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Money and risk in a DSGE
framework: a bayesian application to
the Eurozone
Jonathan Benchimoland André Fourçansy
March, 2012
Abstract
We present and test a model of the Eurozone, with a special em-
phasis on the role of risk aversion and money. The model follows
the New Keynesian DSGE framework, money being introduced in the
utility function with a non-separability assumption. Money is also in-
troduced in the Taylor rule. By using Bayesian estimation techniques,
we shed light on the determinants of output, in‡ation, money, inter-
est rate, ‡exible-price output and ‡exible-price real money balance
dynamics. The role of money is investigated further. Its impact on
output depends on the degree of risk aversion. Money plays a minor
role in the estimated model. Yet, a higher level of risk aversion would
imply that money had signi…cant quantitative e¤ects on business cycle
‡uctuations.
Keywords: Euro Area, Money, Risk, Bayesian Estimation, DSGE.
JEL Classi…cation: E31, E51, E58.
Please cite this paper as:
Benchimol, J., Fourçans, A., 2012. Money and risk in a DSGE
framework: a Bayesian application to the Eurozone. Journal of Macro-
economics 34 (1), 95–111.
Economics Department, ESSEC Business School and CES, University Paris 1
Panthéon-Sorbonne, 106-112 Boulevard de l’Hôpital, 75647 Paris Cedex 13. Email:
jonathan.benchimol@essec.edu
yEconomics Department, ESSEC Business School and THEMA, Avenue Bernard
Hirsch, 95021 Cergy-Pontoise Cedex 2, France. Corresponding author Phone: +33-1-
34433017, Fax: +33-1-34433689. Email: fourcans@essec.edu
1
1 Introduction
Standard New Keynesian literature analyses monetary policy practically with-
out reference to monetary aggregates. In this now traditional framework,
monetary aggregates do not explicitly appear as an explanatory factor nei-
ther in the output gap and in‡ation dynamics nor in interest rate determi-
nation. In‡ation is explained by the expected in‡ation rate and the output
gap. In turn, the output gap depends mainly on its expectations and the
real rate of interest (Clarida et al.,1999;Woodford,2003;Galí and Gertler,
2007;Galí,2008). Finally, the interest rate is established via a traditional
Taylor rule in function of the in‡ation gap and the output gap.
In this framework, monetary policy impacts aggregate demand, and thus
in‡ation and output, through changes in the real interest rate. An increase
in the interest rate reduces output, which decreases the output gap, thus
decreases in‡ation until a new equilibrium is reached. The money stock and
money demand do not explicitly appear. The central bank sets the nominal
interest rate so as to satisfy the demand for money (Woodford,2003;Ireland,
2004).
The money transmission mechanism may also emphasize the connections
between real money balances and risk aversion. First, there may exist a real
balance e¤ect on aggregate demand resulting from a change in prices. Second,
as individuals re-allocate their portfolio of assets, the behavior of real money
balances induces relative price adjustments on …nancial and real assets. In the
process, aggregate demand changes, and thus output. By a¤ecting aggregate
demand, real money balances become part of the transmission mechanism.
Hence, interest rate alone is not su¢ cient to explain the impact of monetary
policy and the role played by …nancial markets (Meltzer,1995,1999;Brunner
and Meltzer,1968).
This monetarist transmission process may also imply a speci…c role to
real money balances when dealing with risk aversion. When risk aversion
increases, individuals may desire to hold more money balances to face the
implied uncertainty and to optimize their consumption through time. Fried-
man alluded to this process as far back as 1956 (Friedman,1956). If this
hypothesis holds, risk aversion may in‡uence the impact of real money bal-
ances on relative prices, …nancial assets and real assets, a¤ecting aggregate
demand and output.
Other considerations as to the role of money are worth mentioning. In
a New Keynesian framework, the expected in‡ation rate or the output gap
may "hide" the role of monetary aggregates, for example on in‡ation deter-
mination. Nelson (2008) shows that standard New Keynesian models are
built on the strange assumption that central banks can control the long-
2
term interest rate, while this variable is actually determined by a Fisher
equation in which expected in‡ation depends on monetary developments.
Reynard (2007) found that in the U.S. and the Euro area, monetary de-
velopments provide qualitative and quantitative information as to in‡ation.
Assenmacher-Wesche and Gerlach (2007) con…rm that money growth con-
tains information about in‡ation pressures and may play an informational
role as to the state of di¤erent non observed (or di¢ cult to observe) variables
in‡uencing in‡ation or output.
How is money generally introduced in New Keynesian DSGE models
? The standard way is to resort to money-in-the-utility (MIU) function,
whereby real money balances are supposed to a¤ect the marginal utility of
consumption. Kremer et al. (2003) seem to support this non-separability
assumption for Germany, and imply that real money balances contribute to
the determination of output and in‡ation dynamics. A recent contribution
introduces the role of money with adjustment costs for holding real balances,
and shows that real money balances contribute to explain expected future
variations of the natural interest rate in the U.S. and the Eurozone (Andrés
et al.,2009). Nelson (2002) …nds that money is a signi…cant determinant of
aggregate demand, both in the U.S. and in the U.K. However, the empiri-
cal work undertaken by Ireland (2004), Andrés et al. (2006), and Jones and
Stracca (2008) suggests that there is little evidence as to the role of money
in the cases of the United States, the Euro area, and the UK.
Our paper di¤ers in its empirical conclusion, resulting in a stronger role to
money, at least in the Eurozone, when risk aversion is high enough. It di¤ers
also somewhat in its theoretical set up. As in the standard way, we resort
to money-in-the-utility function (MIU) with a non-separability assumption
between consumption and money. Yet, in our framework, we specify all the
micro-parameters. This speci…cation permits extracting characteristics and
implications of this type of model that cannot be extracted if only aggregated
parameters are used. We will see, for example, that the coe¢ cient of relative
risk aversion plays a signi…cant role in explaining the role of money. We test
the model and estimate the risk aversion parameter over the sample period.
As risk aversion can be very high in short periods of time, but cannot be
estimated over such short periods, we test the model again by calibrating a
higher risk aversion parameter (twice the previously estimated value). This
strategy allows us to compare the impact of both levels of risk aversion on
the dynamics of the variables.
