TS1414275

Università Commerciale Luigi Bocconi
Faculty of Economics
M.Sc. in Discipline Economiche e Sociali
The Role of Confidence in Identification of
Macroeconomic Shocks
Advisor: Luca Sala
Discussant: Carlo Ambrogio Favero
Master of Science thesis of Paolo Meola (1414275)
academic year 2010/2011

I thank professors Luca Sala and Carlo Ambrogio Favero for helpful comments and
advice and my colleague Luca for valuable discussion.
Any remaining errors are my own.
page 1 out of 158

Table of contents
Abstract 7
A brief introduction to our work 8
1 UNDERSTANDING NON-FUNDAMENTALNESS 11
1.1 What econometricians want (to do, by estimating a VAR) 11
1.2 Limited dataset, VARMA and non-invertibility 14
1.3 When a VARMA can be inverted into a VAR 16
1.3.1 From a MA(1) to an AR 16
1.3.2 FROM a MA(p) to an AR 17
1.3.3 From a VMA(p) to a VAR 18
1.3.4 Non-invertibility (neither in the past nor in the future)
1.3.4 of a VMA(p) 19
1.3.5 Re-stating conditions for non-invertibility in the past 19
1.4 Stating what fundamentalness is 21
1.5 How non-fundamentalness can affect the identification of
1.5 the shocks 22
1.6 The economic meaning of non-fundamentalness 27
1.6.1 Theoretical cases in which the problem of
1.6.1 non-fundamentalness arises 28
1.6.1.1 A first example: innovation slowly spreading 29
1.6.1.2 A second example: posticipated implementation
1.6.1.2 of taxes 33
1.7 How to come with non-fundamentalness in our
1.7 theoretical models 36
1.8 Why might confidence matter? 38
2 EVALUATING THE IMPORTANCE OF CONFIDENCE 41
2.1 The empirical methodology 41
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2.2 Our measure of consumer confidence 42
2.3 Confidence and the identification of fiscal shocks 44
2.3.1 Replicating BP2002 44
2.3.2 Augmenting BP2002 with confidence 51
2.3.3 Evaluating the importance of confidence in the
2.3.3 estimation of fiscal shocks 57
2.3.4 Doing it with a Granger-causality analysis 57
2.3.5 Doing it brute force 60
2.3.6 Doing it with a large factor model 66
2.4 Confidence and the identification of monetary shocks 72
2.4.1 Replicating CEE1999 72
2.4.2 Augmenting CEE1999 with confidence 74
2.5 Confidence in the identification of technology shocks 81
2.5.1 Replicating GAL1999 and CEV2004 81
2.5.2 Augmenting GAL1999 and CEV2004 with confidence 86
2.5.2.1 Long-run restrictions 86
2.5.2.2 Short-run restrictions 87
2.5.2.3 Mixing short-run and long-run restrictions 88
2.5.3 Does confidence matter in the identification of
2.5.3 technology shocks? 91
3 INTERPRETING THE ROLE OF CONFIDENCE 101
3.1 Does confidence anticipate future taxes 101
3.1.1 Anticipated taxes in a VAR system 101
3.1.2 Confidence and anticipated RR dates 104
3.2 Does confidence anticipate future level of productivity? 106
3.3 Anticipated technology shocks VS animal spirits 111
3.4 Fiscal shocks, technology and animal spirits 116
3.5 Consumer confidence and uncertainty 127
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4 VALIDITY OF OUR METHODS AND ROBUSTNESS OF OUR RESULTS 133
4.1 Validity of the "brute force" bootstrap method 133
4.2 Robustness of our results 133
5 MAIN FINDINGS AND CONCLUSIONS 135
Bibliography 137
Appendix1: the dataset 140
Appendix2: the University of Michigan Survay of Consumers 146
Appendix3: tables of p-values for the fundamentalness test 148
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Abstract ♠
Recent works by Forni & Gambetti (May2011i
and Jun2011ii
) and Forni, Gambetti,
Sala (2011iii
) point out how structural VARs, typically used by econometricians in
order to identify structural macroeconomic shocks and to study their impulse
responses on macroeconomic variables, are often affected by non-fundamentalness
problem and may lead to estimated responses which are biased. On the other hand,
Bachmann & Sims (2010iv
) show how consumer confidence can play a relevant role in
estimating impulse responses to fiscal shocks; moreover Barsky & Sims (2010v
and
2011vi
) find that consumer confidence might also be relevant in identifying
technology shocks since it can contain important information about the future level of
productivity. The two main questions we aim to answer in this work are: a) can the
inclusion of consumer confidence in classical structural VARs help in coping with the
non-fundamentalness problem, thus improving the estimation of impulse responses
to fiscal, monetary and technological shocks? b) why may the inclusion of consumer
confidence in the VAR system have a relevant effect in the estimation of structural
shocks? In particular, as to the second question, we will examine three main
perspectives proposed in literature through which one may look at consumer
confidence: the informative, the animal spirits and the sit-and-wait perspectives.
