WorkIng PaPer serIes
Working PaPer SerieS
no 1089 / SePTeMBer 2009
The effecTS of
MoneTary Policy
on uneMPloyMenT
dynaMicS under
Model uncerTainTy
evidence froM The
uS and The euro
area
by Carlo Altavilla
and Matteo Ciccarelli
WORKING PAPER SERIES
NO 1089 / SEPTEMBER 2009
This paper can be downloaded without charge from
http://www.ecb.europa.eu or from the Social Science Research Network
electronic library at http://ssrn.com/abstract_id=1467788.
In 2009 all ECB
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feature a motif
taken from the
€200 banknote.
THE EFFECTS OF MONETARY POLICY
ON UNEMPLOYMENT DYNAMICS
UNDER MODEL UNCERTAINTY
EVIDENCE FROM THE US
AND THE EURO AREA
1
by Carlo Altavilla
2
and Matteo Ciccarelli
3
Workshop, Rome; and the Italian Congress of Econometrics and Empirical Economics, Ancona, for comments and suggestions.
Part of the paper was written while the first author was visiting Columbia Business School, whose hospitality is gratefully
acknowledged. This paper should not be reported as representing the views of the European Central Bank,
or ECB policy. Remaining errors are our own responsibilities.
2 University of Naples “Parthenope”, Via Medina, 40 - 80133 Naples, Italy; e-mail: [email protected];
Phone: (+)39 0815474733, fax (+)39 0815474750
3 European Central Bank, Kaiserstrasse 29, D-60311 Frankfurt am Main, Germany;
e-mail: [email protected]; Phone: (+)49 6913448721,
fax (+)49 6913446575
1 We are particularly grateful to Ken West and two anonymous referees for extensive comments which substantially improved content
and exposition of the paper. We would also like to thank Efrem Castelnuovo, Mark Giannoni, Gert Peersman, Frank Smets, and
the participants at the CESifo Area Conference on Macro, Money, and International Finance, Munich; the 3rd Piero Moncasca
© European Central Bank, 2009
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ISSN 1725-2806 (online)
3
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September 2009
Abstract
4
Non-technical summary
5
1 Introduction
7
2 Model uncertainty and optimal monetary
policy: the macroeconometric framework
9
2.1 The model space
10
2.2 The Central Bank’s problem
13
3 From the models to the data
15
3.1 Data and transformations
15
3.2 Estimation algorithm
16
3.3 Properties of model space and rules
19
4 Effects of policy on unemployment
21
4.1 Impulse response dispersion
22
4.2 Variance decomposition
24
4.3 Transmission mechanism
26
5 Conclusive remarks
27
Appendix
29
References
32
Tables and fi gures
35
45
CONTENTS
4
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September 2009
Abstract
This paper explores the role that the imperfect knowledge of the structure of the
economy plays in the uncertainty surrounding the effects of rule-based monetary
policy on unemployment dynamics in the euro area and the US. We employ a
Bayesian model averaging procedure on a wide range of models which differ in
several dimensions to account for the uncertainty that the policymaker faces when
setting the monetary policy and evaluating its effect on real economy. We find
evidence of a high degree of dispersion across models in both policy rule parameters
and impulse response functions. Moreover, monetary policy shocks have very similar
recessionary effects on the two economies with a different role played by the
participation rate in the transmission mechanism. Finally, we show that a policy
maker who does not take model uncertainty into account and selects the results on the
basis of a single model may come to misleading conclusions not only about the
transmission mechanism, but also about the differences between the euro area and the
US, which are on average essentially small.
Keywords: Monetary policy, Model uncertainty, Bayesian model averaging,
Unemployment gap, Taylor rule
JEL Classification: C11, E24, E52, E58
5
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September 2009
Non-technical summ ary
The pervasive uncertainty that central banks face precludes monetary policy from ne tuning
the level of economic activity. This paper explores the role that the imperfect knowledge of the
structure of the economy plays in the uncertainty surrounding the eects of rule-based monetary
policy on unemployment dynamics in the euro area and the US.
An extended literature has described how central banks should take uncertain ty into account
in their decision-making process. A large part of this literature has focused on the robustness of
policy actions. Other studies have recently accounted for model uncertainty with a Bayesian model
averaging approach that, unlike the robust control methodology, usually considers a model space
with theoretically distinct models. The idea is that, given considerable uncertainty about the true
structure of the economy, policymakers aim at identifying measures that perform well across a wide
range of non-local models.
This paper follows the latter approach. Moreover, unlike most of the literature which only
focuses on how monetary policy should systematically react to changes in unemployment and
in ation (i.e. the policy rules), our work goes further and also analyzes how the uncertainty about
the policy rule translates into the uncertainty surrounding the responses of the economy (and in
particular of unemployment) to policy shocks.
Our ndings support the view that in order to overcome sev ere policy mistakes, decisions could
be based on a wide range of possible scenarios about the structure of the economy. As a result,
when allowing for model uncertainty, policy advice may look signicantly dierent from the one
that would be optimal based on few selected models.
We also show that, although a monetary policy shock might be less importan t than other struc-
tural shocks to explain unemployment dynamics, it has a stable recessionary eect. Moreover, the
average unemployment responses for the US and the euro area are qualitatively and quantitatively
very similar. The analysis of the transmission mechanism also indicates that other labour market
variables such as participation rate play a signicantly dierent role in the transmission mechanism
of the two economies.
One of the main policy implications of our results is that combining results from alternative
representations of the structure of the econom y represents a useful strategy to account for model
uncertainty when assessing the risks for price stabilit y or when deciding a given policy. In particular,
our results show that a policymaker who selects the results on the basis of a single model may come
to misleading conclusions not only about the transmission mechanism — picking up models where,
for instance, the price puzzle is more marked or the eect on unemployment has a wrong sign —
but also about the dierences between the euro area and the US, whic h on average are tiny. By
6
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September 2009
allowing for model uncertainty, instead, results are on average closer to what we expect from a
theoretical point of view, and put the policymaker in a favourable position to calibrate the policy
interventions in a more appropriate way, that is, more consistently with the economic theory and
less distorting for the economy.
7
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September 2009
1Introduction
The pervasive uncertainty that central banks face precludes monetary policy from ne tuning the
level of economic activity. This paper explores the role that the imperfect knowledge of the structure
of the economy plays in the uncertainty surrounding the eects of rule-based monetary policy on
unemployment dynamics in the euro area and the US.
An extended (empirical and theoretical) literature has described how central banks should take
uncertainty into account in their decision-making process. A large part of this literature has focused
on the robustness of policy actions. Since the seminal papers of Hansen and Sargent (e.g. 2001
and 2007), many researchers (e.g. Giannoni 2002; Onatski and Stock 2002; Brock al. 2003) have
studied monetary policy uncertainty with a ‘robust control methodology.’ Within this framework,
the uncertain ty surrounding the eect of policy actions is measured by rst constructing a model
space where each model is obtained as a local perturbation to a given baseline model and then
applying a minimax rule. As a result, a policy with the smallest possible maximum risk is preferred.
Other studies (e.g. Levin and Williams 2003; Brock et al. 2007) have recently accounted
for model uncertain ty with a Ba yesian model averaging approach that, unlike the robust control
methodology, usually considers a model space with theoretically distinct models. The idea is
that, given considerable uncertainty about the true structure of the economy, policymakers aim at
identifying measures that perform well across a wide range of non-local models. Results are then
obtained as w eighted averages across models, with weights given by the relative marginal likelihood
of the models.
Our paper follows the latter a pproach. Moreover, unlike most of the literature which only focuses
on ho w monetary policy should systematically react to changes in unemployment and in ation (i.e.
the policy rules), we go further and also analyze how the uncertainty about the policy rule translates
into the uncertain ty surrounding the responses of the economy (and in particular of unemployment)
to policy shocks. We assume that the monetary authority minimizes expected losses of a social loss
function subject to the economy, and sets up a policy rule. In turn, the economy is alternatively
summarized b y a wide range of multivariate models that dier in the assumptions regarding the
persistence of in ation and unemployment, the measuremen t of the natural rate of unemployment,
the number and types of variables entering the model, and the lag structure.
The perspectiv e adopted in this paper is Bayesian, meaning that a complete model involving
unobservables (e.g. parameters), observables (e.g. data) and variables of interest (e.g. policy rule,
impulse response functions) is identied by a join t distribution of these elements. If P denotes
amodel,
P
denotes unobservable parameters, G denotes the observables, and $ is a vector of
8
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September 2009
interest, then the model P species the joint distribution
s (
P
>G>$ | P)=s (
P
| P) s (G |
P
>P) s ($ | G>
P
>P) (1)
The object of inference, then, is expressed as the posterior density of $:
s ($ | G> P)=
Z
s ($ | G>
P
>P) s (
P
| G> P) g
P
(2)
which is the relevant density for the decisionmakers. In this framework, two sources of uncertainty
are considered. Model uncertainty is accounted for with the incorporation of several competing
models P
1
>P
2
>===>P
M
which might have generated the available sample of data. Parameter uncer-
tainty is re ected in a series of informative priors on the unobservables s
¡
P
m
| P
m
¢
.Weevaluate
the degree of dispersion of s ($ | G> P
m
) between models and quantify the eects which policy
prescriptions coming from dierent models have on unemployment.
The paper can be considered as an extended application of the methodological approac h sugges-
ted, for instance, by Brock et al (2007). As in their w ork, all models are equally likely a priori; unlike
their assumption, w e specify informative priors for the model parameters and compare models on
the basis of their marginal likelihoods.
Using data for the US and the euro area, we show that simple linear autoregressive models which
dier in several dimensions may produce a signicant degree of uncertain ty in the distribution of
optimal policy parameters, expected losses and impulse responses.
Cross-country comparison corroborates the ndings of Sauch and Smets (2008) and Smets and
Wouters (2005) that the dierences in the monetary policy reaction function in the US and the euro
area are small. Moreover, although a monetary policy shock might be less important than other
structural shock s to explain unemployment dynamics, we show that on average it has a stable
recessionary eect in both economies. We also nd that the average unemployment responses
are qualitatively and quantitatively very similar in the two economies, with results for the euro
area being more dispersed than those for the US. The analysis of the transmission mechanism also
indicates that other labor market variables, such as participation rate, play an important distinctive
role in the two economies.
Our results have signicant policy implications. The high degree of dispersion across models
suggests that the eects of a given policy measure are model dependent, and therefore policy
decisions should be based on a wide range of possible scenarios about the structure of the economy
in order to overcome policy mistakes. We show that a policy maker who selects the results on
the basis of a single model may come to misleading conclusions not only about the transmission
mechanism — picking up models where, for instance, the price puzzle is more marked or the response
9
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September 2009
of unemployment has a wrong sign — but also about the dierences between the euro area and the
US — which may only result as an outcome of model selection. A combination procedure, instead,
helps dampen out this uncertainty. By taking into account model uncertainty and averaging across
models, results are more consistent with the economic theory and provide the policymaker with a
robust environment to calibrate interventions in a less distorting way for the economy.
The remainder of the paper is structured as follows. Section 2 describes the general framework
with the model space and the solution to the central bank’s problem. Section 3 reports the empir-
ical ndings in terms of expected loss and policy parameters. Section 4 discusses the eects of a
monetary policy shock on unemployment in the designed uncertain environment. Section 5 sum-
marizes the paper’s main ndings and provides conclusive remarks. A technical appendix presents
the model space and derives the posterior distributions for the Bayesian inference.
2 Modeluncertaintyandoptimalmonetarypolicy: themacroe-
conometr ic framework
In this section we illustrate the empirical framework, which comprises: (i) a set of monetary policy
rules; (ii) a monetary policymaker who chooses the parameters of the rules by minimizing a loss
function; (iii) a set of models which summarize the constrain ts faced by the policymaker in the
minimization problem.
A wide set of models is used to account for the uncertainty surrounding the representation
of the economy. As described in Brock et al. (2007) model uncertainty results from sources as
dierent as economic theory, specication conditional on theory, and heterogeneity regarding the
data generating process. We will generate the model space by limiting the analysis to multivariate
dynamic linear models (VARs) which entail policy and non-policy variables, with dierent prior
assumptions on both sets of variables, as well as on the lag structure.
