρr rbt−1 + ρy ŷt + ρπ π̂ t + εrt

(22)
(23)

Steady state values and parameters in equations (17) to (23) are also defined

 
∂u
η
∂y
∂u
θ
13 ∂u
= (r − 1)e ∂u,=
as r = π/β, c = y, r ∂m
∂y
∂y
θ−1
z , ω1 = − y ∂2u ,

 ∂y2


m ∂2 u∂2 u2yrω 2e ∂y∂mr∂m
γ 2 , γ 2 = (r−1), γ3 =
ω2 = − ∂ 2 u , γ 1 = mω1 + r−1
m
∂2 u
∂2u
ω1
y ∂y2

e

(r−1)e ∂y∂m −ru ∂m2

1 − (r − 1)γ 2 and ψ = θ−1
φ .
In correctly specified model from agent’s optimisation behaviour as above,
real money holdings must have the direct effect on output and inflation as inequations (19) and (22) only when ω 2 is positive, namely utility is non-separable
between consumption and real balances. Therefore, the proper evaluation of realbalance effect on economic activities which is not captured fully in short-term
nominal interest rate can only be possible with estimation under this crossrestriction.

4

Estimation Procedure

In this section, the estimation procedure, namely how to conduct maximum
likelihood estimation on the DSGE model in state space form derived above, isintroduced.14
Three optimality conditions: (19), (20) and (22), one decision rule: (23) andthree distribution process for shocks: (17), (18), and (21) are represented as alinear system of equations:
Aft0 = Bs0t + Cvt ,13 This is due to the fact that in steady state c = y.

14 Again, this part is just a brief summary of Ireland (2002b).

11

(24)

0
DEt s0t+1 + F Et ft+1
= Gs0t + Hft0 + Jvt ,

(25)

(26)
vt = P vt−1 + εt ,0where ft0 = m̂t r̂t , s0t = ŷt−1 m̂t−1 π̂ t−1 r̂t−1 ŷt π̂ t , vt =


0
0
ât êt ẑt εrtand εt = εat εet εzt εrt .
Ireland (2002b) demonstrates that following Blanchard and Kahn (1980),
systems of equations (24) to (26) are transformed into state space form as follows:0

st+1 = Πst + W εt+1

f t = U st ,

(27)

0



(28)

where st = ŷt−1 m̂t−1 π̂ t−1 r̂t−1 ât êt ẑt εrt , ft = m̂t r̂t ŷt0and εt = εat εet εzt εrt .
Linear equation system (27) is so-called the state equation while (28) is theobservation equation and therefore vector st is the state vector. As shown in
Hamilton (1994) and Hansen and Sargent (2000), since the system is now represented in state space form and data in vector ft are available, the coefficientscan be obtained by maximum likelihood estimation.
First, conditional expectations, linear projections here and MSE matrix ofstate vector are defined as follows.
ŝt|t−j = E(st |ft−j , ft−j−1 , ..., f1 )(29)

fbt|t−j = E(ft |ft−j , ft−j−1 , ..., f1 )(30)

Σt|t−j = E(st − ŝt|t−j )(st − ŝt|t−j )0(31)

From (27), (29) and (31), the initial vector and MSE matrix with whichrecursion starts can be expressed as follows:ŝ1|0 = Es1 = 0(32)

vec(Σ1|0 ) = vec(Es1 s01 ) = [I − Π ⊗ Π]−1 vec(W V W 0 ).

(33)

V is the variance-covariance matrix of vector εt+1 .
Further from (28) and (30) linear projection of vector ft becomes
fbt|t−1 = U ŝt|t−1 .

(34)

If we define the forecast error as ut , then from (34)ut = ft − fbt|t−1 = U (st − ŝt|t−1 ),
12

(35)

π̂ t

0

with MSE
Eut u0t = U Σt|t−1 U 0 .