♠ keywords: consumer confidence, fundamentalness, structural VAR, fiscal shocks, technology
shocks, informative perspective, animal spirits uncertainty
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A brief introduction to our work
Recent works by Forni & Gambetti (May2011vii
and Jun2011viii
) and Forni, Gambetti,
Sala (2011ix
) point out how structural VARs, typically used by econometricians in
order to identify structural macroeconomic shocks and to study their impulse
responses on macroeconomic variables, are often affected by non-fundamentalness
problem and may lead to estimated responses which are biased. On the other hand,
Bachmann & Sims (2010x
) show how consumer confidence can play a relevant role in
estimating impulse responses to fiscal shocks; moreover Barsky & Sims (2010xi
and
2011xii
) find that consumer confidence might also be relevant in identifying
technology shocks since it can contain important information about the future level of
productivity. The two main questions we aim to answer in this work are: a) can the
inclusion of consumer confidence in classical structural VARs help in coping with the
non-fundamentalness problem, thus improving the estimation of impulse responses
to fiscal, monetary and technological shocks? b) why may the inclusion of consumer
confidence in the VAR system have a relevant effect in the estimation of structural
shocks? In particular, as to the second question, we will examine three main
perspectives proposed in literature through which one may look at consumer
confidence: the informative, the animal spirits and the sit-and-wait perspectives.
Our work will be structured as follows:
1) In the first section, we will briefly illustrate the theory behind the non-
fundamentalness problem and show how it can arise while using the structural VAR
approach in macroeconomics in order to identify macroeconomic shocks.
2) In the second section, we will empirically investigate about the effects of including
consumer confidence in the VAR system, and understand if it can help in coping with
the non-fundamentalness problem.
In particular, we will reproduce and extend classical works which uses the structural
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VAR approach to the sample 1960Q1-2010Q2 (using U.S. Data:)
– Blanchard & Perotti (2002 xiii
,) for the identification of fiscal shocks;
– Christiano, Eichenbaum, Evans (1999 xiv
,) for the identification of monetary
shocks;
– Galì (1999 xv
) and Christiano, Eichenbaum, Vigfusson (2004 xvi
,) for the
identification of technology shocks.
Following an approach similar to Barsky & Sims (2010xvii
) and Bachmann & Sims
(2010 xviii
) we will augment the classical structural VARs with a measure of consumer
confidence and verify with a brute-force bootstrap method if the inclusion of
confidence can play an important role.
Moreover, similarly to Ramey (2011xix
,) we will test if consumer confidence Granger-
causes the other variables in the structural VARs.
Another useful toolbox we will use in the understanding of the importance of
confidence in identifying the correct economic shocks is an adaptation of the tests of
fundamentalness, based on large factor model, proposed by Forni, Gambetti, Sala
(2010xx
) and Forni & Gambetti (May2011xxi
and Jun2011xxii
) in order to test if the
inclusion of consumer confidence in the VAR can make us reject the hypothesis of
non-fundamentalness for the identified shocks.