The structural behavior of the non-policy variables is assumed to be given by the estimates of
the model. Using this estimated structure, the solution to the minimization problem yields the
values of the loss function under alternative policy parameters. A given set of these parameters
will then minimize the expected loss for each model. The interest rate policy which results from
this optimization problem can be of two t ypes: (i) a linear optimal feedback rule (OFR) where the
nominal interest rate depends on all observable variables included in the model and which appear
to have a closed-form solution; and (ii) an optimized Taylor rule (TR) where the interest rate only
reacts to the current value of the unemployment gap and the in ation rate, similarly to the original
work of Taylor (1993), and where the weights attached to both variables are obtained with a grid
search procedure.
10
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September 2009
Finally, the optimal or optimized rule becomes part of the interest rate equation in a structural
VAR, and its disturbance is used to quantify the uncertainty surrounding the eect of a monetary
policy shock on the unemplo yment gap using a standard Impulse Response Function (IRF) analysis
as, e.g., in Stock and Watson (2001).
In the following, we detail these elements backwards, starting from the model and then turning
to the policymakers and the rules.
2.1 The model space
We start by specifying a comprehensive range of multivariate linear dynamic models which span
the model space. The class of simultaneous equation models considered here tak es the following
general VAR form:
]
w
=
s
X
m=1
A
m
]
w3m
+
s
X
m=0
b
m
l
w3m
+
}
w
l
w
=
s
X
m=0
c
0
m
]
w3m
+
s
X
m=1
g
m
l
w3m
+
l
w
(3)
where ]
w
is a vector of non-policy variables; l
w
is the policy variable; A> b> c>g are conformable
matrices and vectors;
]
w
and
l
w
are vectors of serially uncorrelated structural disturbances. In
section 3 we will explain in more details the estimation algorithm and the impulse response analysis.
For the purpose of this section, it is su!cient to remark here that the structural coe!cients can
be easily recovered with some iden tication scheme. We will use the same scheme throughout the
paper (both for the optimal policy derivation and for the impulse response analysis) and impose the
timing assumption that the central bank reacts contemporaneously to all variables in the economy,
whereas the policy rate does not contemporaneously aect the rest of the economy. In terms of the
above VAR, this assumption imposes a Choleski scheme by setting b
0
=0.
1
The non-policy block ]
w
contains at least the in ation rate (
w
) and the (negative) unemploy-
ment gap (x
w
), calculated as the dierence between the natural rate of unemployment (x
W
w
) and its
actual value (ex
w
). Other non-policy variables enter the specication in the form we will explain
below.
Four broad sets of prior beliefs shape the dimensions of model uncertainty that characterize the
model space.
1
The set up is similar to the one used e.g. by Sack (2000) in a di erent context.
11
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September 2009
2.1.1 Priors on in ation dynamics
In the rst set of priors, we deal with assumptions about the way the in ation rate is modelled.
Concretely, four general prior assumptions are made according to whether in ation is (i) left un-
restricted, or whether it is treated in the system as (ii) a random w alk, (iii) an autoregressive
process of order s, or (iv) a white noise. In all cases we take a Ba yesian perspective and place the
exclusion restrictions through the allocation of probability distributions to the model’s coe!cients.
The starting point is always a Minnesota-type of prior: in the unrestricted case we complement
the autoregressive representation with the specication of a vague prior distribution and a loose
tightness on all coe!cients; in the other three setups, instead, we assume that in ation follows one
of the three processes by setting the mean of own-lag coe!cients, and allow for a much tighter
precision placed on all coe!cients of the in ation equation as compared to the precision placed on
the coe!cients of other equations. In other words, priors are alw ays informative and dier in the
relative tightness placed on the coe!cients in the equation for
w
.
2
2.1.2 Priors on labor market variables
The second set of priors re ects dierent assumptions on the dynamics of the labor mark et variables.
We distinguish two types of prior, according to (i) the degree of persistence of the variables and
(ii) the computation of the natural rate of unemployment.
Analogously to the treatment of in ation, w e model the degree of persistence of unemployment
(and participation rate, when included in the specication) either in an unrestricted way — by
placing a general unit root Minnesota prior and a loose tightness — or restricting the variables to
have a lower degree of persistence. In the latter case, as for the in ation dynamics, we set the mean
of own-lag coe!cients to a value lower than one, while allowing for a much tighter precision placed
on the variance of these coe!cients.
Regarding the uncertainty about the natural rate of unemplo yment, there has been an extensive
debate in the literature on the implications of natural rates mismeasurement for monetary policy.
Staiger et al. (1997a,b) and Laubach (2001) found that estimates of a time-varying natural rate of
unemployment are considerably imprecise. The same results are documented by Orphanides and
van Norden (2005) when analyzing the output gap. Finally, Orphanides and Williams (2007) suggest
that policymakers should consider policy rules that react to changes in economic activity either than
2
Note that while the Random Walk and the A utore gressive hypotheses are relatively standard in the VAR literature
(see e.g. Doan et al., 1984; Stock and Watson, 2007), the White Noise (WN) assumption has been recently validated
in studies on in ation persistence that cover especially the last 10-15 years of sample observations. Benati (2008), for
instance, shows that on recent samples the WN assumption might have become a reasonable one in several countries,
including UK and the e uro area, the latter e specially after the creation of EMU. Our sample choice for the empirical
analysis is consistent with this prio r (see section 3.1).
12
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September 2009
reacting to the uncertain estimates of the natural rate. We do not pretend to be exhaustive here
and limit the scope of our analysis to two t ypes of detrending methods: (i) a “statistical” approach
which uses the Baxter and King (1999) band pass lter; and (ii) a robust alternative that measures
the natural rate with a Phillips-curve method and incorporates some “economic” content. The
details of both approaches are given in the section on data transformation (Section 3.1).
2.1.3 Priors on other variables
In the third set of priors, we enlarge the model space by changing the model specication of the
non-policy block, and considering all combinations of three additional endogenous variables: the
labor force participation rate (su
w
); the exchange rate (h
w
), and a commodity price in ation rate
(fs
w
).
The inclusion of the participation rate is motivated by the possibility of shaping more compre-
hensive dynamics of the labor market, as a negative impact of an increase in the nominal interest
rate on output may have diverse eects on the labor force and, ultimately, on the unemployment
rate. The inclusion of the participation rate would account for these eects and provide a cleaner
picture of the transmission mechanism. As observed above in the description of the rst set of
priors, when the participation rate is included in the specication, it enters either with a vague
Minnesota (unit root) prior, or with a lower degree of persistence.
While the inclusion of an exchange rate might not be suitable for the US (e.g. Taylor, 2001),
it might nonetheless be appropriate for the Euro area (e.g. Peersma n and Smets, 2003; Altavilla,
2003). In any case, its inclusion is intended to re ect the external environment, as well as its
conditionality role for monetary policy, as it is an important part of the monetary transmission
mechanism in an open economy. Moreover, some researchers provide empirical evidence that ex-
change rates are statistically signicant in monetary policy rules summarizing the reaction functions
of several major central banks (e.g. Clarida et al., 1998; Svensson, 2000).
Finally, we include a commodity price in ation rate which should control for the expected future
in ation, as it has become customarily in recent applied works on the transmission mechanism of
monetary policy shocks (see e.g Sack 2000.)
2.1.4 Priors on the lag structure
In the last set of prior assumptions, the dynamics of the system is described by alternative lag
structures. The Wold theorem implies that VAR residuals must be white noise. Sometimes this
feature happens to be veried with a parsimonious representation of the lag structure, perhaps
with a rich number of endogenous variables. The VAR, however, easily becomes overparametrized,
13
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September 2009
since the number of coe!cien ts grows as a quadratic function of the number of variables and
proportionately to the n umber of lags. To trade-o between parsimonious and realistic assumptions,
we com bine dogmatic with exible priors and consider models with s lags, where s =1> 2> 3 or 4.
Then, for models where sA1, a tight Minnesota prior on coe!cients dieren t from the own lag is
used.
Summing up, should we account for all possible combinations of the features described above,
we would be dealing with a very large number of models. The model space would in fact be
composed of 1024 models, as a result of the product of 4 priors on in ation persistence, 2 priors
on the persistence of unemployment, 2 priors on the persistence of participation, 2 priors on the
detrending methods, 2
3
=8ways to combine variables in a model with a xed block [x> > l] and
three additional non-policy variables, and 4 lag assumptions.
We take a shortcut, instead, and restrict the analysis to a comprehensive subset spanned by
224 models. The composition of the models can be summarized as follows:
1. A group of models focuses on in ation dynamics and combines the three restrictive priors on
in ation persistence with unrestricted labor market variables and a band-pass estimation of
the natural rate of unemployment. This combination produces therefore 96 models given by
the product of 3 alternative priors on in ation, 8 ways to add the other non-policy variables
and 4 lag assumptions.
2. The remaining 128 models are c haracterized by assumptions on the labor market combined
with unrestricted in ation dynamics, and are obtained from the product of 2 prior assumptions
on the persistence of labor market variables, 2 detrending methods for the natural rate, 8
ways to add the other non-policy variables, and 4 lag assumptions.
Details of the model space are reported in the appendix (Table A1). The priors on other
unknown of the system which have not been described above will be described in Section 3.
2.2 The Cen tral Bank’s Problem
The central bank minimizes an intertemporal loss function that has a positive relation with the
deviation between the goal variables and their target levels:
O
w
= H
w
(
"
X
=0
h
&x
2
w+
+
2
w+
+ (l
w+
l
w+31
)
2
i
)
(4)
where H
w
denotes the expectations conditional upon the available information set at time t; is a
given discount factor, 0 ??1;and&, ,and are non-negative weights.
14
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September 2009
The variable x
w
has already been dened above as the gap betw een the natural rate of unem-
ployment and its actual value. We also interpret here
w
as the deviation from a constant in ation
target. As a benchmark for our analysis, we take & =4, =1,and =0=5. Based on the Okun’s
law, the variance of the unemployment gap is about 1/4 of the variance of the output gap, so this
choice of & is consistent with an equal weight on in ation and output gap variability.
3
As shown in Rudebush and Sv ensson (1999), for =1, the loss function can be written as the
weighted sum of the unconditional variances of the target variables:
H [O
w
]=&Y du [x
w
]+Y du [
w
]+Y du [l
w
l
w31
] (5)
The aim is to minimize this function subject to
[
w+1
= [
w
+ l
w
+
w+1
(6)
which is the State space representation of the VAR (Eq.3). The dynamics of the state are governed
by the matrix and the vector , whose values are given by the point estimates of the corres-
ponding VAR coe!cients, and depend on the particular model considered in the estimation. As a
consequence, w e ha ve 224 state-space representations for each country. For example, in a model
with 4 non-policy variables and two lags, the state space has the following represen tation:
[
w
=
5
9
9
9
9
9
9
9
9
9
9
9
9
7
x
w
x
w31
su
w
su
w31
h
w
h
w31
w
w31
l
w31
6
:
:
:
:
:
:
:
:
:
:
:
:
8
> =
5
9
9
9
9
9
9
9
9
9
9
9
9
7
d
1
11
d
2
11
d
1
12
d
2
12
d
1
13
d
2
13
d
1
14
d
2
14
e
2
15
100000000
d
1
21
d
2
21
d
1
22
d
2
22
d
1
23
d
2
23
d
1
24
d
2
24
e
2
25
001000000
d
1
31
d
2
31
d
1
32
d
2
32
d
1
33
d
2
33
d
1
34
d
2
34
e
2
35
000010000
d
1
41
d
2
31
d
1
42
d
2
42
d
1
43
d
2
43
d
1
44
d
2
44
e
2
45
000000100
000000000
6
:
:
:
:
:
:
:
:
:
:
:
:
8
> =
5
9
9
9
9
9
9
9
9
9
9
9
9
7
e
1
15
0
e
1
25
0
e
1
35
0
e
1
45
0
1
6
:
:
:
:
:
:
:
:
:
:
:
:
8
>
w
=
5
9
9
9
9
9
9
9
9
9
9
9
9
7
x
w
0
SU
w
0
h
w
0
w
0
0
6
:
:
:
:
:
:
:
:
:
:
:
:
8
For the policy rules, we follow Rudebush and Svensson (1999) and consider a general linear
feedback instrument rule
l = i[
w
(7)
where i is a conformable row v ector.
The problem of minimizing in each period the loss function in (4) subject to (6) is standard
and results in an optimal linear feedback rule (OFR) which, under the limit assumption of =1,
converges to a closed-form solution for the vector i (e.g. Rudebush and Svensson, 1999, p.240 ).