(36)

Combining the formula for updating a linear projection with (27), (35) and(36) leads tost+1 − ŝt+1|t = Π(st − ŝt|t−1 ) + W εt+1 − ΠΣt|t−1 U 0 (U Σt|t−1 U 0 )−1 ut ,and therefore
Σt+1|t = W V W 0 + ΠΣt|t−1 Π0 − ΠΣt|t−1 U 0 (U Σt|t−1 U 0 )−1 U Σt|t−1 Π0 .
If we denote ŝt = ŝt|t−1 = E(st |ft−1 , ft−2 , ..., f1 ) and Σt = Σt|t−1 = E(st −ŝt|t−1 )(st − ŝt|t−1 )0 , then the system can be summarised as follows.
ŝt+1 = Πŝt + Kt ut ,(37)

ft = U ŝt + ut ,(38)

ut = ft − E(ft |ft−1 , ft−2 , ..., f1 ),(39)

Eut u0t = U Σt U 0 = Ωt ,(40)

Kt = ΠΣt U 0 (U Σt U 0 )−1 ,(41)

Σt+1 = W V W 0 + ΠΣt Π0 − ΠΣt U 0 (U Σt U 0 )−1 U Σt Π0 .

(42)

where

This system consists of equations (37) to (42) are updated recursively with
the initial condition of ŝ1 and Σ1 provided by (32) and (33).
The innovations {ut }Tt=1 updated as above can then be used to form the log
likelihood function as
ln L = −2T ln(2π) −TT
1 X 0 −1
1X
ln |Ωt | −u Ω ut .
2 t=1
2 t=1 t t

(43)

Parameters are estimated as the values which minimise the log likelihood
function (43).

135Results

In this section, I first estimate the DSGE model which consists of equations (17)and (23) by minimising equation (43). Then, the impulse responses are shown,and the variance decomposition and simulated autocorrelation obtained fromthe DSGE model are compared to those from identified vector autoregression(VAR) model.

5.1

Parameter Estimates

Estimation is conducted using four variables contained in vector ft , namely,
real money balances (mt ), interest rate (rt ), output (yt ) and inflation rate (πt ).
Real money balances are measured by dividing seasonally adjusted M2+CD
by seasonally adjusted CPI excluding perishables and population of age 16 andover, interest rate is LIBOR three month rate, output is seasonally adjusted real
GDP divided by population of age 16 and over, and inflation rate is measured
by changes in seasonally adjusted CPI excluding perishables. All data are onquarterly bases.
As a distinct upward trend is found in real money balances and output
measured as above, two measures are taken in this paper in order to transfrom
the data series stationary.15 First, the HP filter advocated by Hodrick andPrescott (1997) is employed. The estimated series are the ones from which the
HP filtered trend is eliminated. In the second approach, linear the time trend
is excluded so that these series become stationary. The ground for this removal
of upward trend is that the shocks considered in this DSGE model do not affectdeterministic trend which should obey TFP and Labour supply growth rate.
There are no conclusive way to eliminate deterministic trend, and therefore two
simple but quite different methods are employed. If results from both measures
produce similar results, the conclusion can be considered to be very persuasive.
Estimation period is set from 1982/Q1 to 1995/Q1. The starting date ischosen so as to avoid some distortional effects from the second oil shock. As
for the ending date, it is set at around the Bank of Japan resumed the defacto zero nominal interest rate policy. Of course it would be possible to set
the ending date as recent as possible, but this causes some problems: 1) The
Bank of Japan should have changed the implicit monetary policy rule because
there is no more room left to further lower the interest rate since around 1995.16
Therefore, it would be very difficult to have a valid single decision rule for theestimation period here examined.17 2) Concerning this point, recent researcheson the stability of monetary transmission mechanism in Japan, such as Miyao
15 Hamilton (1994) recommends staionarity in data when estimating a state space model.

16 It is true that zero nominal interest rate policy has started literally in Februally 1999 andofficial annoucement of the change in monetary policy regime by the Bank of Japan is madein March 2001 when the Bank has changed the monetary policy to target not the overnight
call rate but base money henceforth, but the room for further easing has almost disappearedin 1995.
17 Examining the ”quantitative easing” policy rule should be intriguing. However, the timeseries data is not long enough to conduct maximum likelihood estimation.