3) In the third section our aim will be about interpreting the role of confidence, the
reason why confidence might matter in identifying structural shocks. In particular we
will try to discriminate if consumer confidence anticipates future taxes, future
technology or if it more fits with the “animal spirits view” assessed (and rejected) by
Barsky & Sims (2010xxiii
) on the hypothesis that a large part of macroeconomic
fluctuations are due to exogenous changes in the behaviour of consumers. Finally,
following an approach similar to Bachmann, Elstner, Sims (2010xxiv
,) we will study the
“wait-and-see” perspective of confidence, analysing the relation between uncertainty
perceived by consumers, economic activity and consumer confidence.
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In order to understand which role consumer confidence actually have, we will make
use of the structural VAR approach: we will make identifications which allow for
these kinds of shocks and interpret their impulse responses on other macroeconomic
variables.
Moreover, a simple correlation analysis will be made between the identified news
shocks in a fiscal VAR and anticipated tax shocks narratively identified by Mertens &
Ravn (2009xxv
.)
4) In the fourth section we will briefly show evidence in favour of the validity of the
methods used in our work.
5) In the fifth section we will sum up our main findings and state our conclusions on
the issue.
A) in Appendix1 we will explain how our dataset was created.
B) in Appendix2 we will provide further information about the Survey of Consumers
of Michigan.
C) in Appendix3 we will show in detail the results from our fundamentalness testing
procedure.
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1 UNDERSTANDING NON-FUNDAMENTALNESS
1.1 What econometricians want (to do, by estimating a VAR)
One main question in economics is what are the effects of exogenous shocks on the
macroeconomic variables: such shocks are changes independent from the state of
the economy, such as changes in the preference of the central bank about
unemployment/inflation ratio, changes in the preferences of government about
taxes/spending ratio, as well as technological innovations that increases productivity.
After seminal work by Kydland & Prescott (1982xxvi
,) the use of Dynamic Stochastict
General Equilibrium (DSGE) models have become a standard approach to convey
macroeconomic reality. The basic idea underlying this approach is that equilibrium
comes out from the solution of an optimization problem faced by agents which try to
maximize their life-time inter-temporal utility.
For instance, we can assume that data are generated by the linear approximation of
the model of Ireland (2004xxvii
:) the linear approximation of the solution of the model
can be written in the state space form
st+1=Πst+W εt+1
f t=U st
, where the first
equation is the state equation (which represents the in-deepth dynamics of the
model,) and the second is the measurement equation (which tell us how the other
variables are determined once the state variables come out.)
This model can be simply rewritten as (assuming that f and s have no variables in
common:)
X t+1=A X t+Dεt+1 , where X t=
[st
f t
], A=
[Π [0]
U Π [0]]and D=[W
U W ].
Indeed, the solution of a DSGE models generally produces dynamic systems that can
be rewritten into a Vector Autoregressive VAR(q) representation X t=Aq(L) X t+ut ,
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where Aq(L)=∑
i=1
q
Ai Li
and ut=Dεt . In particular, X is a matrix which contains
the k macroeconomic variables, ut is the k-dimensional vector of the impacts of the
l non-observable structural shocks εt affecting the economy, which we assume to
be a zero mean stationary white noise process εt∼w.n.(0l , Σ) , uncorrelated with
each other, so that Σ is a diagonal matrix.
Since we have assumed that the linear relation ut =Dεt holds (and that
εt∼w.n.(0l , Σ) ,) we also have E(ut ut ')=V=D Σ D' .
Let us then assume that the model underlying economy, thus generating the data,
can be rewritten as X t=Aq(L) X t+Dεt , and that we can observe the whole
dataset X t so that the econometricians’ information set coincide with the
information set of the model.
If this is the case, we could estimate Aq( L) by simply running ordinary least
squares, and then we can recover an estimation for V from the fitted residuals.
Once we have recovered the estimations ̂Aq(L) and ̂V , we still have to identify ̂D
and ̂Σ in order to estimate the structural shocks and their impact on macroeconomic
variables. In particular, to find estimates for D and Σ we can substitute in relation
V =D Σ D' the equivalent estimated matrices.
In general, it is convenient to choose some normalization for D and Σ : in general,
one can impose that Σ=Il . In our work (although differently stated,) we choose to
normalize coefficients in D such that the elements on the diagonal are equal to 1:
doing so, an unitary i -th shock implies an immediate positive effect of 1 on the i -th
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variable.