3
We also checked how sensitive are results to alternative settings. In particular we were able to con rm the
previous ndings of the literature that th e poster ior distribution of the policy reaction to bot h unemployment and
interest rate shifts monotonically w ith the values of these parameters in a reasonab le range. T h ese changes in the
p olicy rules, however, do not seem to have a signicant eect on the shap e or the magnitude of the impulse resp onse
functions.
15
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September 2009
This rule is less restrictive than a classical Taylor rule, as the interest rate is a function of all
current and lagged values of the non-policy variables and lagged values of the interest rate.
We also derive results under an optimized classical Taylor rule (TR) that allows the interest
rate to react only to current values of unemployment gap and in ation, that is:
l
w
= i ·
μ
x
w
w
¶
i =[i
x
(> ) i
(> )] (8)
In this case the parameters of the rule depend on the VAR coe!cients in an open form, and need
to be recovered with an optimization routine.
In our empirical exercise w e also allow for the presence of a lagged interest rate, capturing an
interest rate smoothing (e.g. Clarida et al. 2000), or other relevant but omitted macroeconomic
variables (e.g. Sack 2000).
3Fromthemodelstothedata
In this section, w e apply our framework to US and euro area data, describe the estimation technique
and characterize the model space discussing its properties.
3.1 Data and transformations
The data are quarterly values of in ation, interest rate, unemployment rate, exchange rate, labor
force participation rate, and a commodit y price index for the euro area and the US, covering 1970:1
to 2007:4. The rst part of the sample (from 1970:1 to 1990:4) is used as a training sample to derive
the prior hyperparameters. The sample 1991:1 to 2007:4 is used for estimation and inference. Main
sources for the data are Datastream and the Area Wide Model (AWM) database (Fagan et al.,
2001).
The in ation rate is calculated as the four-quarter percentage change of CPI. The US interest
rate is the Federal Funds rate; the euro area interest rate is the short-run rate of the AWM database.
The unemployment gap is calculated as the dierence between the natural rate of unemployment
(x
W
w
) and its actual value (ex
w
). To account for some model uncertainty about the natural rate, as
said in section 2, we compute (x
W
w
) using both a “statistical” and an “econom ic” approach. For
the former, the national unemployment series were detrended using the Baxter and King (1999)
band pass lter. We extract cycles of length comprised between 6 and 32 quarters along with a
truncation of 12 lags. As the lter uses a centered moving average method, we pad the series at
the start and at the end with observations derived from AR(4) backcasts and forecasts.
16
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September 2009
The other approach, which incorporates some economic content, is based on a system of equa-
tions which comprises a Phillips curve, an Okun law and a set of equations dening the stochastic
law of motions of the unobservable variables included in the system, namely potential output and
the natural rate. For the Phillips curve we use a simple relationship between CPI in ation and
lagged in ation, the state of aggregate demand as summarized by the unemployment gap, and
a supply-side shock as summarized by import prices. A s in ationisassumedtodependonlyon
nominal factors in the long run, the coe!cients of lagged in ation are constrained to add up to one.
The Okun law relates output gap to unemployment gap. The system is estimated with standard
Kalman-lter techniques.
4
Exchange rates and commodity price are used in standardized four-quarter growth rates. The
exchange rate is dened as the price of foreign currency in terms of domestic currency, therefore
an exchange rate increase is a depreciation. Finally, the participation rate enter all models in gap
form, with the trend computed using the Baxter and King lter. All series are demeaned to omit
the constant term and ease the computations.
3.2 Estimation algorith m
The reduced form of (3) is estimated using Bayesian techniques and informative priors. If denotes
the vector of all VAR coe!cients and denotes the variance-covariance matrix of the reduced
form disturbances, then
P
m
=(> | P
m
). Given the data as summarized by the likelihood
s
¡
G |
P
m
>P
¢
, and a prior distribution s
¡
P
m
| P
m
¢
, the Bayesian algorithm implies obtaining
the posterior s
¡
P
m
| G> P
m
¢
. In turn, given the estimated dynamic behavior of the non-policy
variables as summarized by the latter posterior distribution, we solve the minimization problem
and recover the distribution of the parameters of the rule that minimize the loss function.
5
If we
denote with $
1
the vector of such parameters, its posterior distribution s ($
1
| G> P
m
) is
s ($
1
| G> P
m
)=
Z
i · s
¡
P
m
| G> P
m
¢
g
P
m
(9)
where i is given by the OFR or the TR.
6
Finally, given the posterior mean of $
1
, we compute the
distribution of the unemployment response to a monetary policy shocks. The algorithm is applied
to each model P
m
, each country and each policy rule.
4
For more technical details the reader can refer to Steiger et al. (1997 a and b), and Fabiani and M estre (2004).
The latter generously shared with us their RATS co des on the Kalman lter ap proach to estimating t he natural rate.
We have used their b aseline specication (see cit., p.320 and appendix A) for both euro area and U S data.
5
Following Sack (2000), the reaction function estimated from the VAR is ignored when solving the central bank’s
minimization problem.
6
Note that the p olicy rule is assumed to be deterministic. Therefore its p osterior unce rtainty fully derives from
the uncertainty of the VAR co e!cients.
17
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September 2009
The following independent prior assumption is specied for each model (now omitting P
m
):
s ()=s () s ()
s ()=Q
³
>Y
´
s
¡
31
¢
= Z
¡
V
31
>
¢
where Z
¡
V
31
>
¢
denotes a Wishart distribution with scale matrix V
31
and degrees of freedom ;
and Q
³
>Y
´
denotes a Normal distribution with mean
and variance-covariance matrix Y
.
The general form of s () in all models is the one of a Minnesota-type, where the prior mean
of coe!cients for the rst own lag is equal to one and the others are set equal to zero; individual
components of are independent of each other, i.e. Y
is a diagonal matrix; and the diagonal
elements of Y
have the structure:
y
lm>o
=
½
(
1
@o)
2
if l = m
(
1
2
l
@o
m
)
2
if l 6= m>
(10)
where y
lm>o
is the prior variance of
lm>o
(coe!cient in equation l relative to variable m at lag o),
1
is the general tightness,
2
is the tightness on “other coe!cients”, and o is the lag.
For all models we assume
1
=0=1 and
2
=1, and estimate the variances
l
and
m
from AR(p)
regressions on the training sample. In all models where we restrict the persistence of in ation or
labor market variables, the o wn-lag coe!cients of the prior mean
are set accordingly, and the
corresponding tightness is set to 10
33
1
. For the AR assumptions of both in ation and labor
variables, the own-lag coe!cients of the prior mean
are estimated on the training sample with
univariate AR(p) regressions.
Regarding the prior for , the prior scale matrix V is set equal to 10
31
L, and the degrees of
freedom equal q +3, thus ensuring an informative but relatively vague prior assumption for .
All in all, the prior assumptions on the unrestricted coe!cients are su!ciently general and not
too tight in order to ensure that the posterior mean of the rst own lag of variables like exchange
rate and commodity price will not necessarily be as persistent as the prior assumption.
Given the independent structure of the prior, a closed form solution for the posterior distribution
of the parameters of interest is not a vailable. It is easy to show, however, that a Gibbs sampler can
be employed because the full conditional distributions s ( | >G) and s ( | > G) are easily derived
(see Appendix). The sampler is initialized using the ML estimate of on the training sample. For
each draw of =(> ), then, the parameters of the rule are derived from the minimization problem.
This algorithm provides the posterior distribution (9).
7
FortheoptimizedTaylorRules,weusea
7
Note that the P in the Gibbs sampler includ es terms from the redu ced form interest rate equation s which are
then zero e d out in the optimal policy computation .
18
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September 2009
grid search procedure to solve for the values of i that minimize the criterion function (5). Because
the computation with high-order models is cumbersome, we solve the optimization problem by
using the posterior median of , instead of grid-searching for each of its draws.
In the case of the optimal feedback rule, instead, the computational burden is not so heavy, for
the optimal values of i and of the loss function are straightforward to compute. Ho wever, in order
to ensure that the parameters of the rule have meaningful signs, we restrict the prior to be
t ()=s () ·=($
1
5 F)
where = ($
1
5 F) is the indicator function that equals 1 if $
1
5 F and 0 otherwise, and F is
the relevant region. The corresponding posterior distribution is therefore t ( | G)=s ( | G) ·
= ($
1
5 F). Strictly speaking, an importance sampling algorithm should be used instead of the
Gibbs sampling, and an importance function elicited. It is easy to show, however, that if the
importance function is the unrestricted posterior distribution we can still use the Gibbs sampling,
drawing from the unrestricted posterior and discarding draws whic h violate the restrictions.
8
Finally, an equal prior probability s (P
m
)=1@M is assigned to each model, therefore the pos-
terior probability of the models is proportional to their marginal likelihood, i.e.
s (P
m
| G)=
s (P
m
) s (G | P
m
)
P
m
s (P
m
) s (G | P
m
)
=
s (G | P
m
)
P
m
s (G | P
m
)
(11)
where s (G | P
m
)=
R
s
¡
G |
P
m
>P
¢
s
¡
P
m
| P
¢
g
P
m
is the marginal lik elihood of model P
m
.An
analytical evaluation of this integ ral is not possible given our prior assumptions. Therefore we
simulate it from the Gibbs output using the harmonic mean of the likelihood values at each draw
of (Newton and Raftery, 1994). Note that the marginal likelihood comparisons and averaging
require the set of left-hand side variables to be the same across models. In the computation of the
harmonic mean, therefore, all marginal likelihoods have been computed on the basis of equations
for the same three endogenous variables, namely unemployment, in ation and interest rate.
9
Results (discussed in the next subsections) are based on 10000 iterations of the Gibbs sampling,
after discarding an initial 5000 burn-in replications and using the remaining 5000 for inference.
8
In particular we assign a zero weight to negative values of the parameters attached to the negative une mp loyment
gap, the in ation gap and the lagged interest rate. Note that a similar approach has b een used by Cogley and Sargent
(2005) and B enati (2008) in dierent contexts, to rule out explosive autoregressive roots in VARs with time-va rying
parameters.
9
If the VAR is written as a linear regression m odel, | = [ + P
1@2
%, under the normality assumption a linear
transformation of |, U|, is also normal. In all mo d els, th erefore , the matrix U selects always the same endogenous
variables when computing the likelihood values.
19
ECB
September 2009
3.3 Properties of m odel space and rules
The properties of the model space can be brie y described by focusing on the Marginal Likelihood,
the parameters of the rules, and the expected losses.
In Figure 1 we plot the Relative Marginal Likelihood (RML) of the models, dened as in (11),
where m goes from 1 to 224. Given an equal prior model probability, s (P
m
), the RML measures how
likely the data believes a given model is the most appropriate one. Models are ordered according
to scheme described in appendix A (Table A1), in ascending number of lags.
Figure 1 about here
The RML turn out to be substantially dierent across models, as shown by the dierence
between the highest and the lowest values, and by the fact that, especially for the euro area, only
for few models the RML is greater than the equal weight (EW).
The data support relatively parsimonious models, and the best models are clustered around
specications with three and four variables, particularly the specications which include 3 lags for
the US and 4 lags for the euro area. More interestingly, there is clear evidence that a specication
which includes (either jointly or alternatively) the participation rate and the exchange rate is highly
supported by both the US and the euro area data, meaning that the inclusion of these variables
in an otherwise standard VAR model may be important to obtain an appropriate inference on
the eects of policy on unemployment. Data also support models with moderate persistence in
the labor market variables, and with an economic-based and a statistical-based detrending of the
unemployment rate for Euro area and US, respectively.
The posterior distributions of the optimal policy parameters and the associated expected losses
across models are summarized in Figure 2 and 3. Figure 2 reports the posterior distributions of
the relevant parameters and of the losses for the OFR and each model. The solid black line that
goes through the areas is the posterior median of each model. The shaded areas comprise the 95
percent of the posterior distribution around it, as in a fan chart representation: there are an equal
number of bands on either side of the central band. The latter covers the interquartile range and is
shaded with the deepest intensity. The next deepest shade, on both sides of the central band, takes
the distribution out to 80%; and so on until the 95% of the distribution is covered. Models on the
x-axes are organized according to two layers of complexity: they are rst sorted in ascending lag
length order and then by number of variables.
Figure 2 about here
20
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September 2009
In Figure 3 we summarize instead the distribution of the optimal policy parameters and expected
losses by only taking the posterior median across models. In this way, we can visually compare
results also across the two rules.