Our model di¤ers also in its in‡ation and output dynamics. Standard
New Keynesian DSGE models give an important role to endogenous inertia
on both output (consumption habits) and in‡ation (price indexation). In
fact, both dynamics may have a stronger forward-looking component than
3
an inertial component. This appears to be the case, at least in the Euro area
if not clearly in the U.S. (Galí et al.,2001). These inertial components may
hide part of the role of money. Hence, our choice to remain as simple as
possible on that respect in order to try to unveil a possible role for money
balances.
Finally, Backus et al. (1992) have shown that capital appears to play a
rather minor role in the business cycle. To simplify the analysis and focus on
the role of money, we therefore do not include a capital accumulation process
in the model, as in Galí (2008).
We di¤er from existing theoretical (and empirical) analyses by specifying
the ‡exible price counterparts of output and real money balances. This im-
poses a more elaborate theoretical structure, which provides an improvement
on the literature and enriches the model.
We also di¤er from the empirical analyses of the Eurozone by using
Bayesian techniques in a New Keynesian DSGE framework, like in Smets
and Wouters (2007), while introducing money in the model. Current liter-
ature attempts to introduce money only by aggregating model parameters,
therefore leaving aside relevant information. Here we estimate all micro-
parameters of the model under average (estimated) and high risk aversion.
This is an important innovation and leads to interesting implications.
In order to assess further the role of money we also incorporate and
estimate di¤erent Taylor rules (without and with di¤erent money variables)
and analyse their impact on the dynamics of the model with the two levels
of risk aversion.
In the process we unveil transmission mechanisms generally neglected
in traditional New Keynesian analyses. Given a high enough risk aversion,
the framework highlights in particular the non-negligible role of money in
explaining output variations.
The dynamic analysis of the model sheds light on the change in the role
of money through time in explaining ‡uctuations in output. It shows that
the impact of money is stronger in the short than in the long run.
Section 2of the paper describes the theoretical set up. In Section 3, the
model is calibrated and estimated with Bayesian techniques and by using
Euro area data. Variance decompositions are analysed in this section, with
an emphasis on the impact of the coe¢ cient of relative risk aversion. Section
4presents alternative introductions of money in the Taylor rule. Section 5
concludes and the Appendix presents additional theoretical and empirical
results.
4
2 The model
The model consists of households that supply labor, purchase goods for con-
sumption, hold money and bonds, and …rms that hire labor and produce and
sell di¤erentiated products in monopolistically competitive goods markets.
Each …rm sets the price of the good it produces, but not all …rms reset their
price during each period. Households and …rms behave optimally: house-
holds maximize the expected present value of utility, and …rms maximize
pro…ts. There is also a central bank that controls the nominal rate of in-
terest. This model is inspired by Galí (2008), Walsh (2017) and Smets and
Wouters (2003).
2.1 Households
We assume a representative in…nitely-lived household, seeking to maximize
Et"1
X
k=0
kUt+k#(1)
where Utis the period utility function and < 1is the discount factor.
We assume the existence of a continuum of goods represented by the
interval [0;1]. The household decides how to allocate its consumption ex-
penditures among the di¤erent goods. This requires that the consumption
index Ctbe maximized for any given level of expenditures. Furthermore, and
conditional on such optimal behavior, the period budget constraint takes the
form
PtCt+Mt+QtBtBt1+WtNt+Mt1(2)
for t= 0;1;2:::, where Wtis the nominal wage, Ptis an aggregate price index
(see Appendix A), Ntis hours of work (or the measure of household members
employed), Btis the quantity of one-period nominally riskless discount bonds
purchased in period tand maturing in period t+ 1 (each bond pays one unit
of money at maturity and its price is Qtwhere it=log Qtis the short
term nominal rate) and Mtis the quantity of money holdings at time t. The
above sequence of period budget constraints is supplemented with a solvency
condition, such as 8tlim
n!1 Et[Bn]0.
In the literature, utility functions are usually time-separable. To intro-
duce an explicit role for money balances, we drop the assumption that house-
hold preferences are time-separable across consumption and real money bal-
ances. Preferences are measured with a CES utility function including real
5
money balances. Under the assumption of a period utility given by
Ut=1
1(1 b)C1
t+be"m
t(Mt=Pt)11
1
1 + N1+
t(3)
consumption, labor, money and bond holdings are chosen to maximize Eq.
1subject to Eq. 2and the solvency condition. This CES utility function
depends positively on the consumption of goods, Ct, positively on real money
balances, Mt=Pt, and negatively on labour Nt.is the coe¢ cient of relative
risk aversion of households (or the inverse of the intertemporal elasticity of
substitution), is the inverse of the elasticity of money holdings with respect
to the interest rate, and can be seen as a non separability parameter, and
is the inverse of the elasticity of work e¤ort with respect to the real wage.
It must be noticed that must be lower than . If =, Eq. 3becomes
a standard separable utility function whereby the in‡uence of real money
balances on output, in‡ation and ‡exible-price output disappears. This case
has been studied in the literature. In our model, the di¤erence between
the risk aversion parameter and the separability parameter, , plays a
signi…cant role.
The utility function also contains a structural money demand shock, "m
t.
band are positive scale parameters.
As described in Appendix A, this setting leads to the following conditions,
which, in addition to the budget constraint, must hold in equilibrium. The
resulting log-linear version of the …rst order condition corresponding to the
demand for contingent bonds implies that
^ct=Et[^ct+1](^{tEt[^t+1]) =(a1()) (4)
(1 a1) ()
a1()Et[ ^mt+1 ^t+1] + t;c
where t;c =(1a1)()
(1)(a1()) Et"m
t+1and by using the steady state of the
…rst order conditions a1
1= 1+ b
1b1
(1 )1
. The lowercase (^) denotes
the log-linearized (around the steady state) form of the original aggregated
variables.
The demand for cash that follows from the household’s optimization prob-
lem is given by
( ^mt^pt) + ^ct+"m
t=a2^{t(5)
with a2=1
exp(1=)1and where real cash holdings depend positively on con-
sumption with an elasticity equal to unity and negatively on the nominal
interest rate. In what follows we will take the nominal interest rate as the
6
central bank’s policy instrument. In the literature, due to the assumption
that consumption and real money balances are additively separable in the
utility function, cash holdings do not enter any of the other structural equa-
tions: accordingly, the above equation becomes recursive to the rest of the
system of equations.
The …rst order condition corresponding to the optimal consumption-
leisure arbitrage implies that
^nt+ (a1()) ^ct() (1 a1) ( ^mt^pt) + t;m = ^wt^pt(6)
where t;m =()(1a1)
1"m
t.