On the other hand, we would like to recover the k⋅l unknown parameters in ̂D and
Σ from the k (k+1)/2 independent elements contained in ̂V , indeed the matrix
equation ̂V = ̂D ̂Σ ̂D' is equivalent to a system of k (k+1)/2 equations in k⋅l
unknowns. In order to make the problem well identified, so that it has an unique
solution, we have to to impose k (k+1)/2−k l additional restrictions.
One largely used approach, become extremely popular after the seminal work by
(1980xxviii
,) which is referred to as the Structural VAR (SVAR) approachxxix
, is to embed
some theory in the model, imposing the restrictions needed according to the
underlying economic theory, so that matrices ̂D and ̂Σ are uniquely determined.
Another possible solution – see, for example, Favero & Giavazzi (2010xxx
) – is to
estimate some coefficient from external information.
Once we have recovered the estimated the model X t= ̂Aq(L) Xt + ̂D ̂εt , by
inverting the VAR representation into a Vector Moving Average (VMA) X t=Ct (L)εt ,
we can easily derive an estimate for the impulse response function X t=C(L ,t )ε0
of the variables in X t for t≥0 and C(L ,t) containing only non-negative powers
of the lag operator L , being ε0 a vector of initial shocks.
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1.2 Limited dataset, VARMA and non-invertibility
Let us imagine now that, given the model X t=Aq(L) X t+Dεt generating the
data, we can only observe a subset xt of X t .
As pointed out by Dufour & Pelletier (2011 xxxi
,) even if we assume that the process
generating X t is a VAR process, for a subset xt of X t it may be impossible to
write a VAR for xt only. Instead, it is always possible to write it as a VARMA. The
problem is also discussed by Fernandez-Villaverde et al (2007 xxxii
.)
For example, let us consider the following VAR process:
[
xt ,1
xt ,2
xt ,3
]=
[
0.1 0.1 0.1
0.1 0.1 0
0.1 0 0 ][
xt−1,1
xt−1,2
xt−1,3
]+
[
1 0
1 1
0 γ][εt ,1
εt ,2
].
If we considered only the couple of observables
[xt ,1
xt ,2
], hence we could rewrite
(substituting the expression for xt ,3 , xt ,3=0.1 xt−1,1+γεt ,2 ) the process as
[xt ,1
xt ,2
]=[0.1 0.1
0.1 0.1][xt−1,1
xt−1,2
]+[0.01 0
0 0][xt−2,1
xt−2,2
]+[1 0
1 1][εt ,1
εt ,2
]+[0 0.1γ
0 0 ][εt−1,1
εt−1,2
]
which can explicitly be rewritten in the VARMA form
[1−0.1 L−0.01 L2
−0.1L
−0.1 L 1−0.1 L][xt ,1
xt ,2
]=[1 0.1 γ L
1 1 ][εt ,1
εt,2
].
The process has been rewritten into a VARMA form, therefore one must check if it
can be rewritten again into a VAR for the two variables xt ,1 and xt ,2 , so that the
model can be correctly estimated by using a VAR. In particular, it is the case if
the matrix
[1 0.1γ L
1 1 ]
−1
exists and contains only positive powers of L .
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This problem of “invertibility” is closely connected with the problem of non-
fundamentalness and prevents, in practice, from estimating unbiased impulse
responses to structural shocks when the model is estimated through a VAR.
In the following sections we will:
– explicit the mathematical conditions under which this VARMA form can be
rewritten as a VAR;
– state the relation between fundamentalness, invertibility and sufficiency;
– explain the economics underlying the non-fundamentalness problem, and
analyse which features of the real model generating the data are more likely
to bring to problems of non-fundementalness;
– propose an empirical solution to cope with non-fundamentalness;
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1.3 When a VARMA can be inverted into a VAR
In the previous paragraph we have shown the basic concepts behind the SVAR
approach to the identification of macroeconomic shocks. In order to use this
approach when the process generating data is represented in VARMA form, as stated
before, we need the VMA part to be inverted, so that we can recovery the impulse
response function through the VAR estimate.
In the next section we will state under which condition a VARMA model
Bq(L) X t=Cp(L)ut can be rewritten in a VAR form.