10
The box plots report the extreme values and the in terquartile
ranges computed using the posterior medians across the 224 models in a given class (OFR or
TR) of the relevant policy parameters and the expected losses. For the TR, where l
w
= i
x
x
w
+
i
w
+ i
l
l
w31
,thecoe!cient on interest rate is simply i
l
, whereas the unemployment and in ation
long-run reaction coe!cients are computed as i
x
@ (1 i
l
) and i
@ (1 i
l
), respectively. For the
OFR, where the policy rate depends also on the lags of the variables, i.e., l
w
=
P
s31
m=0
i
m
x
x
w3m
+
P
s31
m=0
i
m
w3m
+
P
s31
m=1
i
m
l
l
w3m
+ i
0
Z
Z,andZ con tains all other non-policy variables, the respective
coe!cients are
P
s31
m=1
i
m
l
,
P
s31
m=0
i
m
x
@
³
1
P
s31
m=1
i
m
l
´
,and
P
s31
m=0
i
m
@
³
1
P
s31
m=1
i
m
l
´
,wheres is the
order of autoregression of the estimated model. The dark squares in the box plot are the weighted
averages of the results, where the w eights are given by the RML. The empty circles represent instead
results associated with the best models (i.e. Model 196 for the euro area and Model 117 for US as
describedinTableA.1).
Figure 3 about here
Some considerations emerge from the charts. The rst immediate feature is the high degree of
uncertainty, as measured by the dispersion of the results both within and between models. The
average ranges of results are, however, consistent with previous literature, as the bulks of the
distributions are concentrated on values in line both with the theory and with previous empirical
ndings, for both classes of rules. The dispersion across models seems to be only marginally larger
for the TR than for the OFR in both countries, and results seem more volatile across models for
the euro area than for the US.
A closer look sho ws that the in terquartile range of the optimal long-run reaction of unemploy-
ment is [1=7 3=5] for the US and [0=7 2=8] for the euro area; the long run reaction of in ation
is in the range [1=1 2=5] for the US and [1=3 2=6] for the euro area; and the lagged interest rate
coe!cient is in the range [0=1 0=65] for both countries, with the variance of the distribution of
coe!cients derived from the OFR signican tly smaller than the one obtained from the TR. The
weighted averages and the results associated with the best models are very much similar to the
median values. These ndings indicate that in both countries the policies have on average been
marginally more aggressive than the original Taylor rule, and that interest rate smoothing is a ro-
bust feature of the policy. Very similar results have been found by, for instance, Brock et al. (2007),
10
Recall that d ue to the complexity of the grid search in the TR, we simulate the posterior distribution of parameters
and losses only for the O FR, whereas for the TR we compute the estimates of i using the posterior mean of =(> P).
21
ECB
September 2009
Levin and Williams (2003), and Clarida et al (2000), for the US; and by Smets and Wouters (2005),
and Gerlach and Schnabel (2000), among others, for the euro area.
Comparing the two economies, the euro area policy rate reacts on average relatively more to
in ation than to unemployment, whereas the opposite seems to be true for the US policy rate
(on this see also Sah uc and Smets 2008). Another interesting nding is the negative relationship
between the optimal policy parameters and the model complexity as the median values in Figure
2 are clearly decreasing by lags and coe!cients spike up with the rst prior and short lag length.
This pattern is more evident for the euro area than for the US, and partly conrms previous results
which relate model complexity and optimal parameters (see e.g. Brock et al. 2007).
Finally, posterior expected losses are also consistent with the existing literature using similar
values for the weights in the loss function. If anything, our estimates seem to be on the lower side
(see e.g. Brock et al., 2007; and Rudebush and Svensson, 1999 for a comparison) and become not-
ably similar to those obtained by previous studies only under the autoregressive prior for in ation.
Interestingly, the posterior losses associated with the best models are overall lower than the average
(except in the Taylor Rule for the US).
In sum, the evidence provided above conrms that simple linear autoregressive models may
giverisetoasignicant degree of uncertainty in the distribution of optimal policy parameters
and expected losses. Simple or weighted averages across models help dampen this uncertainty and
provide a reasonable representation of the policy rules. Our results would also suggest the choice
of a relatively parsimonious representation of the economy, regardless of the country and the policy
rules.
4Eects of policy on une m ploym ent
The successful conduct of monetary policy requires policymak ers not only to specify a set of object-
ives for the performance of the economy but also to understand the eects of policies designed to
attain these goals. In this section, therefore, we will answer the following questions: Given the set
of objectives and rules, what are the eects of policy prescriptions that come from dierent models
on the unemployment gap? What is the role of model uncertainty and what are the consequences
for policymakers of allowing for it?
The estimation algorithm directly follows from the one described in Section 3. Using the
structural VAR in Eq. (3), we assume that the central bank sets the policy variables l
w
according
to the two policy rules OFR and TR as estimated in the previous step. The estimated equation error
l
w
can be interpreted as a monetary policy shoc k, as also discussed e.g. by Stoc k and Watson (2001),
or Sack (2000). The shock is identied by (i) replacing the parameters of the policy equation with
22
ECB
September 2009
the posterior means of the i estimated above, while leaving unrestricted all the other parameters of
the VAR; and (ii) imposing the timing assumption that the central bank reacts contemporaneously
to all variables in the economy, whereas the policy rate does not contemporaneously aect the
rest of the economy. The former restriction is placed in the form of a normal distribution with a
very tight variance. The latter restriction is a pure zero-restriction. A relatively vague Minnesota
prior is assumed on the rest of parameters in the two blocks. Results are reported in terms of
the probability distributions of the responses to the identied monetary policy shock (Figure 4
and Tables 1-2); in terms of variance decomposition (Figure 5); and in terms of the transmission
mechanism (Figure 6).
4.1 Impulse response dispersion
Figure 4 reports the responses of unemploym ent gap to a 100-basis-point contractionary monet-
ary policy for both countries and rules. Since the unemploym ent gap has been computed as the
dierence between the natural rate of unemployment (x
W
w
) and its actual value (x
w
),aslowdown
correspond to a negative response.
To jointly visualize the “average” eect and the dispersion within and between models we
report the posterior distribution of the IRF obtained from the Mark ov Chain Monte Carlo (MCMC)
simulation by ‘fan-charting’ separately three quantiles of such distributions — the median responses,
the 16wk percentile and the 84wk percentile — for all models. Therefore, in the c harts with the title
‘median’, for instance, we plot the distribution across models of the median responses. In each
chart, the shaded areas represent the dispersion across models. The principle has already been
described for Figure 2: there is an equal number of bands on either side of the central band. The
latter covers the interquartile range across models and is shaded with the deepest intensity. The
next deepest shade, on both sides of the central band, takes the distribution out to 80%;andso
on up to the 95%. The solid black line that goes through the areas is the weighted average of each
quantile (median, 16wk and 84wk percentile) across models, where the weights are given by the RML
of each model.
A detailed quantication of the responses is also reported in Table 1, which displays the impacts
computed from the median of Figure 4 and reports the 10wk and the 90wk percentile, the median
and the weighted average across the 224 models.
Figure 4 and Table 1 about here
Four preliminary comments are in order.
23
ECB
September 2009
First, impulse responses look reasonably well behaved and give rise to the usual hump-shaped
dynamics. Their pattern is fairly robust across models, countries and rules. One dimension of
such robustness is that, although model responses are very much dispersed — and therefore any
statement on statistical signicance w ould require some caution, especially for the euro area — the
68% posterior probability intervals do not include the 0 at the horizons of the peak eects, and
this, on average, appears to be a stable feature.
Second, regarding the dynamics, most of the signicant economic slowdown occurs in the rst
two years after the rate hike, when the cumulative impact on the unemployment gap is between
-0.2 and -0.3 percentage points, on average across models, rules and countries. Measured on the
weighted average response across models (the dark line in the c harts), the (negative) unemployment
gap reaches a maximum decline of around 5 basis points 5-6 quarters after the contractionary
monetary policy shock for the US, and of 4 basis points 4-5 quarters after the rise in interest rate
for the euro area. Half of the maximum eect on the gap disappears after about 9 to 11 quarters
for both economies.
It is important to note that, the timing of the peak eect obtained by the previous literature is
very consistent with our results (see e.g. Christiano et al, 1996; Stock and Watson, 2001; Bernanke
et al. 2005). The size of our eects appears to be more subdued than in other studies, most likely
because our responses are measured on the unemployment gap and not on the unemployment rate.
Intuitively, as the natural rate of unemployment is not constant in the measurement of the gap, a
contractionary monetary policy shock might lead to an increase in the natural rate itself, after an
initial increase in the actual unemployment rate, and this in turn w ould explain the mu"ed eect
on the gap.
11
Third, impulse responses are only marginally sensitive to the policy rule used in the identication
of the structural VAR. Visual inspection, however, seems to show that results based on the TR are
to some extent less dispersed than those based on OFR, and also that with a TR the average peak
eects might be dela yed of one or two quarters with respect to the OFR, in both economies. These
results do not come entirely as a surprise for, even if both rules are backward-looking, the OFR
is less restrictive than the TR, being a function of all current and lagged values of the non-policy
variables beside the lagged values of the interest rate.
Fourth, there is a substantial degree of uncertainty across models, for a given rule or country.
The dispersion is signicant for both economies and regardless of the policy rules in particular
11
A more extensive analysis of this point go es clearly beyond the scope of this paper. We have, however, run a
subset of models with the (dem e an ed) actual unemployment rate instead of the u nem ployment gap . Coeteris paribus,
the average responses were doubled, thus conrming our intuition that the transformation used is in part resp onsible
for the result. In a companion pap er (Altavilla and Ciccarelli, 2007), where we use the actual rate instead of the gap,
our impulse resp onses are the same as, e.g., those obtained by Stock and Watson (2001).
24
ECB
September 2009
around the peak values of the responses, between one and two years. Nonetheless, overall results
for the US are m uch less dispersed than those for the euro area where some models can even show
puzzling positive eects of monetary policy on the (negative) unemployment gap at the crucial
horizons. Moreover, for the euro area the weighted average provides more muted responses at the
peak than a simple average, meaning that models which receive more support b y the data — and
therefore are weighted more in the average — tend to dampen the response of unemployment to a
monetary policy shock relatively to the other models.
Two conclusions can be drawn from the comparison between US and euro area results. First,
the elevated dispersion across models implies that policy decisions based on few selected models
— as opposed to a combination from several of them — may potentially give a twisted picture of
the policy eects, and this, in turn, might lead to policy mistakes. Second, while the degree of
uncertainty can dier considerably across countries, as Figure 4 shows, the average impacts hardly
exhibit meaningful dierences across the two economies, both in terms of timing and in terms
of magnitude. We interpret this evidence as a warning for other comparative studies which ma y
nd signicant dierences in the reaction of unemployment to a monetary policy shock across the
two economies. Our results suggest that major discrepancies could mainly arise as an outcome
of conditioning the analysis on few specic models, instead of accounting for model uncertainty.
Consequent policy decisions taken on the basis of presumed dierences between the two economies
could therefore lead to distorting eects.
4.2 Variance decomposition
So far the discussion seems to indicate that, albeit a mute one, monetary policy shocks play a
similar recessionary role for unemployment uctuations in both economies. In order to examine
from a dierent perspectiv e the relative importance of the identied shock for the v olatility of the
unemployment gap, we have also inspected the forecast error variance decomposition. Results —
reported in Figure 5 using the same fan-chart approach — show that at short and long horizons
only a small fraction of the forecast error variance of the unemployment gap is accounted for by
the monetary policy shock which, beyond the one year horizon, is never contributing with more
than 10 percent on average across countries and rules.
Interestingly, the dispersion across models of the percentage of variance explained by the iden-
tied shock is very tiny when compared with the dispersion of the portion of variance explained by
other non-policy variables. This, in turn, leads to two additional considerations. On the one hand,
it seems that there are important sources of variability in unemploym ent that are not identied
by the monetary policy shock and are re ected in the portion of variance explained by the other
25
ECB
September 2009
variables of the model space. On the other hand, it suggests that such a muted contribution of
the monetary policy shock would an yway be a robust feature, should we not account for model
uncertainty.
Figure 5 about here
Clearly, any speculation about the role that other structural shocks might have played goes
beyond the scope of this paper. It is nonetheless interesting to remark that the selected non-policy
variables can help explain up to 40 percent of the movements in unemployment gap bey ond the
three-year horizon. Incidentally, this high percentage amply justies our prior variable selection to
construct the model space.