Finally, these equations represent the Euler condition for the optimal in-
tratemporal allocation of consumption (Eq. 4), the intertemporal optimality
condition setting the marginal rate of substitution between money and con-
sumption equal to the opportunity cost of holding money (Eq. 5), and the
intratemporal optimality condition setting the marginal rate of substitution
between leisure and consumption equal to the real wage (Eq. 6).
2.2 Firms
We assume a continuum of …rms indexed by i2[0;1]. Each …rm produces
a di¤erentiated good but uses an identical technology with the following
production function,
Yt(i) = AtNt(i)1(7)
where At= exp ("a
t)is the level of technology, assumed to be common to all
…rms and to evolve exogenously over time, and is the measure of decreasing
returns.
All …rms face an identical isoelastic demand schedule, and take the ag-
gregate price level Ptand aggregate consumption index Ctas given. As in
the standard Calvo (1983) model, our generalization features monopolistic
competition and staggered price setting. At any time t, only a fraction 1
of …rms, with 0< < 1, can reset their prices optimally, while the remaining
…rms index their prices to lagged in‡ation.
2.3 Central bank
The central bank is assumed to set its nominal interest rate according to a
generalized smoothed Taylor rule such as:
^{t= (1 i)(^tc) + x^yt^yf
t+mf
Mt;k+i^{t1+"i
t(8)
7
where ,xand mare policy coe¢ cients re‡ecting the weight on in‡ation,
on the output gap and on a money variable; the parameter 0< i<1cap-
tures the degree of interest rate smoothing; "i
tis an exogenous ad hoc shock
accounting for ‡uctuations of the nominal interest rate. cis an in‡ation
target and e
Mt;k is a money variable: when k= 0, money does not enter
the Taylor rule; k= 1 to 3corresponds respectively to the real money gap
(di¤erence between real money balances and its ‡exible-price counterpart),
the nominal money growth and the real money growth.
2.4 DSGE model
Solving the model (Appendix A) leads to six micro-founded equations and
six dependent variables: in‡ation, nominal interest rate, output, ‡exible-price
output, real money balances and its ‡exible-price counterpart.
Flexible-price output and ‡exible-price real money balances are com-
pletely determined by shocks. Flexible-price output is mainly driven by
technology shocks (whereas ‡uctuations in the output gap can be attributed
to supply and demand shocks). The ‡exible-price real money balances are
mainly driven by money shocks and ‡exible-price output.
^yf
t=y
a"a
t+y
mcmpf
ty
c+y
sm"m
t(9)
cmpf
t=m
y+1Eth^yf
t+1i+m
y^yf
t+1
"m
t(10)
^t=Et[^t+1] + x;t ^yt^yf
t+m;t cmptcmpf
t(11)
^yt=Et[^yt+1]r(^{tEt[^t+1]) (12)
+mpEtcmpt+1 +smEt"m
t+1
cmpt= ^yti^{t+1
"m
t(13)
^{t= (1 i)(^tc) + x^yt^yf
t+mf
Mt;k+i^{t1+"i
t(14)
where
y
a=1+
(()a1)(1)++
y
m=(1)()(1a1)
(()a1)(1)++
y
c=(1)
(()a1)(1)++log "
"18
y
sm =()(1a1)(1)
((()a1)(1)++)(1)
m
y+1 =a2
(()a1)
m
y= 1 + a2
(()a1)
x;t =()a1++
1(1)(1
)(1)(1+("1)"p
t)
1+(+"p
t)("1)
m;t = () (1 a1)(1)(1
)(1)(1+("1)"p
t)
1+(+"p
t)("1)
r=1
a1()
mp =()(1a1)
a1()
sm =(1a1)()
(a1())(1)
i=a2=
with a1=1
1+(b=(1b))1= (1)(1)= and a2=1
exp(1=)1.
The structural money shock and the markup shock1,"m
tand "p
t, the ex-
ogenous component of the interest rate, "i
t, and of the technology, "a
t, are
assumed to follow a …rst-order autoregressive process with an i.i.d.-normal
error term such as "k
t=k"k
t1+!k;t where "k;t N(0; k)for k=fm; p; i; ag.
As can be seen, and in‡uence all macro-parameters. This in‡uence
highlights the fact that separability and risk aversion are prominent factors
involved in output, in‡ation, real money balances and nominal interest rate
dynamics, as well as in ‡exible-price output and ‡exible-price real money
balances. Moreover, as far as money is concerned, it is the three macro-
parameters, y
m,mand mp, that are essential to highlight its possible role
in the dynamics of the model: these coe¢ cients determine the weight of
money in Eq. 9, Eq. 11 and Eq. 12.
3 Empirical results
As in Smets and Wouters (2003) and An and Schorfheide (2007), we ap-
ply Bayesian techniques to estimate our DSGE model. Contrary to Ireland
(2004) or Andrés et al. (2006), we did not opt to estimate our model by us-
ing the maximum of likelihood because such computation hardly converges
toward a global maximum. First, we estimate the risk aversion level and all
parameters over the sample period. Second, we re-estimate all parameters
but with a constant risk aversion level calibrated to approximately twice its
estimated value. We also test four speci…cations of the Taylor rule under
these two alternatives risk aversion levels.
1The markup shock is introduced and explained in Appendix A.
9
3.1 Euro area data
In our model of the Eurozone, ^tis the log-linearized detrended in‡ation
rate measured as the yearly log di¤erence of detrended GDP De‡ator from
one quarter to the same quarter of the previous year; ^ytis the log-linearized
detrended output per capita measured as the di¤erence between the log of
the real GDP per capita and its trend; and ^{tis the short-term (3-month) de-
trended nominal interest rate. These Data are extracted from the Euro Area
Wide Model database (AWM) of Fagan et al. (2001). cmptis the log-linearized
detrended real money balances per capita measured as the di¤erence between
the real money per capita and its trend, where real money per capita is mea-
sured as the log di¤erence between the money stock per capita and the GDP
De‡ator. We use the M3monetary aggregate from the Eurostat database.
^yf
t, the ‡exible-price output, and cmpf
t, the ‡exible-price real money balances,
are completely determined by structural shocks. To make output and real
money balances stationary, we use …rst log di¤erences, as in Adolfson et al.
(2008).