14

(2000) and Fujiwara (2003a) suggest that there is a structural break around 1995
and since then the effect of monetary policy has become weaker than before.
3) Woodford (2002) discusses that even if the utility function is non-separable
in consumption and real balances, there is no direct effect of real balances on
output and inflation when the steady state interest rate is zero. It is true that
almost zero nominal interest rate as in Japan now does not have to imply thatthe steady state interest rate is zero. However, in the process of maximum
likelihood estimation, it is quite possible to obtain the result which tells thatthe steady state interest rate is almost zero. For these reasons, I have chosen
the estimation period from 1982/Q1 to 1995/Q1. Unless any support for the
existence of non-separability of utility from the estimation result from 1982/Q1
to 1995/Q1 could be obtained, we can conclude that there is almost no direct
effect of real balances in Japanese macroeconomic dynamics.
When conducting estimation, some parameters are pre-fixed. As claimed in
Ireland (2001), preliminary attempts to estimate all the parameters in equations (17) and (23) are not successful. This leads to implausibly low ω 1 and ψ.
Since the aim of this paper is to examine the existence of the direct effect ofmoney on real economic activities, I estimate the parameters with constrained
maximum likelihood estimates with ω 1 and ψ pre-fixed as exactly examined in
Ireland (2002a). ω 1 can be considered as the parameter for intertemporal elasticity of substitution. Fujiwara (2003b) estimates this parameter with GMM
and obtained the value of 0.66. As for ψ, this can be considered as the parameter on the output gap in New Keynesian Philiips curve and implicitly contains
the information on how frequently firms can change the price. As for Japanesedata, Kimura and Kurozumi (2002) estimate New Keynesian Phillips curve18
for annual inflation rate and obtains the value of 0.2 for this parameter. Further, several researches on Phillips curve conducted by the Bank of Japan, suchas Price Developments in Japan–A Review Focusing on the 1990s released in
October 2000 which report that this value for annual inflation is around 0.2.
Transforming this into quarterly inflation rate bases with considering that theintertemporal elasticity of substitution is now 0.66, leads to that ψ should be
set at 0.075.
Tables 4 and 5 show the estimation results when the HP filtered and thetime trends are eliminated respectively, where standard errors are computedfrom inverse of Hessian matrix.

18 Although it estimates the hybrid New Keynesian Phillips curve as appeared in Fuhrerand Moore (1995).

15

Table 4: Parameter Estimates when the HP trends are eliminatedParameter
β
ω1
ω2
γ1
γ2
γ3
ψ
ρr
ρy
ρπ
log(y)
log(m)log(π)
log(r)
ρa
ρe
ρz
σa
σe
σz
σr

Estimate
0.99150.66000.01120.00000.00000.99990.07500.81680.0142
0.33628.150310.30180.00410.01260.8172
0.94390.19780.01700.00630.02770.001416

Standard Error
0.00100.06020.00100.04860.0006
0.05740.01970.03110.00250.01210.00070.00130.04820.03560.14540,00290.00060.00450.0001

Table 5: Parameter Estimates when the time trends are eliminatedParameter
β
ω1
ω2
γ1
γ2
γ3
ψ
ρr
ρy
ρπ
log(y)
log(m)log(π)
log(r)
ρa
ρe
ρz
σa
σe
σz
σr

Estimate
0.99170.66000.00170.00000.00000.99990.07500.78690.00030.3266
8.165610.3246
0.00390.0122
0.95440.9874
0.96890.02300.01020.01390.0014Standard Error
0.00590.00970.00220.11590.00140.03370.01420.04730.0262
0.09620.00920.01510.03980.01320.07470,01010.00100.00280.0001

Estimation results show that reasonable values are obtained for all the parameters19 and almost the same results are obtained in both cases. A minor
difference can be found in the persistence of technology shock represented as
ρz . As the time span when the variables are away from their steady state value
are naturally quite different whether the HP or the time trend is considered to
be steady state pass, the parameter which decides this time span, namely thepersistence of shocks are different in the two cases. Technology shock is very
persistent in the time trend case. Other than this minor difference, the size ofthe parameters and their significance are almost the same.
The parameter which is of most interest most naturally ω 2 which decides
the non-separability in consumption and real balances in utility function andtherefore defines the magnitude of direct effect of money holdings on output andinflation. In both cases, this parameter is very small and insignificant. To ensure
this finding further, I examined likelihood ratio test for the null hypothesis H0 :
ω2 =0. Under this null hypothesis, likelihood ratio for the HP trend case is1.3962e-008 and 7.0825e-009 for the time trend case. As the 95% significant
level for the cumulated Chi-square distribution when the degree of freedom is
19 Impulse responses obtained from estimated dynamic stochastic general equilibrium model

here show reasonable properties as understood by standard macroeconomic theory and empirical intuition.