Asignicant role is played in particular by the labor force participation rate whose variability
helps explain 20 to 25 percent of the variability of unemployment gap across countries, models and
rules after a two-year horizon.
12
This is a remarkable result and reinforces the nding - previously
discussed in Section 3 - that including labor force participation often increases the relative posterior
probabilities (and lowers the value of the loss function), meaning that the data at hand support
the importance of this variable to understand the eects of policy on unemployment.
One might want to ask, therefore, whether the impact of a monetary policy shock measured with
our model space may change (and by how m uch) depending on the presence of the participation
rate in the specication. An attempt to describe and quantify a plausible answer could be based on
the same kind of inference discussed so far, only dividing the set of models in two groups, according
to the presence of the labor force participation rate among the non-policy variables.
In table 2 we report the evidence on unemployment gap. Given the model space described in
table A.1 we ha ve 112 models including participation rate (denoted as “with” in table 2) and 112
models which do not include participation rate (denoted as “without” in table 2). The quantiles
and the weighted averages have been computed from the median responses of all models as in
Table 1. Although the dierences might not be impressive, they point out that on average the
monetary policy eect is slightly more muted in models that contain the participation rate. This
is easily rationalized and it is in line with the evidence reported in table 1 showing that the simple
average across models provides a deeper impact than the weighted average, as the models with
the highest RML always contain the participation rate. Intuitively, in models with participation a
contractionary monetary policy shock eventually has a negative impact on the participation rate
(see below) and this, in turn, reduces the initial impact on unemployment.
12
Note that the average variances explained by each variable as shown in Figure 6 cannot sum up to one as not all
variables appear a lways in the sam e m odels. Therefore the variance attributed to the single variables refers to the
fraction of the variance explained by these variables only in mo dels whose specication contains them.
26
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September 2009
4.3 Transmission mechanism
The dierence in impacts displayed in table 2 is similar in the two economies and seems only
marginally more pronounced for the Euro area than for the US. This turns out to be related to
adierent transmission mechanism and in particular to dierent dynamics that the participation
rate shows in the two countries in response to a monetary policy shock. To better analyze this
point, Figure 6 reports the distributions (o ver all models) of the median responses of all variables.
Figure 6 about here
Average responses (the darkest areas of the charts or the black lines inside them) presen t the
expected signs and patterns, and, except for the somewhat uncertain response of the exchange rate
for the US in the OFR, which migh t depend on the subsequent dynamics of the interest rate after
the initial hike, they are also qualitatively similar in the two economies.
More interestingly, in both countries the response function for the in ation rate obtained by
combining all models with the Bayesian averaging scheme (the black line in all graphs) exhibits a
small price-puzzle. This evidence might suggest that the positive reaction of the in ation rate to
a monetary policy shock is a model-dependent phenomenon that tends to disappear when taking
into account model uncertainty.
One of the main dierences in the transmission mechanisms of the two economies is clearly
related to the dierent responses of the participation rate. In the US the response of labor force
participation follows with some lag the inverted U-shape of the unemployment response, has broadly
the same magnitude, and peaks 8 to 9 quarters after the rise in interest rate. The fall in participation
rate is therefore consequent to the initial impact on unemployment.
In the euro area, on the contrary, the participation rate not only reacts earlier than unemploy-
ment, but it also displays an initial positive response which, depending on the policy rule, may
last more than one year. Afterwards the response becomes negative, with a maximum decrease
reached only around 9 to 14 quarters after the initial increase in the interest rate. This pattern
may help explain the greater uncertainty around the unemployment responses in the euro area,
and is consistent with a lower degree of exibility of the European labor market with respect to
the one exhibited in the US, where a contractionary shoc k directly in uences the unemployment
gap without being transmitted through the participation (see, e.g. Blanchard and Katz, 1992; and
Blanc hard, 2006).
The initial positive response of participation in the euro area —which is also responsible for the
slightly more persistent dynamics of the unemployment response and for some positive responses
of unemployment right after the shock— is not necessarily unreasonable and can be rationalized
27
ECB
September 2009
from a theoretical perspective. After a contractionary monetary policy shock, unemployed workers
may stop actively looking for a job and exit the labor force. This eect —that the literature has
typically denominated “discouraged w orker eect”— gives rise to a net reduction of the labor force
participation rate. On the other hand, the same contractionary monetary policy can force workers
who are currently outside the labor force to start actively looking for a job, and this, in turn, may
result in a positive eect on the participation rate. In fact, secondary workers (women and youths)
might start seeking employment because of drop in primary workers’ wages and employment. The
literature has typically referred to this phenomenon as “added work eect”. Theoretically, models
of family utility maximization indicate that a decrease in family income due to the earnings losses
of one family member might be oset by increases in the labor supply of others (e.g. Stephens,
2002).
In the comparison between the US and the euro area the relative importance of the two ef-
fects, in combination with dierent degrees of exibility in the labor market, provides a reasonable
explanation for the dierent transmission mechanisms.
5Conclusiveremarks
We have shown that model uncertainty plays a crucial role in determining the eects of monetary
policy shocks on unemployment dynamics in the euro area and the US.
Our ndings support the view that in order to overcome severe policy mistakes, decisions could
be based on a wide range of possible scenarios about the structure of the economy. As a result,
when allowing for model uncertainty, policy a dvice ma y look signicantly dierent from the one
that would be optimal based on few selected models.
With the help of a Ba yesian model averaging procedure to account for the uncertainty inherent
to the model selection process, we have specied a range of 224 BVAR models that dier in several
dimensions according to assumptions regarding in ation, persistence of labor market variables,
measurement of the natural rate of unemployment, number of variables and lag structure. Each
model represents a constraint for the central bank which sets the interest rate minimizing a social
loss function. Given the solution in terms of policy rule, we have quantied the impact of a
monetary policy shock on unemployment and measured the degree of uncertainty as represented b y
the dispersion of both the policy rule parameters and the impulse response functions across models.
The comparative evidence from the US and the euro conrms that simple linear autoregressive
models that dier in several dimensions may give rise to a signicant degree of uncertainty in the
distribution of optimal policy parameters, expected losses and impulse response functions.
We have shown that, although a monetary policy shock might be less important than other
28
ECB
September 2009
structural shocks to explain unemployment dynamics, it has a stable recessionary eect. Moreover,
the average unemployment responses for the US and the euro area are qualitatively and quantit-
atively very similar, with results for the euro area being more dispersed than those for the US.
The analysis of the transmission mec hanism also indicates that other labor market variables suc h
as participation rate pla y a signicantly dierent role in the transmission mechanism of the two
economies.
One of the main policy implications of our results is that combining results from alternative
representations of the structure of the econom y represents a useful strategy to account for model
uncertainty when assessing the risks for price stability or when deciding a given policy. In particular,
our results show that a policymaker who selects the results on the basis of a single model may come
to misleading conclusions not only about the transmission mechanism —pic k ing up models where,
for instance, the price puzzle is more marked or the eect on unemployment has a wrong sign— but
also about the dierences between the euro area and the US, which on average are tiny. By allowing
for model uncertainty, instead, results are on average closer to what we expect from a theoretical
point of view, and put the policymaker in a favorable position to calibrate the policy interventions
in a more appropriate w ay, that is, more consistently with the economic theory and less distorting
for the economy.
Some extensions that enrich the previous analysis are feasible in the same framework. Another
dimension of uncertainty could be explored perhaps in a unied framework that considers model
and data uncertainty. First-released data are often no isy, as incomplete or mismeasured initial
information has been used in their construction and it may take sev eral years of revisions before
data are considered as nal. All relevant information for monetary policy is, therefore, measured
with error and the dierence between the responses obtained with real-time vs nal data might be
sizable.
The model space can also be enlarged by considering several alternative economic models in
the estimation of the natural rate of unemployment based on the Phillips curve. We have taken a
shortcut and considered, instead, only one possible specication ignoring further sources of uncer-
tainty.
Finally, the set of models could be further expanded by including additional labor market
variables such as wages, which would provide a more complete dynamics of the labor market and a
richer transmission mechanism. Wages would in fact re ect the conditions on which the equilibrium
in the labor market is established and might, at the same time, give some indications on the price
formation process or the existence of nominal pressures on the path of prices.
29
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September 2009
Appendix
AModels
The table A1 describes the 224 models that span the model space. The rst column reports the
model number. In the second column the models specication is detailed with the number and the
t ype of variables used; the third column reports the codication. Eac h model is characterized by
4 elements: the number of variables (V), the number of lags (L), the type of prior (P), and the
type of detrending method used in the calculation of the natural rate of unemployment (U). The
estimated VAR can have three (3V) to six (6V) endogenous variables, and one (1L) to four (4L)
lags. Five types of priors are possible. With the rst prior (1P), both the in ation persistence
and the persistence of the labor mark et variables are unrestricted, in the sense that a very loose
prior assumption is assumed. With the second (2P), third (3P) and fourth (4P) prior, in ation is
assumed to be a Random Walk, an Autoregressive process and a White Noise respectively, while
the persistence of the labor market variables is unrestricted. With the fth prior (5P), there is
no restriction on the in ation persistence and the labor market variables are assumed to follow an
Autoregressive process. Two types of detrending methods are used to compute the natural rate
of unemployment. The rst one (1U) uses the Baxter and King band pass lter. The second one
(2U) is a Phillips-curve-based method estimated with Kalman Filter techniques. As an example, in
model 117 (coded as 3V_3L_5P_1U) there are three variables (unemployment, in ation rate and
interest rate), three lags, the prior on in ation and unemployment is the fth one, and the natural
rate of unemployment has been computed with Baxter and King’s method.
Table A1 he re
B Derivation of the posterior
By stacking appropriately variables and coe!cients in the VAR (3), we can re-write it as:
|
w
=(L
q
Z
w
) + %
w
(12)
where, |
w
is the (q × 1) vector of endogenous variables []
0
w
l
0
w
]
0
, Z
w
=
¡
|
0
w31
>===>|
0
w3s
¢
0
is n × 1, is
the qn × 1 vectorization of all coe!cien ts, %
w
is the (q × 1) vector of reduced form innovations, and
n = qs is the number of parameters in each equation.
Because by assumption it is s (%
w
)=Q (0> ), the likelihood is proportional to
O (G | > ) 2 ||
3W@2
exp
(
1
2
X
w
[|
w
(L
q
Z
w
) ]
0
31
[|
w
(L
q
Z
w
) ]
)
(13)
30
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September 2009
where, as in the text, G represents the stacked data.
Given the joint prior distribution on the parameters, s (> ), the joint posterior distribution of
the parameters conditional on the data is obtained through the Bayes rule
s (> | G)=
s (>) O (G | > )
s (G)
2 s (> ) O (G | > ) >
We have assumed an independent Normal-Wishart distribution for the prior, with
s ()=Q
³
>Y
´
2
¯
¯
¯
Y
¯
¯
¯
31@2
exp
½
1
2
¡
¢
0
Y
31
¡
¢
¾
(14)
and
s
¡
31
¢
= Z
¡
V
31
>
¢
2 ||
3(3q31)@2
exp
½
1
2
wu
¡
V
31
¢
¾
(15)
As remarked above (Section 3), the chosen hyperparameters V and ensure a relatively vague prior
assumption for and therefore for most terms of the Cholesky decomposition. The joint posterior
density for (> ) is proportional to the product of (13), (14), and (15). Given the independency
assumption, such posterior does not take the form of a standard distribution and cannot be directly
used for inference. A Gibbs sampling algorithm is instead available, for the conditional posterior
of both and aresimpletoderive. Theconditionalposteriorof is derived by multiplying (13)
and (14), and ignoring the terms that in the product do not involve .Itisgivenby
s ( | G> )=Q
¡
¯
>
¯
Y
¢
2 exp
½
1
2
¡
¯
¢
0
¯
Y
31
¡
¯
¢
¾
(16)
where
¯
Y
=
Ã
X
w
(L
q
Z
w
)
0
31
(L
q
Z
w
)+Y
31
!
31
¯
=
¯
Y
Ã
X
w
(L
q
Z
w
)
0
31
|
w
+ Y
31
!
Similarly, the conditional posterior for is derived by multiplying (13) and (15). Ignoring the
terms that do not involve ,wehave
s
¡
31
| G>
¢
= Z
¡
V
W31
>
W
¢
2 ||
3(
W
3q31)@2
exp
½
1
2
wu
¡
V
W
31
¢
¾
(17)
31
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September 2009
where
V
W
= V +
X
w
[|
w
(L
q
Z
w
) ][|
w
(L
q
Z
w
) ]
0
W
= + W
Starting from arbitrary values of , a Gibbs algorithm samples alternately from (16) and (17). For
each draw of the posterior the minimization problem is solved and the empirical distributions of
the policy rules parameters and the losses are computed.