3.2 Calibration
Following standard conventions, we calibrate beta distributions for parame-
ters that fall between zero and one, inverted gamma distributions for pa-
rameters that need to be constrained to be greater than zero, and normal
distributions in other cases.
The calibration of is inspired by Rabanal and Rubio-Ramírez (2005)
and by Casares (2007). They choose, respectively, a risk aversion parameter
of 2:5and 1:5. In line with these values, we consider that = 2 corresponds
to a standard risk aversion while values above that level imply higher and
higher risk aversion, hence our choice of = 4 to represent a high level of
risk aversion, twice the standard value. Excepted for risk aversion, we adopt
the same priors in the two models.
As in Smets and Wouters (2003), the standard errors of the innovations
are assumed to follow inverse gamma distributions and we choose a beta
distribution for shock persistence parameters (as well as for the backward
component of the Taylor rule) that should be lesser than one.
The calibration of ,,,, and "comes from Smets and Wouters
(2003,2007), Casares (2007) and Galí (2008). The smoothed Taylor rule
(i,,xand m) is calibrated following Gerlach-Kristen (2003), Andrés
et al. (2009) and Barthélemy et al. (2011), analogue priors as those used by
Smets and Wouters (2003) for the monetary policy parameters. In order to
observe the behavior of the central bank, we assign a higher standard error
10
(0:50) and a Normal prior law for the Taylor rule’s coe¢ cients except for the
smoothing parameter, which is restricted to be positive and below one (Beta
distribution). The in‡ation target, c, is calibrated to 2% and estimated.
v(the non-separability parameter) must be greater than one. i(Eq. 13)
must be greater than one as far as this parameter depends on the elasticity
of substitution of money demand with respect to the cost of holding money
balances, as explained in Söderström (2005); while still informative, this
prior distribution is dispersed enough to allow for a wide range of possible
and realistic values to be considered (i.e. > v > 1).
Our prior distributions are not dispersed to focus on the role of risk aver-
sion. The calibration of the shock persistence parameters and the standard
errors of the innovations follows Smets and Wouters (2003) and Fève et al.
(2010). All the standard errors of shocks are assumed to be distributed ac-
cording to inverted Gamma distributions, with prior means of 0:02. The
latter ensures that these parameters have a positive support. The autore-
gressive parameters are all assumed to follow Beta distributions. All these
distributions are centered around 0:75, except for the autoregressive parame-
ter of the monetary policy shock, which is centered around 0:50, as in Smets
and Wouters (2003). We take a common standard error of 0:1for the shock
persistence parameters, as in Smets and Wouters (2003).
3.3 Results
As already said, we calibrate …rst the level of risk aversion to its standard
value, = 2, and we estimate it. This model version is considered as a
benchmark speci…cation. In the second version, we set = 4, about twice
this estimated value. This set-up is motivated by Holden and Subrahmanyam
(1996). They show that acquisition of short-term information is encouraged
by high risk aversion level, and that the latter can cause all potentially in-
formed investors in the economy to concentrate exclusively on the short-term
instead of the long-term. Risk aversion is generally low in the medium and
long run while it could be very high in short periods. As we can’t estimate
risk aversion in the short run, we choose to estimate our model also by setting
= 4, i.e. a high risk aversion level.
In this section, we present only results with a Taylor rule incorporating the
real money gap (f
Mt;1=cmptcmpf
t), the most signi…cant money variable as
shown in Section 4. The model is estimated with 117 observations from 1980
(Q4) to 2009 (Q4) in order to avoid high volatility periods before 1980 and
to take into consideration the core of the global crisis. The estimation of the
implied posterior distribution of the parameters under the two con…gurations
of risk (Table 1) is done using the Metropolis-Hastings algorithm (10 distinct
11
chains, each of 50000 draws; see Smets and Wouters (2007) and Adolfson
et al. (2007)). Average acceptation rates per chain for the benchmark model
(estimated) are included in the interval [0:256; 0:261] and for the high
risk aversion model (= 4) are included in the interval [0:248; 0:252]. The
literature has settled on a value of this acceptance rate around 0:25.
Priors and posteriors distributions are presented in Appendix B. To assess
the model validation, we insure convergence of the proposed distribution to
the target distribution (Appendix C)
Table 1: Bayesian estimation of structural parameters
Priors Posteriors
estimated = 4
Law Mean Std. 5% Mean 95% 5% Mean 95%
beta 0.33 0.05 0.282 0.378 0.473 0.384 0.484 0.589
beta 0.66 0.05 0.657 0.710 0.764 0.673 0.726 0.777
vnormal 1.25 0.05 1.380 1.447 1.518 1.491 1.528 1.568
normal 2.00 0.50 1.771 2.157 2.545
bbeta 0.25 0.10 0.085 0.252 0.410 0.084 0.246 0.399
normal 1.00 0.10 0.895 1.053 1.218 0.957 1.120 1.281
"normal 6.00 0.10 5.807 5.978 6.143 5.815 5.979 6.141
ibeta 0.50 0.10 0.449 0.573 0.700 0.502 0.614 0.726
normal 3.00 0.50 2.856 3.494 4.104 2.874 3.491 4.145
xnormal 1.50 0.50 1.133 1.872 2.614 1.175 1.923 2.632
mnormal 1.50 0.50 0.320 1.011 1.681 0.276 0.964 1.635
cnormal 2.00 0.10 1.733 1.903 2.071 1.739 1.908 2.071
abeta 0.75 0.10 0.987 0.992 0.997 0.991 0.994 0.998
pbeta 0.75 0.10 0.960 0.973 0.987 0.958 0.972 0.986
ibeta 0.50 0.10 0.377 0.460 0.540 0.490 0.560 0.631
mbeta 0.75 0.10 0.952 0.971 0.991 0.974 0.984 0.995
ainvgamma 0.02 2.00 0.011 0.013 0.016 0.015 0.019 0.022
iinvgamma 0.02 2.00 0.013 0.018 0.023 0.009 0.012 0.015
pinvgamma 0.02 2.00 0.003 0.004 0.006 0.003 0.004 0.006
minvgamma 0.02 2.00 0.023 0.026 0.029 0.024 0.027 0.030
12
3.4 Variance decompositions
In this section we analyse the forecast error variance of each variable following
exogenous shocks, in two di¤erent ways. The analysis is conducted …rst
via an unconditional variance decomposition (long term), and second via a
conditional variance decomposition (short term and over time).