Without loss of generality, we will consider the VMA(p) representationxxxiii
X t=C p(L)ut , where C p(L)=∑
i=0
p
Ci Li
, and explain how (and under which
conditions) one can invert this model in the past (and, equivalently, the moving
average part of a VARMA).
1.3.1 From a MA(1) to an AR
We will first consider an univariate MA(1) process with one single lag xt=ut −cut−1 .
This representation can be inverted into a AR(+ ∞ ) as follows:
ut =(1−c L)−1
xt=xt ∑
i=0
+∞
ci
Li
=xt+c xt−1+c2
xt −2+... . Of course, in order to make
this operation possible, we must have −1<c<1 , such that lim
i →∞
ci
=0 , and so the
last terms of the series disappear and thus can be ignored, and ∑
i=0
+∞
ci
converges.
We call this operation "inversion in the past" xxxiv
since the inverted model depends
only on non-negative powers of L, that means that we can express the present
values of xt as a linear combination of the past values of it-self and the present
value of the shock ut . By estimating the model xt=−c xt−1−c2
xt−2−...+ut
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through (constrained) ordinary least squares, one can also recover an estimation for
the shock impact ut on xt as the residual of the regression.
If, otherwise, we had ∣c∣>1 we wouldn't be able to invert the model in the past,
and we would only be able to express ut as a function of future values of xt :
indeed, rearranging −c−1
L−1
xt−1=−c−1
L−1
εt−1+εt−1 one can obtain
ut=−xt ∑
i=0
+∞
(c L)−i−1
=−c−1
xt+1−c−2
xt+2−c−3
xt +3−... where the the last terms
c−i
ut+i disappear as i →+∞ , and −c−1
∑
i=0
+∞
c−i
converges.
In such a case, the only way to recovery the impact of the shocks on economy would
be looking at the future realization of the observable.
1.3.2 From a MA(p) to an AR
Let us now add some lags in the previous MA model: in order to invert in the past a
MA(p) model xt=cp(L)ut , with cp(L)=1−∑
i=1
p
ci Li
, it is convenient to rewrite it
as xt , p=ut , p−C ut−1, p , where:
xt , p=
[ xt
0p−1
]
'
, C=
[c1 c2 .. cp
I p−1 0 p−1
] and ut , p=
[
ut
ut−1
..
ut− p+1
].
From ut , p=ut , p−C ut−1, p , we can write M−1
xt , p=M−1
ut , p−N M−1
ut−1, p ,
where A is decomposed in C=M N M −1
. Denoting with xt , p the linear
combination M−1
xt , p , and ̃ut , p the linear combination M−1
t , p , we can rewrite
the following system of linear equations:
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̃xt , p=N ̃xt−1, p+̃ut , p , or equivalently
̃xt=̃ut−N1,1 L ̃ut
̃xt−1=̃ut−1−N 2,2 L ̃ut−1
...
̃xt− p+1=̃ut −p−N 3,3 L ̃ut− p
which can be
inverted in the past if and only if ∣N i ,i∣<1, ∀i=1,2,.., p , and we will have
ut , p=∑
i=0
+∞
Ci
Li
xt , p . Again, if we have ∣N j , j∣>1 for even just one j , the model
xt=cp(L)ut could not lead to an explicit form in which the present value of xt can
be expressed as a linear combination of only the past values of it-self and the
present value of the shock impact ut .
Let us note that is important to consider the eigenvalues in absolute value, since the
fact that C is not symmetric implies that some eigenvalues may be complex xxxv
.
1.3.3 From a VMA(p) to a VAR
Let us first consider a VMA(p) model X t=C p(L)ut (so that
X t=[xt ,1 , xt ,2 ,.., xt ,k ]
'
,
ut =[ut ,1 ,ut ,2 ,..,ut ,k ]
'
) with C p(L)=Ik−∑
i=1
p
Ci Li
.
Again, we can rewrite X t , p=ut , p−C ut−1, p where X t , p=
[ X t
0pk−k
],
C=
[C1 C2 .. C p
I pk−k [0]( p k−k )×k
], and ut , p=
[
ut
ut−1
..
ut− p+1
].