32
ECB
September 2009
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Table 1: Properties of Impulse Response Functions. Eects on unemployment gap of a 100 basis-
point contractionary monetary policy shock
10th median wgt. ave. 90th 10th median wgt. ave. 90th
1 Quarter -0.035 -0.022 -0.014 0.003 -0.029 -0.013 -0.014 -0.006
2 Quarters -0.063 -0.044 -0.027 -0.001 -0.047 -0.030 -0.031 -0.018
3 Quarters -0.075 -0.053 -0.034 -0.004 -0.059 -0.042 -0.043 -0.028
4 Quarters -0.076 -0.053 -0.035 -0.006 -0.065 -0.048 -0.048 -0.034
5 Quarters -0.071 -0.045 -0.033 -0.007 -0.066 -0.048 -0.048 -0.035
6-8 Quarters -0.053 -0.028 -0.024 -0.003 -0.058 -0.038 -0.040 -0.027
9-12 Quarters -0.021 -0.005 -0.008 0.013 -0.033 -0.016 -0.019 -0.010
cumulative impact after 2 years -0.479 -0.303 -0.197 -0.026 -0.439 -0.297 -0.270 -0.203
variance decom
p
osition 2.31
3
4.322 3.885 5.248 3.182 4.949 5.248 7.216
10th median wgt. ave. 90th 10th median wgt. ave. 90th
1 Quarter -0.036 -0.016 -0.011 0.002 -0.032 -0.015 -0.018 -0.001
2 Quarters -0.056 -0.030 -0.022 -0.005 -0.050 -0.027 -0.031 -0.013
3 Quarters -0.067 -0.038 -0.029 -0.012 -0.062 -0.034 -0.038 -0.018
4 Quarters -0.070 -0.044 -0.033 -0.019 -0.066 -0.038 -0.042 -0.021
5 Quarters -0.070 -0.046 -0.035 -0.023 -0.063 -0.042 -0.043 -0.023
6-8 Quarters -0.061 -0.042 -0.034 -0.023 -0.056 -0.039 -0.038 -0.021
9-12 Quarters -0.041 -0.022 -0.024 -0.010 -0.035 -0.023 -0.022 -0.012
cumulative impact after 2 years -0.482 -0.298 -0.201 -0.127 -0.442 -0.271 -0.251 -0.140
variance decomposition
2.304 3.285 4.605 5.084 3.344 5.141 5.869 8.156
Euro Area US
Euro Area US
Optimal Feedback rule
Taylor Rule
Note: The table reports the posterior impulse responses of unem ployment gap to a 100 basis-point contractionary m onetary policy
shock. The top and the bottom panel refer to the responses under the Optimal Feedback Rule and the Taylor Rule, respectively.
C olum n (2 ) to (4) refer to th e eu ro area resu lts. Colum n (6) to (9 ) refer to the U S results. R ow s from (1 ) to (5) rep ort th e qu antiles
of the simple responses. Rows (6) and (7) report a time average of the quantiles over the second half of the second year and over the
third year respectively. R ow (8) reports the cumulative impact after 8 quarters. Row (9) reports the p ercentage of the variance of the
unemployment gap 24-quarter-ahead forecast errors explained by the m onetary p olicy shock. The reported quantiles (10th, median,
average and 90th) are com puted over the distribution of the posterior median responses across the 224 m odels. The column "average"
repo rts a w eig hted ave ra ge ove r a ll m o d els with w eig hts g iven by th e r elat ive m a rg ina l like liho o d c om p u ted as in E q. 11 o f t he p a p er.
36
ECB
September 2009
Table 2: Change in inference due to labor force participation. Eects on the responses of unem-
ployment gap to a 100 basis-point contractionary monetary policy shock
steps
with without with without with without with without with without with without
1 Quarter -0.035 -0.035 -0.020 -0.013 0.006 0.001 -0.028 -0.029 -0.015 -0.013 -0.007 -0.005
2 Quarters -0.063 -0.064 -0.036 -0.025 0.004 -0.005 -0.047 -0.048 -0.033 -0.030 -0.020 -0.018
3 Quarters -0.074 -0.080 -0.043 -0.032 0.000 -0.009 -0.058 -0.059 -0.044 -0.042 -0.029 -0.028
4 Quarters -0.073 -0.082 -0.042 -0.034 -0.001 -0.012 -0.065 -0.064 -0.049 -0.047 -0.033 -0.034
5 Quarters -0.066 -0.078 -0.037 -0.033 0.000 -0.013 -0.066 -0.066 -0.048 -0.048 -0.035 -0.035
6-8 Quarters -0.043 -0.058 -0.022 -0.025 0.002 -0.007 -0.058 -0.057 -0.039 -0.041 -0.027 -0.028
9-12 Quarters -0.014 -0.025 -0.001 -0.011 0.015 0.007 -0.031 -0.034 -0.017 -0.021 -0.010 -0.010
cumulative impact after 2 years -0.440 -0.513 -0.244 -0.213 0.014 -0.060 -0.438 -0.437 -0.305 -0.303 -0.205 -0.204
cumulative impact after 6 years -0.547 -0.730 -0.150 -0.226 0.338 0.278 -0.546 -0.602 -0.297 -0.339 -0.104 -0.108
variance decomposition
2.176 2.689 3.275 4.097 4.504 6.004 2.996 4.202 4.475 5.800 6.040 7.567
steps
with without with without with without with without with without with without
1 Quarter -0.039 -0.033 -0.010 -0.010 0.003 0.000 -0.030 -0.033 -0.018 -0.018 0.000 -0.003
2 Quarters -0.059 -0.054 -0.021 -0.021 -0.004 -0.007 -0.050 -0.049 -0.030 -0.031 -0.010 -0.014
3 Quarters -0.068 -0.065 -0.029 -0.028 -0.011 -0.015 -0.062 -0.061 -0.037 -0.039 -0.015 -0.021
4 Quarters -0.069 -0.071 -0.034 -0.032 -0.018 -0.019 -0.066 -0.067 -0.041 -0.042 -0.018 -0.026
5 Quarters -0.070 -0.071 -0.037 -0.034 -0.023 -0.023 -0.062 -0.063 -0.042 -0.044 -0.021 -0.027
6-8 Quarters -0.062 -0.060 -0.036 -0.033 -0.024 -0.023 -0.055 -0.058 -0.037 -0.039 -0.020 -0.024
9-12 Quarters -0.044 -0.039 -0.025 -0.023 -0.011 -0.010 -0.033 -0.036 -0.021 -0.023 -0.011 -0.013
cumulative im
p
act after 2
y
ears -0.490 -0.476 -0.239 -0.225 -0.125 -0.133 -0.434 -0.447 -0.278 -0.292 -0.123 -0.161
cumulative im
p
act after 6
y
ears -0.880 -0.790 -0.358 -0.309 0.078 0.135 -0.615 -0.649 -0.306 -0.332 0.000 -0.042
variance decomposition
2.266 2.323 4.080 4.176 5.091 4.990 3.061 3.950 6.145 7.335 7.108 8.361
weighted average 90th10th weighted average 90th 10th
weighted average 90th
Taylor Rule
Euro Area US
10th weighted average 90th 10th
Optimal Feedback rule
Euro Area US
Note: The table reports the impulse responses of unemploym ent gap to a 100 basis-point contractionary monetary policy shock. The
top a nd the b ottom p anel refer to th e resp onses un der the O ptim al Feedb ack R ule an d the Taylor R ule, resp ectively. C olum n (2) to
(7) refer to the euro area results. C olum n (8) to (13) refer to th e U S results. R ows from (1) to (5) rep ort the qu antiles of the sim ple
responses. Rows (6) and (7) report a time average of the quantiles over the second half of the second year and over the third year
resp ectively. R ow (8) and (9 ) rep ort the cum ulative im pa ct after 8 and 24 quarters resp ectively. R ow (10) rep orts the p ercenta ge o f
the varian ce of the unem ploym ent gap 24-q uarter-ahea d forecast errors ex plained by th e m on etary p olicy sh ock. T he rep orted qu antiles
(10th, weighted average and 90th) are computed over the distribution of the posterior m edian responses across the 224 m odels. The
we igh ted ave rag e is t ake n ove r a ll m o d els w it h we igh ts g ive n by t he r elat ive m a rgin al like liho o d co m p ut ed a s in E q . 11 o f th e p ap er .
Results for each quantile are reported for two classes of m odels, according to whether the model includes (column "with") or does not
include (colum n "without") the labor force participation rate in the speci cation. Note that, given the model space described in table
A.1, there are 112 m odels with participation rate and 112 models without.