3.4.1 Long term analysis
Table 2: Unconditional variance decomposition (percent)
estimated = 4
"p
t"i
t"m
t"a
t"p
t"i
t"m
t"a
t
^yt1.65 1.09 3.07 94.18 0.83 0.28 10.38 88.51
^t97.66 2.14 0.09 0.12 97.64 1.79 0.24 0.33
^{t78.53 19.64 0.64 1.19 74.41 20.67 1.86 3.07
cmpt1.85 0.91 52.49 44.75 0.83 0.26 60.87 38.04
^yf
t0.00 0.00 3.06 96.94 0.00 0.00 10.23 89.77
cmpf
t0.00 0.00 54.42 45.58 0.00 0.00 62.04 37.96
The unconditional variance decomposition (Table 2) shows that, what-
ever the risk aversion level, the variance of output essentially results from
the technology shock, the remaining from the other shocks. If money plays
some role, this role is rather minor (an impact of 3:07%) under an estimated
standard risk aversion.
Yet, as Table 2shows, the money shock contribution to the business cycle
depends on the value of agents’risk aversion. The estimation with the higher
risk aversion (= 4) gives interesting information as to the role of money,
and more generally as to the role of each shock in the long run.
These results indicate that a higher coe¢ cient of relative risk aversion
increases signi…cantly the impact of money on output. Yet it does not really
change the impact of money on in‡ation dynamics, essentially explained by
the markup shock whatever the level of risk aversion. The very small role of
the money shock on in‡ation dynamics is a consequence of the low value of
m;t in Eq. 11, whatever the level of risk aversion, even though m;t increases
13
with . By comparison, the value of mp in Eq. 12 is signi…cantly higher,
and increases no less signi…cantly with (see Table 5in Appendix D).
If more than 88% of the variance of output is still explained by the tech-
nology shock with the high risk calibration (= 4), the role of the interest
rate shock and the role of the markup shock decrease whereas the impact
of the money shock increases from about 3% to 10:4%, i.e. is multiplied by
a factor of 3:4. Similarly, the impact of shocks on ‡exible-price output also
depends on the risk aversion level. The role of the money shock increases
with the risk level from about 3% to 10:2%.
Although money enters the Taylor rule, it does not have a signi…cant role
in the dynamics of the interest rate, whatever the level of risk aversion.
Furthermore, following an increase in the risk aversion level, the dynamics
of real money balances and its ‡exible-price counterpart are to a lesser extent
explained by the technology shock. Unsurprisingly, the variance of these
variables are mainly explained by the money shock.
3.4.2 Short term and through time analysis
The analysis through time (conditional variance) of the di¤erent shocks on
output (Figure 1) shows that the impact of the money shock decreases with
the time horizon, as for the interest rate shock2. Under high risk aversion,
the role of money in the …rst periods remains around 22%, i.e. twice the
value in the long term (10:38%).
As far as in‡ation variance is concerned, the markup shock not only
dominates the process but its impact does not change through time in both
risk con…gurations.
In the short term, as shown in Table 3, the monetary policy shock ex-
plains around 83% of the nominal interest rate variance whereas the markup
shock explain less than 17% for the two con…gurations of risk. For longer
terms, there is an inversion: whatever the risk aversion level, the interest rate
variance is dominated by the price-markup shock and the monetary policy
shock explains around 20% of the interest rate variance. Although money is
introduced in the Taylor rule, the money shock has a minor impact on the
nominal interest rate variance at any time horizon.
2The conditional variances decompositions …gures for the other variables are not shown
here but are available upon request.
14
Figure 1: Conditional forecast error variance decomposition of Output
010 20 30 40 50 60
0
20
40
60
80
100 σ estimated
%
010 20 30 40 50 60
0
20
40
60
80
100 σ = 4
Quarter
%
Markup Shoc k
Inter est R ate Shock
Money Shock
Technology Shock
Table 3: First period variance decomposition (percent)
estimated = 4
"p
t"i
t"m
t"a
t"p
t"i
t"m
t"a
t
^yt2.16 31.17 7.50 59.16 2.23 11.19 22.38 64.20
^t77.72 22.16 0.08 0.03 83.73 16.08 0.13 0.06
^{t16.35 83.44 0.14 0.07 16.66 82.99 0.23 0.13
cmpt1.28 13.76 69.46 15.49 1.09 5.46 77.25 16.20
^yf
t0.00 0.00 10.56 89.44 0.00 0.00 24.89 75.11
cmpf
t0.00 0.00 81.72 18.28 0.00 0.00 82.62 17.38
15
The role of monetary policy on real money balances is di¤erent in the
short term: the monetary policy shock explains almost 14% of the variance
of real money balances in the short term under the standard risk aversion
(and around 5% under high risk aversion), whereas, under the two con…g-
urations of risk, it has a very small role at longer horizons. Similarly, the
technology shock explains around 16% of the real money balances variance
in both con…gurations of risk, whereas at longer horizons it explains around
45% of the real money variance under the estimated risk aversion (and around
38% under the high risk aversion).
It is interesting to notice that the same type of analysis applies to the
‡exible-price output variance decomposition. In the short term as well as in
the long term, technology is the main explanatory factor. The role of money
increases with the relative risk aversion coe¢ cient in the short term (from
a weight of less than 11% under standard calibration to close to 25% under
high risk aversion calibration), as in the long term, whereas the monetary
policy and the price-markup shocks play no role in the ‡exible-price output
and the ‡exible-price real money balances dynamics.
3.5 Interpretation
The estimates of the macro-parameters (aggregated structural parameters)
for estimated and high risk aversions are reported in Appendix D(Table
5). These estimates suggest that a change in risk aversion implies signi…cant
variations in the value of several macro-parameters, notably y
m,mand
mp - respectively the weight of money in the ‡exible-price output, in‡ation
and output equations. Moreover, the weight of the money shock on output
dynamics, sm, and on ‡exible-price output, y
sm, increases with risk aversion,
thus reinforcing the role of money in the dynamics of the model. It is also
worth mentioning that the smoothing parameter in the Taylor rule equation,
i, increases with risk aversion. This may re‡ect the idea that the central
bank strives for …nancial stability in crisis periods.
The comparison between the variance decompositions (Table 2and 3)
of the two model versions illustrates the fact that the role of the money
shock on output and ‡exible-price output depends crucially on the degree
of agents’risk aversion, increases accordingly, and becomes signi…cant when
risk aversion is high, whatever the time horizon. This result highlights the
role of real money to smooth consumption through time, especially when risk
aversion reaches certain levels.