17

set to the number of restrictions, namely one in this case, is 3.75, this null
hypothesis is not rejected at all in both cases. This finding suggests that there
is no direct role of real money balances in Japan, which is consistent with theones obtained for US data in Ireland (2002a) and Euro area in Andres, LopezSalido and Valles (2001). Hence, it is reasonable to assume separable utility
function as examined in the analysis using a DSGE model with money in utility
such as Fukunaga (2002) and Fujiwara (2003b) on the Japanese macroeconomic
dynamics.
Another intriguing finding is that real money balances are following almost
independent dynamics from other macroeconomic variables. Parameters γ 1 , γ 2
and γ 3 which defines the dynamics in the money demand function, or LM curvein other words, indicate that dynamics in real money balances are solely defined by independently distributed shocks. This is consistent with the findingin Miyao (1996), which finds no robust co-integration relationship in money demand function when M2+CD is included,20 and the current informal argumentthat money’s relationship with macroeconomic variables such as inflation andoutput has become less significant.

5.2

Impulse Responses

In this and following subsections, we check whether the estimated DSGE model
has reasonable simulation properties to determine the estimated model is appropriate tool for the analysis on Japan and therefore ensure the above results
for non-existence of direct effect of money.
First, we evaluate the impulse responses of the estimated DSGE model sothat we could judge that the models constructed above are considered to be agood approximation of macroeconomic dynamics of Japanese economy. Figure 1
shows the impulse responses of output, real balances, inflation rate and nominalinterest rate to each shock assigned in the model.

5.2.1

Impulse responses to a preference shock, at

As seen in equations (1) and (19), this preference shock can be considered as ademand shock. Figure 1-a shows the impulse responses to this demand shock.
A demand shock naturally increases the output level and from the Phillips curve
relationship, this results in higher inflation. Against these backgrounds, monetary authority raises nominal interest rate. Although money demand function
is included in this system of equations, the effect from expanding demand andhigher interest rate on real balances is extremely limited.
5.2.2

Impulse responses to a preference shock, etThis preference shock determines the weight of preferences between consumption and real balances as in equation (1) and therefore can be called the money
20 Sekine (1998) finds the cointegration relationship in money demand function with M2+CD
by including wealth as a scale variable.

18demand shock. The larger this preference shock becomes, the more the representative consumer increases the real balances compared to consumption. As
shown in figure 1-b, after this shock, real balances increase. However, although
the direction of the responses of other endogenous variables are consistent withtheory such that increase in real money balances will raise the nominal interest rate and therefore these altogether lead to lower inflation and output, themagnitude of the responses of these three variables are minuscule.
5.2.3

Impulse responses to a technology shock, zt

Figure 1-c demonstrates the impulse responses to a positive technology shock.
A positive technology shock increases the output. However, as the demand does
not catch up to the supply side immediately, output gap widens and this leads
to lower inflation. In response to lower inflation, monetary authority lowers thenominal interest rate. Again, the response of real balances is extremely small,which implies how independent the movements of real balances in Japanesemacroeconomy is.
The difference in the persistence of technology shock represented as ρz results in more graudal adjustment towards steady state in the time trend casecompared to the HP filtered case. This is due to the the fact that the difference in the time span when the variables are away from their steady state value
are naturally quite different depending on whether the HP or the time trend isconsidered to be steady state pass.
5.2.4

Impulse responses to a monetary policy shock, εrt

A shock term included in equation (23) can be considered as a monetary policy
shock as it deviates the nominal interest rate from the level suggested by themonetary policy rule. A positive shock on nominal interest rate lowers theoutput and therefore inflation. Again, the effect on real balances is extremely
small.
Thus far, we have learned that the impulse responses of these two estimated
DSGE model have reasonable properties and the real balances’ dynamics arealmost independently determined from other macroeconomic variables in Japan
before 1995.

5.3

Variance Decomposition

In this subsection, I evaluate the model property from a different perspective,the variance decomposition. The variance decomposition separates the variationin an endogenous variable into the component shocks and therefore, provides information about the relative importance of each random innovation in affecting
the variables employed in the system.
Tables 6 and 7 illustrate the forecast error variance decompositions for output, real balances, inflation rate and nominal interest rate.