37
ECB
September 2009
Table A.1: Mapping of the model numbers
Model
number
Variables Code
Model
number
Variables Code
Model 1
3 Variables
3V_1L_1P_1U Model 57
3 Variables
3V_2L_1P_1U
Model 2 3V_1L_2P_1U Model 58 3V_2L_2P_1U
Model 3 3V_1L_3P_1U Model 59 3V_2L_3P_1U
Model 4 3V_1L_4P_1U Model 60 3V_2L_4P_1U
Model 5 3V_1L_5P_1U Model 61 3V_2L_5P_1U
Model 6 3V_1L_1P_2U Model 62 3V_2L_1P_2U
Model 7 3V_1L_5P_2U Model 63 3V_2L_5P_2U
Model 8
4 Variables
4V_1L_1P_1U Model 64
4 Variables
4V_2L_1P_1U
Model 9 4V_1L_2P_1U Model 65 4V_2L_2P_1U
Model 10 4V_1L_3P_1U Model 66 4V_2L_3P_1U
Model 11 4V_1L_4P_1U Model 67 4V_2L_4P_1U
Model 12 4V_1L_5P_1U Model 68 4V_2L_5P_1U
Model 13 4V_1L_1P_2U Model 69 4V_2L_1P_2U
Model 14 4V_1L_5P_2U Model 70 4V_2L_5P_2U
Model 15
4 Variables
4V_1L_1P_1U Model 71
4 Variables
4V_2L_1P_1U
Model 16 4V_1L_2P_1U Model 72 4V_2L_2P_1U
Model 17 4V_1L_3P_1U Model 73 4V_2L_3P_1U
Model 18 4V_1L_4P_1U Model 74 4V_2L_4P_1U
Model 19 4V_1L_5P_1U Model 75 4V_2L_5P_1U
Model 20 4V_1L_1P_2U Model 76 4V_2L_1P_2U
Model 21 4V_1L_5P_2U Model 77 4V_2L_5P_2U
Model 22
4 Variables
4V_1L_1P_1U Model 78
4 Variables
4V_2L_1P_1U
Model 23 4V_1L_2P_1U Model 79 4V_2L_2P_1U
Model 24 4V_1L_3P_1U Model 80 4V_2L_3P_1U
Model 25 4V_1L_4P_1U Model 81 4V_2L_4P_1U
Model 26 4V_1L_5P_1U Model 82 4V_2L_5P_1U
Model 27 4V_1L_1P_2U Model 83 4V_2L_1P_2U
Model 28 4V_1L_5P_2U Model 84 4V_2L_5P_2U
Model 29
5 Variables
5V_1L_1P_1U Model 85
5 Variables
5V_2L_1P_1U
Model 30 5V_1L_2P_1U Model 86 5V_2L_2P_1U
Model 31 5V_1L_3P_1U Model 87 5V_2L_3P_1U
Model 32 5V_1L_4P_1U Model 88 5V_2L_4P_1U
Model 33 5V_1L_5P_1U Model 89 5V_2L_5P_1U
Model 34 5V_1L_1P_2U Model 90 5V_2L_1P_2U
Model 35 5V_1L_5P_2U Model 91 5V_2L_5P_2U
Model 36
5 Variables
5V_1L_1P_1U Model 92
5 Variables
5V_2L_1P_1U
Model 37 5V_1L_2P_1U Model 93 5V_2L_2P_1U
Model 38 5V_1L_3P_1U Model 94 5V_2L_3P_1U
Model 39 5V_1L_4P_1U Model 95 5V_2L_4P_1U
Model 40 5V_1L_5P_1U Model 96 5V_2L_5P_1U
Model 41 5V_1L_1P_2U Model 97 5V_2L_1P_2U
Model 42 5V_1L_5P_2U Model 98 5V_2L_5P_2U
Model 43
5 Variables
5V_1L_1P_1U Model 99
5 Variables
5V_2L_1P_1U
Model 44 5V_1L_2P_1U Model 100 5V_2L_2P_1U
Model 45 5V_1L_3P_1U Model 101 5V_2L_3P_1U
Model 46 5V_1L_4P_1U Model 102 5V_2L_4P_1U
Model 47 5V_1L_5P_1U Model 103 5V_2L_5P_1U
Model 48 5V_1L_1P_2U Model 104 5V_2L_1P_2U
Model 49 5V_1L_5P_2U Model 105 5V_2L_5P_2U
Model 50
6 Variables
6V_1L_1P_1U Model 106
6 Variables
6V_2L_1P_1U
Model 51 6V_1L_2P_1U Model 107 6V_2L_2P_1U
Model 52 6V_1L_3P_1U Model 108 6V_2L_3P_1U
Model 53 6V_1L_4P_1U Model 109 6V_2L_4P_1U
Model 54 6V_1L_5P_1U Model 110 6V_2L_5P_1U
Model 55 6V_1L_1P_2U Model 111 6V_2L_1P_2U
Model 56 6V_1L_5P_2U Model 112 6V_2L_5P_2U
>@
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S
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S
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S
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38
ECB
September 2009
Model 113
3 Variables
3V_3L_1P_1U Model 169
3 Variables
3V _4L_1P_1U
Model 114 3V _3L_2P_1U Model 170 3V _4L_2P_1U
Model 115 3V _3L_3P_1U Model 171 3V _4L_3P_1U
Model 116 3V _3L_4P_1U Model 172 3V _4L_4P_1U
Model 117 3V _3L_5P_1U Model 173 3V _4L_5P_1U
Model 118 3V _3L_1P_2U Model 174 3V _4L_1P_2U
Model 119 3V _3L_5P_2U Model 175 3V _4L_5P_2U
Model 120
4 Variables
4V_3L_1P_1U Model 176
4 Variables
4V _4L_1P_1U
Model 121 4V _3L_2P_1U Model 177 4V _4L_2P_1U
Model 122 4V _3L_3P_1U Model 178 4V _4L_3P_1U
Model 123 4V _3L_4P_1U Model 179 4V _4L_4P_1U
Model 124 4V _3L_5P_1U Model 180 4V _4L_5P_1U
Model 125 4V _3L_1P_2U Model 181 4V _4L_1P_2U
Model 126 4V _3L_5P_2U Model 182 4V _4L_5P_2U
Model 127
4 Variables
4V_3L_1P_1U Model 183
4 Variables
4V _4L_1P_1U
Model 128 4V _3L_2P_1U Model 184 4V _4L_2P_1U
Model 129 4V _3L_3P_1U Model 185 4V _4L_3P_1U
Model 130 4V _3L_4P_1U Model 186 4V _4L_4P_1U
Model 131 4V _3L_5P_1U Model 187 4V _4L_5P_1U
Model 132 4V _3L_1P_2U Model 188 4V _4L_1P_2U
Model 133 4V _3L_5P_2U Model 189 4V _4L_5P_2U
Model 134
4 Variables
4V_3L_1P_1U Model 190
4 Variables
4V _4L_1P_1U
Model 135 4V _3L_2P_1U Model 191 4V _4L_2P_1U
Model 136 4V _3L_3P_1U Model 192 4V _4L_3P_1U
Model 137 4V _3L_4P_1U Model 193 4V _4L_4P_1U
Model 138 4V _3L_5P_1U Model 194 4V _4L_5P_1U
Model 139 4V _3L_1P_2U Model 195 4V _4L_1P_2U
Model 140 4V _3L_5P_2U Model 196 4V _4L_5P_2U
Model 141
5 Variables
5V_3L_1P_1U Model 197
5 Variables
5V _4L_1P_1U
Model 142 5V _3L_2P_1U Model 198 5V _4L_2P_1U
Model 143 5V _3L_3P_1U Model 199 5V _4L_3P_1U
Model 144 5V _3L_4P_1U Model 200 5V _4L_4P_1U
Model 145 5V _3L_5P_1U Model 201 5V _4L_5P_1U
Model 146 5V _3L_1P_2U Model 202 5V _4L_1P_2U
Model 147 5V _3L_5P_2U Model 203 5V _4L_5P_2U
Model 148
5 Variables
5V_3L_1P_1U Model 204
5 Variables
5V _4L_1P_1U
Model 149 5V _3L_2P_1U Model 205 5V _4L_2P_1U
Model 150 5V _3L_3P_1U Model 206 5V _4L_3P_1U
Model 151 5V _3L_4P_1U Model 207 5V _4L_4P_1U
Model 152 5V _3L_5P_1U Model 208 5V _4L_5P_1U
Model 153 5V _3L_1P_2U Model 209 5V _4L_1P_2U
Model 154 5V _3L_5P_2U Model 210 5V _4L_5P_2U
Model 155
5 Variables
5V_3L_1P_1U Model 211
5 Variables
5V _4L_1P_1U
Model 156 5V _3L_2P_1U Model 212 5V _4L_2P_1U
Model 157 5V _3L_3P_1U Model 213 5V _4L_3P_1U
Model 158 5V _3L_4P_1U Model 214 5V _4L_4P_1U
Model 159 5V _3L_5P_1U Model 215 5V _4L_5P_1U
Model 160 5V _3L_1P_2U Model 216 5V _4L_1P_2U
Model 161 5V _3L_5P_2U Model 217 5V _4L_5P_2U
Model 162
6 Variables
6V_3L_1P_1U Model 218
6 Variables
6V _4L_1P_1U
Model 163 6V _3L_2P_1U Model 219 6V _4L_2P_1U
Model 164 6V _3L_3P_1U Model 220 6V _4L_3P_1U
Model 165 6V _3L_4P_1U Model 221 6V _4L_4P_1U
Model 166 6V _3L_5P_1U Model 222 6V _4L_5P_1U
Model 167 6V _3L_1P_2U Model 223 6V _4L_1P_2U
Model 168 6V _3L_5P_2U Model 224 6V _4L_5P_2U
>@
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S
>@
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S
>@
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>@
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S
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Note: The table reports the composition of the model space, with the total number of m odels (column 1), the variables included (column
2) and the codi cation (column 3). Each m odel is characterized by 4 elements: number of variables (V), number of lags (L), type of
prior (P), and type of detrending method used in the calculation of the natural rate of unemployment (U). The VAR can have from
three (3V ) to six (6 V) e ndog enou s variables, and from o ne (1L ) to fo ur (4L ) lags. F ive typ es of p riors are p ossible. W ith the rst p rior
(1P ), b oth th e in ation persistence and the persistence of the labor market variables are unrestricted. W ith the second (2P), third
(3P) and fourth (4P) prior, in a tion is a ssum ed to b e a Ra ndom Walk, an A utore gressive p rocess and a W hite N oise, resp ectively,
w hile th e p ersistence of th e lab or m arket variab les is u nrestricted. W ith th e fth prior (5P ), th ere is no restriction o n the in ation
persistence and the labor market variables are assumed to follow an Autoregressive process. Two types of detrending methods are used
to compute the natural rate of unemployment. The rst one (1U) uses the Baxter and King band pass lter. T he second one (2U) is
a P hillip s- cu rve-b a sed me th od es tim a ted w it h K alm a n F ilter t ech niq ue s. T h er efor e, in m o d el 1 96 (co d ed a s 4 V_4 L_5 P_ 2 U ) th er e a re
four variables (unemployment, in ation rate and interest rate and exchange rate), four lags, the prior on in ation and unemployment
is th e fth on e, a nd t he n at ura l r ate of u n em p loym e nt h as b e en co m p ut ed w it h a P h illip s-cu rve -ba sed me tho d e stim a ted w ith K a lm an
F ilter t ech niq ue s.
39
ECB
September 2009
Figure 1: Relative Marginal Likelihoods
Euro area
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
1 17 33 49 65 81 97 113 129 145 161 177 193 209
Model Number
RML
posterior w eight equal weight
US
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
1 17 33 49 65 81 97 113 129 145 161 177 193 209
Model Number
RML
post erior wei ght equal weight
N ote: T he charts rep ort the R elative M arginal Likeliho od (R M L) o f the 22 4 m odels (b ars) an d the xe d e qu al w eig ht (h oriz ont al lin e) .
The RML is de n ed a s th e ra tio o f th e M a rg ina l L ikelih o o d (M L ) o f a give n m o d el ove r th e su m o f all M L s (E q . 1 1 in th e p ap e r). T h e
M L is n um e rica lly c om p u ted fr om t he G ib b s o ut pu t u sin g t he h a rm on ic m e an of th e likelih o o d va lue s a t ea ch d raw of th e p o ste rior
d istrib u tion o f th e p a rame ter ve cto r. In th e co m p ut atio n o f t he h arm o n ic m e an a ll m a rgin a l like liho o d s h ave b e en co m p u ted o n t he
basis of equations for the same three endogenous variables, namely unemployment, in a tion a nd inte res t rat e. T h e m o d els o n th e x -ax is
are o rdered a ccord ing to th e schem e d escrib ed in Table A 1.
40
ECB
September 2009
Figure 2: Posterior distributions of policy parameters and expected losses
Eu r o Ar e a US
Unemployment Gap
50 100 150 200
0. 0
2. 5
5. 0
7. 5
10. 0
Inflation Rate
50 100 150 200
0. 0
2. 5
5. 0
7. 5
10. 0
Inte rest Rat e
50 100 150 200
0. 0 0
0. 2 5
0. 5 0
0. 7 5
1. 0 0
Losses
25 50 75 100 125 150 17 5 200
0
5
10
15
20
Unemployment Gap
50 10 0 150 200
0. 0
2. 5
5. 0
7. 5
10. 0
Inflation Rate
50 10 0 150 200
0. 0
2. 5
5. 0
7. 5
10. 0
In terest Rat e
50 10 0 150 200
0. 00
0. 25
0. 50
0. 75
1. 00
Losses
25 50 75 10 0 125 150 175 200
0
5
10
15
20
N ote: T he charts rep ort th e p osterior distribu tion s o f th e long ru n rea ction co e! cient of unemployment gap and in ation rate, the
smoothing parameter of the interest rate and the loss values, for the OFR and all models. Column (1) refers to the euro area results.
Column (2) refers to the US results. The solid black line that goes through the areas is the posterior median of each model. The shaded
ar eas co m p rise t he 95 p erc ent of th e p o ste rior dis trib utio n aro un d it, a s i n a fan ch art r ep rese nta tion : th ere is a n eq u al nu m b er o f
bands on either side of the central band. The latter covers the interquartile range and is shaded with the deepest intensity. The next
deepest shade, on both sides of the central band, takes the distribution out to 80% ; and so on, until the 95% of the distribution is
cove red . Mo d els o n th e x -a xes a re o rga n ized a cc ord in g to tw o laye rs o f co m p lex ity: th ey a re rstsortedinascendinglaglengthorder
and then by the number of variables. Therefore, the models with one lag com e rst, then the models with two lags, and so on. Among
the sp eci cations with the same number of lags, the m odels with three variables come rs t, fo llowe d b y t he mo d els w ith fo ur var iab les,
an d so o n. T he rea fter , th e or der ing is t he s am e a s in ta ble A 1 , i.e ., rstwehavethespeci cations w ith priors from 1 to 5 a nd the rst
detrend ing m ethod, and then the sp eci cations with priors 1 and 5 and the second detrending method.
41
ECB
September 2009
Figure 3: Distributions across models of the median policy parameters and expected losses
0.00
2.00
4.00
6.00
8.00
OFR
TR
OF R
TR
EA US
Unemployment gap
0.00
2.00
4.00
6.00
OFR
TR
OFR
TR
EA US
Inflation Rate
0.0 0
0.2 5
0.5 0
0.7 5
1.0 0
OFR
TR
OF R
TR
EA US
Interest Rate
0.00
2.00
4.00
6.00
8.00
10.00
OF R
TR
OFR
TR
EA US
Loss
Note: The b ox plots report the extreme values, the m edian and the interquartile ranges of the relevant (long-run) policy parameters
and the expected losses computed over the posterior medians of the 224 m odels for the Optimal Feedback Rule (OFR) and the Taylor
Rule (TR). Each chart is divided in two parts: on the left hand side the euro area box plots are reported, and on the right hand side the
US box plots are reported. The interest rate coe! c ient is sim p ly th e smo o thin g p ar am e ter in th e T R , a nd th e s um over s 1 la gs
of the autoregressive coe! cie nts in t he O F R . T h e lo ng -run r esp o ns e c oe ! cients for unemployment gap and in ation rate are computed
as
i
x
@ (1 i
l
), i
@ (1 i
l
), for the TR and as
P
s31
m=0
i
x
@(1
P
s31
m=1
i
l
) an d
P
s31
m=0
i
@(1
P
s31
m=1
i
l
) fo r
the O FR, respectively, w here s represents the order of autoregression of the estimated m odel. The dark squares in the box plot are
th e w eig hted ave ra ge s o f th e r esu lts, w h er e th e w eig hts a re give n b y th e R M L . T h e e m pty cir cles re pr ese nt th e re su lts a sso c iate d w it h
the best models (i.e. Model 196 and M odel 117 of Table A.1, for the Euro Area and for the US, respectively).