Impulse response functions for the two con…gurations of risk (Appendix
E) highlights the role of risk aversion on the dynamics of several of the
model’s variables. These results also demonstrate the predominant role of
16
the risk aversion level on the impact of the money shock on output, in‡ation,
and real and nominal interest rates. The higher the risk aversion level, the
greater the reactions to the shocks.
4 Money in the Taylor rule
To evaluate further the role of money we analyse di¤erent speci…cations of
the Taylor rule ( e
Mt;k for k=f0;1;2;3g, as exposed in Section 2.3), …rst
without money, then with money introduced in three di¤erent ways. We
thus test both models with four types of Taylor rules:
- With no money (e
Mt;0= 0);
- With a real money gap (e
Mt;1=cmptcmpf
t);
- With a nominal money growth ( e
Mt;2=bmtbmt1);
- With a real money growth ( e
Mt;3=cmptcmpt1).
Table 4: Alternative ECB’s Taylor rules
estimated = 4
e
Mt;0e
Mt;1e
Mt;2e
Mt;3e
Mt;0e
Mt;1e
Mt;2e
Mt;3
i0.527 0.573 0.561 0.547 0.545 0.614 0.546 0.5469
(1 i)1.594 1.491 1.463 1.537 1.579 1.345 1.585 1.575
(1 i)x1.066 0.799 1.018 1.042 1.034 0.741 1.038 1.039
(1 i)m0.431 0.1360.0840.371 -0.012-0.018
ST y
m(%) 7.05 7.50 2.23 3.66 22.61 22.38 23.20 23.28
LT y
m(%) 2.75 3.07 2.24 2.36 9.56 10.38 9.29 9.15
LMD -629.8 -618.2 -634.9 -635.3 -639.8 -626.5 -646.1 -646.1
estimations are not signi…cant in terms of student tests (t<1.645)
As shown in Table 4, all the coe¢ cients of the in‡ation and the output
gap variables, as well as the interest rate smoothing coe¢ cients are signi…cant
(Student tests superior to 1:96) whatever the risk. Yet, the money coe¢ cient
is signi…cant only with the money gap variable ( e
Mt;1).
Furthermore, the log marginal density (LMD) of the data measured through
a Laplace approximation indicates that the Taylor rule including this real
17
money gap performs better than the others, followed by the no-money ( e
Mt;0)
case.
These results suggest that if money has to be introduced in the ECB
monetary policy reaction function, it should rather be a real money gap
variable than a money growth variable (contrary to what Andrés et al. (2009)
and Barthélemy et al. (2011) found, whereas Fourçans (2004,2007) didn’t
…nd such a role for money growth).
Either way, it is interesting to notice that whatever the formulation of the
Taylor rule, the estimated parameters of the whole model are quite similar.
This is true with both levels of risk aversion.
The impact of a money shock on output, as shown through the short term
(ST y
m, in the …rst period) and the long term (LT y
m) variance decomposition
of output with respect to a money shock, are also rather similar whatever the
Taylor rule (Table 3). The impact of money increases with the risk aversion
coe¢ cient, and is stronger in the short run than in the long run, especially
when risk aversion is high.
5 Conclusion
We built and empirically tested a model of the Eurozone, with two levels of
risk aversion and with a particular emphasis on the role of money. The model
follows the New Keynesian DSGE framework, with money in the utility func-
tion whereby real money balances a¤ect the marginal utility of consumption.
By using Bayesian estimation techniques, we shed light not only on the
determinants of output and in‡ation dynamics but in addition on the interest
rate, real money balances, ‡exible-price output and ‡exible-price real money
balances variances. We further investigated how the results are a¤ected when
intertemporal risk aversion changes, especially as far as money is concerned.
Money plays a minor role in the estimated model with a moderate risk
aversion. Most of the variance of output is explained by the technology
shock, the rest by a combination of markup, interest rate and money shocks,
a result in line with current literature (Ireland,2004;Andrés et al.,2006,
2009). However, another calibration with a higher risk aversion (twice the
estimated value) implies that money plays a non-negligible role in explaining
output and ‡exible-price output ‡uctuations. We also found that the short
term impact is signi…cantly stronger than the long run one. These results
di¤er from existing literature using New Keynesian DSGE frameworks with
money, insofar as it neglects the impact of a high enough risk factor.
On the other hand, the explicit money variable does not appear to have
a notable direct role in explaining in‡ation variability. The overwhelming
18
explanatory factor is the price-markup whatever the level of risk aversion.
Another outcome concerns monetary policy. The higher the risk aversion,
the stronger the smoothing of the interest rate. This re‡ects probably the
central bankers’objective not to agitate markets.
Our results suggest that a nominal or real money growth variable does
not improve the estimated ECB monetary policy rule. Yet, a real money gap
variable (the di¤erence between the real money balances and its ‡exible-price
counterpart) signi…cantly improves the estimated Taylor rule. This being
said, the introduction, or not, of a money variable in the ECB monetary
policy reaction function does not really appear to change signi…cantly the
impact of money on output and in‡ation dynamics.
All in all, one may infer from our analysis that by changing economic
agents’perception of risks, the last …nancial crisis may have increased the
role of real money balances in the transmission mechanism and in output
changes.
A Solving the model
Price dynamics
Let’s assume a set of …rms not reoptimizing their posted price in period t.
Using the de…nition of the aggregate price level and the fact that all …rms re-
setting prices choose an identical price P
t, leads to Pt=hP 1t
t1+ (1 ) (P
t)1ti1
1t,
where t= 1 + 1
1
"1+"p
tis the elasticity of substitution between consumption
goods in period t, and t
t1is the markup of prices over marginal costs (time
varying). Dividing both sides by Pt1and log-linearizing around P
t=Pt1
yields
t= (1 ) (p
tpt1)(15)
In this setup, we don’t assume inertial dynamics of prices. In‡ation results
from the fact that …rms reoptimizing in any given period their price plans,
choose a price that di¤ers from the economy’s average price in the previous
period.