19

Table 6: Variance Decomposition on DSGE: the HP filtered case

Output (y)
εe
εz
εr
1y
0.0 8.0 15.5
2y
0.0 8.2 15.4
Inflation (π)
εa
εe
εz
εr
1y 51.9 0.0 37.7 10.5
2y 52.0 0.0 37.5 10.5
εa
76.5
76.4

Real Balances (m)
εa
εe
εz
εr
0.0 100.0 0.0 0.0
0.0 100.0 0.0 0.0
Interest Rate (r)
εa
εe
εz
εr
41.9 0.0 11.2 46.9
59.9 0.0 8.1 32.0

Table 7 Variance Decomposition on DSGE: the time trend case

1y
2y

1y
2y

Output (y)
εa
εe
εz
εr
13.8 0.0 59.7 11.4
8.0 0.0 89.2 2.8
Inflation (π)
εa
εe
εz
εr
74.2 0.0 18.6 7.2
74.7 0.0 19.6 5.7

Real Balances (m)
εa
εe
εz
εr
0.0 100.0 0.0 0.0
0.0 100.0 0.0 0.0
Interest Rate (r)
εa
εe
εz
εr
48.3 0.0 11.8 39.9
66.1 0.0 17.0 17.0

Both the HP filtered and the time trend cases show very similar results.
Money demand shock only help to forecast real balances but they have almost
no predictive power on other endogenous variables. As far as output and inflation are concerned, the demand and technology shocks are the most importantsources of fluctuations. Therefore, these two shocks are also important driverof fluctuations in interest rate, but monetary policy shock itself has significant
effects on interest rate formation.
Are those results from the forecast error variance decomposition consistent
with the ones derived from more objective method, such as VAR? To examine
this, I here estimate four variable VAR. The structural VAR estimated here isvery standard as in Christiano, Eichenbaum and Evans (1999) and Teruyama(2001), which consists of a constant and four variables, output, price level, nominal interest rate and money supply.21 Estimated period is set at 1982/Q1 to
1995/Q1 as the DSGE model estimated above and lag length is set at four.
Impulse responses are derived by assuming recursive structure on contemporaneous relationship between variables where the order of Choleski decompositionis output, price level, nominal interest rate and money supply. The forecast
error variance decomposition from this structural VAR is shown in table 8.
21 Output: real GDP, price level: CPI excluding perishables and utility prices, Nominalinterest rate: Libor three month rate, Money supply; M2+CD.

20

Table 8: Variance Decomposition on structural VAR
Outputεyεpεrεm1y 58.2 4.0 1.5 36.2
2y 22.2 7.1 13.8 56.8
Price Level
εy
εp
εr
εm
1y 2.6 84.2 10.3 2.9
2y 4.4 70.0 12.2 13.4

Money Supply
εy
εp
εr
εm
1.9 12.3 1.4 84.3
8.0 23.5 9.3 58.9
Interest Rate
εy
εp
εr
εm
1.4 25.2 72.4 1.0
5.3 35.1 43.3 16.3

Although the detailed definitions of shocks in this case are quite different
from the estimated DSGE models, these shocks can be defined as follows: εy : ademand shock, εp : a supply shock or a technology shock, εr : a monetary policy
shock, εm : a money demand shock.22
As in the case of estimated DSGE models, money demand shock is the mostimportant source of money supply’s fluctuations. Although money demand
shock has some predictive power on output, demand and supply shocks areimportant sources of fluctuations on output and price level respectively, andfurther, a monetary policy shock is important in explaining the dynamics of
nominal interest rate in Japan.
Although a consistent comparison is impossible as the variables employed are
different from those of estimated DSGE models, we can conclude that the forecast error variance decompositions resembles the one obtained from a structural
VAR.

5.4

Simulated Autocorrelations

Simulated autocorrelations contain another information concerning the model’splausibility. Figure 2 shows the simulated autocorrelation of four endogenous
variables in the estimated DSGE models and the ones obtained from VAR estimation. The estimated VAR here is again forth order and the variables employedis the same as used for the DSGE estimation in the time trend case.
The results show that simulated autocorrelations obtained from VAR lie
mostly between those from the HP filtered and the time trend case of the estimated DSGE models. As already mentioned, difference in the persistence of thetechnology shock results in the less persistent autocorrelations in the HP filteredcase. From the evaluation with these simulated autocorrelations, it can be concluded that the estimated DSGE models mimic the VAR’s simulated properties
well.
22 Miyao (2000) and Miyao (2002) consider this shock as a money demand shock rather thana liquidity shock.