As before, it can be shown that the model can be inverted in the past if and only if
the eigenvalues of C are inside the unit circle, and in this case we will have
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ut , p=(I −C L)−1
xt , p=∑
i=0
+∞
Ci
Li
X t , p . Again, if some eigenvalue is greater than
one, the model could not be rewritten such that X t is only function of present and
past values of u .
1.3.4 Non-invertibility (neither in the past nor in the future) of a VMA(p)
We said that the VMA(p) model X t=C p(L)ut can be inverted in the past if and
only if all the eigenvalues of C=
[C1 C2 .. C p
I pk−k [0]( p k−k )×k
] are inside the unit circle.
What happens when some eigenvalue lies on the unit circle?
If we have some eigenvalue equal to one, we would not have neither invertibility in
the past nor invertibility in the future.
In order to show this, let us take the MA(1) model xt=ut −ut −1 : from this model we
can equally obtain two different AR representations, ut =xt +xt−1+xt−2+xt−3+...
and ut =−xt+1−xt+2−xt+3−xt+4+... , that are composed by an infinite number of
terms which are all equally relevant in recovering the value of ut , and which do not
vanish as they are distant from ut with respect to time.
If some eigenvalues- of C are on the unit circle, we say that the VMA(p)
representation X t=C p(L)ut cannot be inverted (neither in the past nor in the
future.)
1.3.5 Re-stating conditions for non-invertibility in the past
We said that the representation X t=C (L)ut is non-fundamental if
C=
[C1 C2 .. C p
I pk−k [0]( p k−k )×k
] has some eigenvalues outside the unit circle, since in this
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case it cannot be rewritten into a VAR representation X t=A(L) Xt +ut .
The eigenvalues of C are computed equating to zero the characteristic function of
C , that is det (C – I λ)=0 .
Let us take, for example, the MA(q) case xt=cp(L)ut
xxxvi
. Here, the characteristic
equation becomes det
([
c1−λ c2 c3 .. cp−1 ap
1 −λ 0 .. 0 0
0 1 −λ .. 0 0
.. .. .. .. .. ..
0 0 0 .. 1 −λ
])=0 that leads to
λp
– c1 λ p−1
−c2 λp−2
− ... −λ1
cp−1−cp=0 , that can be rewritten as
1– c1 z−c2 z2
− ... −cp zp
=0 (with z=λ−1
), that is essentially cp(z)=0 .
Hence, we will say that the representation xt=cp(L)ut cannot be inverted in the
past if the equation c p(z)=0 has one root inside the unit circle.
Similarly one can proceed in order to state a condition for the more general VMA(p)
case: on can prove that the representation X t=C p(L)ut cannot be inverted in the
past if the equation det (C p(z))=0 has even one root inside the unit circle.
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1.4 Stating what fundalmentalness is
Given a vector of white noise structural shocks εt , a process X t=C (L)εt lies in
the space generated by present and past values of εt . We say that εt is X-
fundamental (or fundamental for the process X t ) if and only if it lies on the space
generated by present and past values of εt .
Any covariance-stationary process can be rewritten into its Wold Representation
X t=B(L)ut
xxxvii
by inverting its VAR representation X t=A( L) X t+ut , where ut
are the innovations process of X t and by definition it belongs to the space
generated by present and past values of X t . Indeed, when in the SVAR approach,
we try to recover the structural shocks εt from the innovations ut we first have to
be sure that they lie on the same space or it would be impossible to find a matrix D
such that ut=Dεt .
Rozanov (1967 xxxviii
) provides a necessary and sufficient condition for εt to be
fundamental for xt (with C(z) matrix of only rational functions):
– it should hold that rank (C (z))=l , ∀z∈ℂ s.t.∣z∣<1 .
In particular, as noticed by Forni, Gambetti, Sala (2010xxxix
), for every square system
such that k =l , the above condition become:
– det (C(z)) must have no roots inside the unit circle (that is the same
condition we stated for the invertibility in the past of εt ).