Price setting
A …rm reoptimizing in period tchooses the price P
tthat maximizes
the current market value of the pro…ts generated while that price remains
19
e¤ective. This problem is solved and leads to a …rst-order Taylor expansion
around the zero in‡ation steady state:
p
tpt1= (1 )
1
X
k=0
()kEtcmct+kjt+ (pt+kpt1)(16)
where cmct+kjt=mct+kjtmc denotes the log deviation of marginal cost from
its steady state value mc =, and = log "
"1is the log of the desired
gross markup.
Equilibrium
Market clearing in the goods market requires Yt(i) = Ct(i)for all i2[0;1]
and all t. Aggregate output is de…ned as Yt=R1
0Yt(i)11
tdit
t1; it
follows that Yt=Ctmust hold for all t. One can combine the above goods
market clearing condition with the consumer’s Euler equation (Eq. 4) to
yield the equilibrium condition
^yt=Et[^yt+1]1
a1()(^{tEt[^t+1]) (17)
+() (1 a1)
a1()(Et[ ^mt+1]Et[^t+1]) + t;c
Market clearing in the labor market requires Nt=R1
0Nt(i)di. By using
the production function (Eq. 7) and taking logs, one can write the following
approximate relation between aggregate output, employment and technology
as
yt="a
t+ (1 )nt(18)
An expression is derived for an individual …rm’s marginal cost in terms
of the economy’s average real marginal cost:
mct= ( ^wt^pt)dmpnt(19)
= ( ^wt^pt)1
1("a
t^yt)(20)
for all t, where dmpntde…nes the economy’s average marginal product of labor.
As mct+kjt= ( ^wt+k^pt+k)mpnt+kjtwe have
mct+kjt=mct+kt
1(p
tpt+k)(21)
20
where the second equality follows from the demand schedule combined with
the market clearing condition ct=yt. Substituting Eq. 21 into Eq. 16
yields
p
tpt1= (1 )
1
X
k=0
t+k()kEt[cmct+k] +
1
X
k=0
()kEt[t+k](22)
where t=1
1+t1is time varying in order to take into account the
markup shock.
Finally, Eq. 15 and Eq. 22 yield the in‡ation equation
t=Et[t+1 ] + mctcmct(23)
where ,mct=(1)(1)
t.mctis strictly decreasing in the index of
price stickiness , in the measure of decreasing returns , and in the demand
elasticity t.
Next, a relation is derived between the economy’s real marginal cost and a
measure of aggregate economic activity. From Eq. 6and Eq. 18, the average
real marginal cost can be expressed as
mct=()a1++
1^yt"a
t1 +
1(24)
+ () (1 a1) ( ^mt^pt) + t;m
Under ‡exible prices the real marginal cost is constant and equal to mc =
. De…ning the natural level of output, denoted by yf
t, as the equilibrium
level of output under ‡exible prices leads to
mc =()a1++
1^yf
t"a
t1 +
1(25)
+ () (1 a1)cmpf
t+t;m
where cmpf
t= ^mf
t^pf
t, thus implying
^yf
t=y
a"a
t+y
mcmpf
ty
c+y
sm"M
t(26)
where
21
y
a=1 +
(()a1) (1 ) + +
y
m=(1 ) () (1 a1)
(()a1) (1 ) + +
y
c=(1 )
(()a1) (1 ) + +log "
"1
y
sm =() (1 a1) (1 )
((()a1) (1 ) + +) (1 )
We deduce from Eq. 17 that ^{f
t= (()a1)Eth^yf
t+1iand by
using Eq. 5we obtain the following equation of real money balances under
‡exible prices
cmpf
t=m
y+1Eth^yf
t+1i+m
y^yf
t+1
"M
t(27)
where m
y+1 =a2(()a1)
and m
y= 1 + a2(()a1)
Subtracting Eq. 25 from Eq. 24 yields
cmct=x^yt^yf
t+mcmptcmpf
t(28)
where cmpt= ^mt^ptis the log linearized real money balances around its
steady state, cmpf
tis its ‡exible-price counterpart, x=()a1++
1
and m= () (1 a1).
By combining Eq. 28 with Eq. 23 we obtain
^t=Et[^t+1] + x;t ^yt^yf
t+m;t cmptcmpf
t(29)
where ^yt^yf
tis the output gap,cmptcmpf
tis the real money balances gap,
x;t =()a1++
1(1 )1
(1 ) (1 + ("1) "p
t)
1+(+"p
t) ("1)
and
m;t = () (1 a1)(1 )1
(1 ) (1 + ("1) "p
t)
1+(+"p
t) ("1)
Then Eq. 29 is our …rst equation relating in‡ation to its one period ahead
forecast, the output gap and real money balances.
The second key equation describing the equilibrium of the model is ob-
tained by rewriting Eq. 17 so as to determine output
22
^yt=Et[^yt+1]r(^{tEt[^t+1]) + mp Etcmpt+1+t;c (30)
where r=1
()a1,mp =()(1a1)
a1()and t;c =smEt"M
t+1where
sp =1
a1()and sm =(1a1)()
a1()
1
1. Eq. 30 is thus a dynamic IS
equation including the real money balances.
The third key equation describes the real money stock. From Eq. 5we
obtain
cmpt= ^yti^{t+1
"m
t(31)
where i=a2=.
23
B Priors and posteriors
The vertical line of Figures 2and 3denotes the posterior mode, the grey line
the prior distribution, and the black line the posterior distribution.