21

6Conclusion

In chapter two, I have estimated IS curves in reduced form and the finding is that
real balances enter the equation significantly. However, using single estimation
for the evaluation of such a vague effect as the direct effect of money tends
to be problematic since the endogeneity of variables is not properly accountedfor. Therefore, in chapters three and onwards, the DSGE model with a direct
effect of money is constructed based on rigid microfoundations, and maximum
likelihood estimates of the direct effects of money support no evidence on the
direct effect of money due to the non-separability of utility in consumption andreal balances. This finding is supported by two cases; the variables are detrended
by the HP filter and by the time trend.
As for the simulated properties to evaluate the plausibility of these models,
impulse responses obtained from these two estimated DSGE models have reasonable properties, which are consistent with our intuition on Japanese macroeconomics and macroeconomic theory. In addition, we have learned that the
dynamics of real balances are almost independent from those of other variables.
Furthermore, the forecast error variance decomposition and simulated autocorrelation from these two models resemble those from VARs.
Taking these findings into account, we can conclude that the direct effectof money is extremely small. Therefore, the effect of the ”quantitative easing”
monetary policy, which is currently employed by the Bank of Japan, on the real
economic activities should be extremely limited, and this finding is consistent
with the ones in Kimura, Kobayashi, Ugai and Muranaga (2002) and Fujiwara(2003a).
Future research replacing the policy rule as in (23) to the BOJ’s current
monetary policy rule, which targets not the nomianl interest rate but the basemoney, with more recent observations included in the estimation period would
make this conclusion more persuasive.

22

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25

Figure 1-a: Impulse responses to a preference shock, a(t)mr
0.00000002

0.16

0

0.14
0.12

-0.00000002

0.1-0.00000004
0.08-0.00000006
0.06-0.00000008

0.04-0.0000001

0.02-0.00000012

0
0

5

10

15

pi

0

5

10

15

0

5

10

15

y

0.25

0.8
0.7

0.2
0.6
0.5

0.15

0.4
0.1

0.3
0.2

0.05
0.1
0

0
0

5

bold line:HP filter basesolid line:tim e trend base

10

15

Figure 1-b: Impulse responses to a preference shock, e(t)mr
0.000000009

1.2

0.000000008
10.000000007
0.000000006

0.8

0.000000005
0.6
0.000000004
0.000000003

0.4

0.000000002
0.2
0.000000001
0

0
0

5

10

15

pi

0

5

10

15

0

5

10

15

y
0.000000005

0.000000002
0.000000001

00.000000000-0.000000001-0.000000005-0.000000002-0.00000001-0.000000003-0.000000004-0.000000015-0.000000005-0.000000006-0.00000002
0

5

10

15

Figure 1-c: Impulse responses to a technology shock, z(t)mr
0.00000007

00.00000006

-0.01
-0.02

0.00000005-0.03
0.00000004-0.04
0.00000003-0.05
0.00000002-0.06

0.00000001

-0.07

0

-0.08
0

5

10

0

15

pi

5

10

15

10

15

y

0.05

0.9
0.8

0
0.7
-0.05

0.6
0.5

-0.1
0.4
-0.15

0.3
0.2

-0.2
0.1
-0.25

0
0

5

10

15

0

5

Figure 1-d: Impulse responses to a monetary policy shock, a(t)
m

r
0

0.16
0.14

-0.00000002

0.12-0.00000004

0.1-0.00000006

0.08
0.06

-0.00000008

0.04-0.0000001

0.02-0.00000012

0
0

5

10

15

pi

0

5

10

15

0

5

10

15

y
0

0
-0.01

-0.05

-0.02
-0.1
-0.03
-0.04

-0.15

-0.05

-0.2

-0.06
-0.25
-0.07
-0.3

-0.08
-0.09

-0.35
0

5

10

15

Figure 2: Simulated Autocorrelation
y

p
1.2

1.2
hp

hp

time

1

1

time

var

var

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

5

6

7

8

r

0

1

2

3

4

5

6

7

8

4

5

6

7

8

m
1.2

1.2
hp
time

1

1

var

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

hp
time

0

var

0
0

1

2

3

4

5

6

7

8

0

1

2

3