The definition of fundamentalness and its necessary and sufficient conditions should
clarify how non-fundamentalness and non-invertibility in the past are closely linked:
in particular, it is not possible to recover the fundamental shocks εt from the
estimated innovations ̂u of a VAR representation which is not invertible in the past.
page 21 out of 158

1.5 How non-fundamentalness can affect the identification of the shocks
Alessi, Barigozzi & Capasso (2011xl
) present a very simple example in order to show
how estimation of the true shocks can be affected by non-fundamentalness.
They consider a simple univariate MA(1) xt=(1−c L)εt with ∣c∣>1 , and
εt∼w.n(0,1) . We know that this representation is non-fundamental, since it cannot
be inverted into a AR representation.
If we tried to estimate the shocks by estimating its AR representation
xt=a xt−1+a2
xt−2+...+ut using data generated which are instead generated by
the real MA representation underlying the process, we would be lead to estimate a
shock that is not the real one. As a result, the impulse response computed
estimating the AR would be misleading with respect to the true process generating
the data.
In order to show this in practice, we simulate 1000 time series of length 100 periods
(with 100 periods of time-burn) from the process xt=(1−c L)εt , with c=1.5,0.5
and εt=w.n.(0,σ) with σ=1 , and then we compare the true impulse response
(represented by the dark line) with the median impulse response together with the
95% and 68% percentiles (represented by the light lines) estimated with a AR(1)
by running OLS.
As one can see by the graph above, the AR estimation systematically (more precisely,
one should say consistently) fails in estimating the impulse response when c=1.5
(in particular, even the lower 95% quartile of the impact is estimated to be almost
twice the true impact on x ,) while it is far more precise in estimating the true
impulse response when c=0.5 (in particular, both the real x0 and x1 are also
contained in the 68% confidence bands.)
page 22 out of 158

MA(1) with c=1.5 estimated through an AR(1)
MA(1) with c=0.5 estimated through an AR(1)
page 23 out of 158
0 0 .5 1 1 .5 2 2 .5 3 3 .5 4
-1
-0 .5
0
0 .5
1
1 .5
im p u ls e re s p o n s e o f x
im p u ls e re s p o n s e s to a s h o c k ε0
= [1 ]', e s tim a te d w ith a V A R w ith # la g s = 1
0 0 .5 1 1 .5 2 2 .5 3 3 . 5 4
-2
-1
0
1
2
3
4
im p u ls e re s p o n s e o f x
= [1 ]', e s tim a t e d w ith a V A R w ith # la g s = 1

Let us go back to the example we have previously presented. We have
considered the VAR process
[
xt ,1
xt ,2
xt ,3
]=
[
0.1 0.1 0.1
0.1 0.1 0
0.1 0 0 ][
xt−1,1
xt−1,2
xt−1,3
]+
[
1 0
1 1
0 γ][εt ,1
εt ,2
] and we
have reduced the dataset to the couple of observables
[xt ,1
xt ,2
]. We know that for
these two variables the process can be rewritten as
[xt ,1
xt ,2
]=[0.1 0.1
0.1 0.1][xt−1,1
xt−1,2
]+[0.01 0
0 0][xt−2,1
xt−2,2
]+[1 0
1 1][εt ,1
εt ,2
]+[0 0.1γ
0 0 ][εt−1,1
εt−1,2
]
and thus made explicit in the VARMA form
[1−0.1 L−0.1L2
−L
−0.1 L 1−0.1 L][xt ,1
xt ,2
]=[1 0.1γ L
1 1 ][εt ,1
εt ,2
].
Thus, since the equation det
([1 0.1γ z
1 1 ])=0 have solution z=10 γ−1
, the MA
part of the VARMA can be inverted (and thus one can rewrite the model as a VAR) if
and only if ∣z∣>1 , that is ∣γ∣<10 . Otherwise the model could not be written
as a VAR, and thus it would be impossible to estimate the shocks
[εt ,1
εt ,2
] from the
residuals from the fitted values of the VAR.
We will make the same experiment made above with the simple AR(1) process. In
particular, we will generate 1000 series from the real model (with γ=20,5 ) and
estimate the structural shocks from a VAR on the variables [xt ,1 , xt ,2]' , using a
Choleski identification (as suggested by the real structure of the model.)