Figure 2: Priors and posteriors (estimated)
0.020.040.060.080.1
0
100
200
300 SE_ua
00.05 0.1
0
50
100
SE_ui
00.05 0.1
0
200
400
SE_up
0.020.040.060.080.1
0
50
100
150
200
SE_um
0.2 0 .4 0 .6 0.8
0
2
4
6
8
alpha
0.5 0 .6 0.7 0.8 0 .9
0
5
10
teta
1.2 1 .4 1 .6
0
5
10 vega
1 2 3
0
0.5
1
1.5
sigma
00.5 1
0
1
2
3
4
b
1.5 22.5
0
1
2
3
4
pb
0.5 11.5
0
1
2
3
4
ne ta
5.5 66.5
0
1
2
3
4
epsilon
00.5 1
0
2
4
li1
2 4 6
0
0.5
1
li2
024
0
0.2
0.4
0.6
0.8
li3
-1 0 1 2 3
0
0.5
1li4
0.4 0.6 0 .8 1
0
50
100
rhoa
0.4 0.6 0 .8 1
0
20
40
rhop
0.2 0 .4 0.6
0
2
4
6
8
rhoi
0.4 0 .6 0.8 1
0
10
20
30
rhom
24
Figure 3: Priors and posteriors (= 4)
0.020.040.060.080.1
0
50
100
150
SE_ua
0.020.040.060.080.1
0
50
100
150
200
SE_ui
0.020.040.060.080.1
0
200
400
SE_up
0.020.040.060.080.1
0
50
100
150
200
SE_um
0.2 0.4 0.6 0.8
0
2
4
6
8
alpha
0.5 0.6 0 .7 0.8 0 .9
0
5
10
teta
1.2 1.4 1 .6
0
5
10
15
vega
-0.2 00.2 0 .4 0.6 0 .8
0
1
2
3
4
b
1.5 22.5
0
1
2
3
4
pb
0.5 11.5
0
1
2
3
4
ne ta
5.5 66.5
0
1
2
3
4
epsilon
0.2 0.4 0 .6 0.8 1
0
2
4
6li1
2 4 6
0
0.5
1
li2
0 2 4
0
0.2
0.4
0.6
0.8
li3
0 2 4
0
0.5
1li4
0.4 0.6 0.8 1
0
50
100
150
200 rhoa
0.4 0 .6 0.8 1
0
20
40
rhop
0.4 0 .6 0.8
0
5
10 rhoi
0.4 0.6 0 .8 1
0
20
40
60
rhom
25
C Model validation
The diagnosis concerning the numerical maximization of the posterior kernel
indicates that the optimization procedure leads to a robust maximum for the
posterior kernel. The convergence of the proposed distribution to the target
distribution is thus satis…ed.
Figure 4: Multivariate MH convergence diagnosis (estimated)
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
6.5
7
7.5
8
8.5 Interval
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
6
8
10
12 m2
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
30
40
50
60
70 m3
A diagnosis of the overall convergence for the Metropolis-Hastings sam-
pling algorithm is provided in Figure 4and Figure 5.
Each graph represents speci…c convergence measures with two distinct
lines that show the results within (red line) and between (blue line) chains
(Geweke,1999). Those measures are related to the analysis of the parameters
mean (interval), variance (m2) and third moment (m3). For each of the three
measures, convergence requires that both lines become relatively horizontal
and converge to each other in both models.
From Figure 4, it can be inferred that the model with standard risk
aversion needs more chain to stabilize m3 (third moment), in comparison
with the case where risk aversion is high (Figure 5).
26
Figure 5: Multivariate MH convergence diagnosis (= 4)
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
6.5
7
7.5
8
8.5 Interval
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
6
8
10
12 m2
0.5 11.5 22 .5 33 .5 44.5 5
x 10
4
20
40
60
80 m3
Diagnosis for each individual parameter (not included but it can be pro-
vided upon request) indicates that most of the parameters do not exhibit
convergence problems.
Moreover, a BVAR identi…cation analysis (Ratto,2008) suggests that all
parameter values are stable.
The estimates of the innovation component of each structural shock, re-
spectively for the estimated and the calibrated = 4, respect the i.i.d.
properties and are centered around zero. This reinforces the statistical va-
lidity of the estimated model (the corresponding …gures can be provided by
the authors).
27
D Macro-parameters
Table 5: Aggregated structural parameters
est. = 4
y
a0,8166 0,6849
y
m-0,1027 -0,3508
y
c0,0452 0,0304
y
sm 0,2294 0,6644
m
y+1 -0,6889 -1,0841
m
y1,6889 2,0841
x;t 0,1057 0,0960
m;t 0,0108 0,0337
r0,5741 0,3457
mp 0,2386 0,7286
sm -0,5328 -1,3799
i0,3956 0,3748
i0,5732 0,6146
(1 i)1,4914 1,3455
(1 i)x0,7991 0,7412
(1 i)m0,4314 0,3717
28
E Impulse response functions
The thin solid line of Figure 6represents the impulse response functions
of the model with estimated risk aversion while the dashed line represents
the impulse response functions of the model with high risk aversion (= 4).
After a markup shock, the in‡ation rate and the nominal interest rise,
then gradually decrease toward the steady state. The output and the output
gap decrease then increase to their steady state values.
After an interest rate shock, in‡ation, output and the output gap fall.
The real rate of interest rises. Real money growth displays an overshoot-
ing/undershooting process in the …rst periods, then rapidly returns to its
steady state value.
After a technology shock, the output gap, the nominal and real interest
rate, and the in‡ation decrease whereas output as well as real money balances
and real money growth rise.
After a money shock, the nominal and the real rate of interest, the output
and the output gap rise. In‡ation increases a bit then decreases through time
to its steady state value.
The ‡exible-price output and the ‡exible-price real money balances in-
crease after a technology shock and after a money shock.
All these results are in line with the DSGE literature, especially with
Smets and Wouters (2003) and Galí (2008).
29
Figure 6: Impulse response functions with both risk con…gurations
0
0.2
0.4
Markup
Shock
Inflation
(%)
-0.4
-0.2
0
Output
(%)
-1
0
1x 10
-15
Flexible
Price
Output (%)
0
0.2
0.4
Nominal
Interest
Rate (%)
-0.2
0
0.2
Real Interest
Rate (%)
-0.4
-0.2
0
Real
Money (%)
-0.5
0
0.5
Real Money
Growth (%)
-2
0
2x 10
-15
Flexible
Price Real
Money (%)
-0.4
-0.2
0
Output
Gap (%)
-0.4
-0.2
0
Real Money
Gap (%)
020 40
0
0.5
Shock
(%)
Quarters
-0.01
-0.0050
Technology
Shock
0.5
1
1.5
0.5
1
1.5
-0.04
-0.020
-0.04
-0.020
0.5
1
1.5
-2
0
2
0.5
1
1.5
-0.01
-0.0050
-5
0
5x 10
-3
020 40
1
1.5
2
Quarters
0
0.01
0.02
Money
Shock
0
0.5
1
0
0.5
1
0
0.05
0
0.02
0.04
0
2
4
-5
0
5
0
2
4
0
0.01
0.02
-0.02
-0.010
020 40
0
2
4
Quarters
-0.2
-0.1
0
Interest
Rate Shock
-1
-0.5
0
-2
0
2x 10
-16
0
0.5
1
0
0.5
1
-1
-0.5
0
-1
0
1
0
1
2x 10
-16
-1
-0.5
0
-1
-0.5
0
020 40
0
1
2
Quarters
30
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