42
ECB
September 2009
Figure 4: Posterior distributions of impulse response functions - Responses of unemployment gap
to a 100 basis-point contractionary monetary policy shock
Median
84th
percentile
16th
percentile
Taylor Rule
EA
US
10 20 30
-0.1
0.0
0.1
0.2
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
Optimal Feedback Rule
EA
US
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.1 0
-0.0 5
0.00
0.05
0.10
10 20 30
-0.1 0
-0.0 5
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
10 20 30
-0.10
-0.05
0.00
0.05
0.10
N ote: T he charts rep ort th ree quantiles — the m edian , the 1 6th p ercentile and th e 8 4th p ercentile — o f the p osterior Im pulse R esp onse
Functions of unem ployment gap to a 100 basis-p oint contractionary monetary p olicy obtained from the Gibbs sam pler. For each quantile
the distribution across the models has been ‘fan-charted’. Results are reported for the Optimal Feedback Rule and the Taylor rule,
and for the euro area and the US. Hence, in the charts with the title ‘median’ we plot the distribution across models of the median
responses. In each chart, the shaded areas represent the dispersion across models. There is an equal number of bands on either side of
the central band. The latter covers the interquartile range across m odels and is shaded w ith the deepest intensity. T he next deepest
shade, on b oth sides of the central band, takes the distribution out to 80% ; and so on up to the 95% . The solid black line that goes
through the areas is the weighted average across m odels, where the weights are given by the relative m arginal likelihoods of each model
computed as in Eq. 11 of the paper.
37
43
ECB
September 2009
Figure 5: Forecast errorr variance decomposition. Percentage of the variance of unemployment gap
explained by all variables
EA
US
u
10 20 30 40
0.00
0.25
0.50
0.75
1.00
10 20 30 40
0.00
0.25
0.50
0.75
1.00
pr
10 20 30 40
0.00
0.25
0.50
0.75
1.00
10 20 30 40
0.00
0.25
0.50
0.75
1.00
S
10 20 30 40
0. 00
0. 25
0. 50
0. 75
1. 00
10 20 30 40
0.00
0.25
0.50
0.75
1.00
cp
10 20 30 40
0.00
0.25
0.50
0.75
1.00
10 20 30 40
0. 00
0. 25
0. 50
0. 75
1. 00
e
10 20 30 40
0. 0 0
0. 2 5
0. 5 0
0. 7 5
1. 0 0
10 20 30 40
0.00
0.25
0.50
0.75
1.00
i
10 20 30 40
0. 0 0
0. 2 5
0. 5 0
0. 7 5
1. 0 0
10 20 30 40
0.00
0.25
0.50
0.75
1.00
N ote: T he charts re port the p osterior m ed ian s of the p ercentag e of the u nem ploym ent gap foreca st errors variance ex plained by th e
monetary policy shock (column “i ”) and by all other endogenous variables of the VAR. More precisely, x is t he u ne m p loyme nt g ap ;
su is th e p articipation ra te; is th e in ation rate; fs is the commodity price in ation; h is the exchange rate; and, l stands for
the n om ina l interest rate. T he d istribu tions — w hich are ob tained u nder th e O ptim al Feed back R ule for b oth eco nom ies — are rep orted
with the same “fan-chart” principle as in Figure 4. Hence, in each chart, the shaded areas represent the dispersion across models of the
portion of variance explained by each variable. There is an equal number of bands on either side of the central band. The latter covers
the interquartile range across m odels an d is shad ed w ith the d eepest intensity. T he next d eepest shade, on b oth sides of the central
b an d, t ake s th e d ist ribu tio n o ut t o 80 % ; a nd s o on , u ntil th e 9 5% is c ove red . T h e so lid b lack lin e th at g o es t hro u gh th e a rea s is th e
we igh ted ave rag e a cro ss m o d els, w h er e th e w eigh ts ar e giv en b y th e re lat ive m a rgin a l likelih o o d s o f ea ch m o d el c om p u ted a s in E q . 1 1
of the paper. The average variances explained by each variable cannot sum up to one as not all variables appear always in the same
m od els. T herefore, the varian ce attribu ted to th e single variables refers to the fraction of the variance e xplaine d by the se variables o nly
in m o d els w h o se spec i cation contains them .
44
ECB
September 2009
Figure 6: The transmission mechanism. Distribution across models of the posterior median impulse
responses of all variables to a 100 basis-point contractionary monetary policy shock
Optimal Feedback Rule
EA
US
pr
10 20 30
-0.02
-0.01
0.0 0
0.0 1
10 20 30
-0 . 02
-0 . 01
0. 00
0. 01
S
10 20 30
-0 . 2
-0 . 1
0.0
0.1
10 20 30
-0 . 2
-0 . 1
0. 0
0. 1
c p
10 20 30
-0 .1 0
-0 .0 5
0.0 0
0.0 5
10 20 30
-0.10
-0.05
0.00
0.05
e
10 20 30
-0.10
-0.05
0.00
0.05
10 20 30
-0. 10
-0. 05
0. 0 0
0. 0 5
i
10 20 30
-0.5
0.0
0.5
1.0
10 20 30
-0 . 5
0. 0
0. 5
1. 0
1. 5
Taylor rule
EA
US
pr
10 20 30
-0.02
-0.01
0.00
0.01
10 20 30
-0 .02
-0 .01
0.00
0.01
S
10 20 30
-0.2
-0.1
0.0
0.1
10 20 30
-0.2
-0.1
0.0
0.1
cp
10 20 30
-0.10
-0.05
0.00
0.05
10 20 30
-0.10
-0.05
0.00
0.05
e
10 20 30
-0.1 0
-0.0 5
0.00
0.05
10 20 30
-0. 10
-0. 05
0.00
0.05
i
10 20 30
-0 . 5
0.0
0.5
1.0
10 20 30
-0.5
0.0
0.5
1.0
1.5
Note: The charts report the p osterior m edians of the Impulse Response Functions of all variables to a 100 basis-point contractionary
m onetary p olicy. T he acrony ms of the variab les are the sam e as in Figu re 5 , tha t is: su is th e p articipation rate; is th e in ation rate;
fs is the comm odity price in ation; h is the exchange rate; l stands for the nominal interest rate. T he distributions across models are
reported for the Optimal Feedback Rule and the Taylor rule, and for the euro area and the US. The ‘fan-chart’ principle is the same as
in Figures 4 and 5. Therefore, in each chart, the shaded areas represent the dispersion across models of the m edian responses. There
is an equal number of bands on either side of the central band. The latter covers the interquartile range across m odels and is shaded
with the deepest intensity. The next deepest shade, on both sides of the central band, takes the distribution out to 80% ; and so on
until the 95% is covered. The solid black line that goes through the areas is the weighted average of each quantile (median, 16th and
84 th p e rce ntile ) a cro ss m o d els, w h ere t he w eig hts a re g iven b y t he re lative m a rgin al like liho o d s o f ea ch m o d el c om p u ted a s in E q . 1 1
of the pap er.
45
ECB
September 2009
For a complete list of Working Papers published by the ECB, please visit the ECB’s website
(http://www.ecb.europa.eu).
1059 “Forecasting the world economy in the short-term” by A. Jakaitiene and S. Dées, June 2009.
1060 “What explains global exchange rate movements during the financial crisis?” by M. Fratzscher, June 2009.
1061 “The distribution of households consumption-expenditure budget shares” by M. Barigozzi, L. Alessi, M. Capasso
and G. Fagiolo, June 2009.
1062 “External shocks and international inflation linkages: a global VAR analysis” by A. Galesi and M. J. Lombardi,
June 2009.
1063 “Does private equity investment spur innovation? Evidence from Europe” by A. Popov and P. Roosenboom,
June 2009.
1064 “Does it pay to have the euro? Italy’s politics and financial markets under the lira and the euro” by M. Fratzscher
and L. Stracca, June 2009.
1065 “Monetary policy and inflationary shocks under imperfect credibility” by M. Darracq Pariès and S. Moyen,
June 2009.
1066 “Universal banks and corporate control: evidence from the global syndicated loan market” by M. A. Ferreira and
P. Matos, July 2009.
1067 “The dynamic effects of shocks to wages and prices in the United States and the euro area” by R. Duarte and
C. R. Marques, July 2009.
1068 “Asset price misalignments and the role of money and credit” by D. Gerdesmeier, H.-E. Reimers and B. Roffia,
July 2009.
1069 “Housing finance and monetary policy” by A. Calza, T. Monacelli and L. Stracca, July 2009.
1070 “Monetary policy committees: meetings and outcomes” by J. M. Berk and B. K. Bierut, July 2009.
1071 “Booms and busts in housing markets: determinants and implications” by L. Agnello and L. Schuknecht, July 2009.
1072 “How important are common factors in driving non-fuel commodity prices? A dynamic factor analysis”
by I. Vansteenkiste, July 2009.
1073 “Can non-linear real shocks explain the persistence of PPP exchange rate disequilibria?” by T. Peltonen, M. Sager
and A. Popescu, July 2009.
1074 “Wages are flexible, aren’t they? Evidence from monthly micro wage data” by P. Lünnemann and L. Wintr,
July 2009.
1075 “Bank risk and monetary policy” by Y. Altunbas, L. Gambacorta and D. Marqués-Ibáñez, July 2009.
1076 “Optimal monetary policy in a New Keynesian model with habits in consumption” by C. Leith, I. Moldovan and
R. Rossi, July 2009.
1077 “The reception of public signals in financial markets – what if central bank communication becomes stale?”
by M. Ehrmann and D. Sondermann, August 2009.
46
ECB
September 2009
1078 “On the real effects of private equity investment: evidence from new business creation” by A. Popov and
P. Roosenboom, August 2009.
1079 “EMU and European government bond market integration” by P. Abad and H. Chuliá and M. Gómez-Puig,
August 2009.
1080 “Productivity and job flows: heterogeneity of new hires and continuing jobs in the business cycle” by J. Kilponen
and J. Vanhala, August 2009.
1081 “Liquidity premia in German government bonds” by J. W. Ejsing and J. Sihvonen, August 2009.
1082 “Disagreement among forecasters in G7 countries” by J. Dovern, U. Fritsche and J. Slacalek, August 2009.
1083 “Evaluating microfoundations for aggregate price rigidities: evidence from matched firm-level data on product
prices and unit labor cost” by M. Carlsson and O. Nordström Skans, August 2009.
1084 “How are firms’ wages and prices linked: survey evidence in Europe” by M. Druant, S. Fabiani, G. Kezdi,
A. Lamo, F. Martins and R. Sabbatini, August 2009.
1085 “An empirical study on the decoupling movements between corporate bond and CDS spreads”
by I. Alexopoulou, M. Andersson and O. M. Georgescu, August 2009.
September 2009.
1087 “Modelling global trade flows: results from a GVAR model” by M. Bussière, A. Chudik and G. Sestieri,
September 2009.
1088 “Inflation perceptions and expectations in the euro area: the role of news” by C. Badarinza and M. Buchmann,
September 2009.
1089 “The effects of monetary policy on unemployment dynamics under model uncertainty: evidence from the US
and the euro area” by C. Altavilla and M. Ciccarelli, September 2009.
1086 “Euro area money demand: empirical evidence on the role of equity and labour markets” by G. J. de Bondt,
Working PaPer SerieS
no 1089 / SePTeMBer 2009
The effecTS of
MoneTary Policy
on uneMPloyMenT
dynaMicS under
Model uncerTainTy
evidence froM The
uS and The euro
area
by Carlo Altavilla
and Matteo Ciccarelli