As before, as γ=20 we find a systematic bias in estimating the impulse responses.
page 24 out of 158

VMA(2) with γ = 20 estimated through a VAR(1): responses to a ε1,0 shock
page 25 out of 158
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
im puls e res pons e of
x
1
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
2
im puls e responses to a s hoc k ε0
= [1 0]', es tim ated with a V A R with # lags = 1
0 1 2 3 4
-0.5
0
0.5
1
1.5
2
2.5
x 1
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x 2
im puls e res pons es to a s hoc k ε0

page 26 out of 158
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
1
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
2
0 1 2 3 4
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x 1
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
x 2

1.6 The economic meaning of non-fundamentalness
In the first paragraphs, we stated that theoretical DSGE models often lead to
dynamic solution (the process generating data) that can be represented as a VAR
process.
These models assume that macroeconomic variables are generated by an optimal
decision rule that governs economic agents’ behaviour in function of their knowledge
about the state of the macroeconomic variables: X t+1=A X t+Dεt+1 can be seen
as a linear approximation of the agents’ optimal decision rule, where X t is the
whole set of economic variables that agents take into account when create their
expectations.
As previously shown, even admitting that the initial model can be expressed as a VAR
for, it is possible that for a subset xt of X t one cannot rewrite the process
generating the data as functions of present and past values of xt , and of the
present values only of εt (that represents the unexpected changes in economy at
time t ). Alessi & Barigozzi (2011xli
) present a short example taken from Hansen &
Sargent (1991xlii
) in order to show how agent’s rational expectations may cause
problems of non-fundamentalness.
Let us imagine a model with two economic variable, say x1,t and x2, t :
– x1 is generated by a fundamental MA process x1,t=εt−θεt−1 , with θ<1 ;
– x2 is the sum of discounted expected value of x1,τ , that is
x2,t =∑
i=1
+∞
βi
x2,i=(1−βθ)εt−θεt−1 .
If econometricians are able to only observe x2,t and it holds that (1+β)−1
<θ , it is
impossible for them to recover the shock εt .
page 27 out of 158

The basic idea underlying this example is that econometricians may be not able to
recover the structural shocks if they only observe agents' actions if they are based on
agents’ rational expectations about future value of some other economic variables
which are not in the econometricians’ information set. Indeed, when agents'
decisions take into account expectations about the value of some variables in the
future (such as prices, interest rates, etc. ), they will use information contained in
their information set in order to create these expectations: sometimes it is possible
that econometricians’ information set may be not large enough to recover structural
shocks driving agents' behaviour xliii
.
We now understand the economic meaning of the inversion in the future: if some
variables are related to expectations about future, and econometricians do not
observe the variables used by agents to create these expectations, they will only be
able to explain xt as functions of future values of the observed variables.
1.6.1 Theoretical cases in which the problem of non-fundamentalness arises
In this section we will show some examples in which the problem of
non-fundamentality arises from rational expectations of agents and from the fact that
econometricians' information set does not contain enough information.
Canova (2007xliv
) proposed two interesting cases in which the non-fundamentalness
problem arises:
– an economy where innovations in the most advanced sectors takes some
periods to spread to the economy;
– an economy where taxes are announced before their implementation.
In the next paragraphs we will show two DSGE models representing these cases,
solve them – following the procedures as described in Malley (2004xlv
) – rewrite them
in a VAR form and see when non-fundamentalness problem arise.
page 28 out of 158

1.6.1.1 A first example: innovation slowly spreading
Canova (2007xlvi
) model the slowly spreading innovation by giving to zt (which is the
level of technology Zt log-linearized around the steady-state) the dynamics
zt=∑
i=1
p
γi εz ,t+1−i so that agents are aware of future technology innovations before
they actually affect production.
A similar effect is obtained by Barsky & Sims (2010xlvii
) by assuming that technology
is a random walk with stochastic drift, where agents knows information about
changes in the drift. In particular: zt=zt−1+gt−1+εz ,t with
gt=(1−ρ)g+ρgt−1+εg ,t .
We will try to highlight the effect of anticipated technology using an even simpler
dynamic for technology. In particular we will assume zt=ρ zt−1+εz ,t −1 , so that the
level of technology at time t is already known by agents at time t−1 .
In particular, we will analyse the results in a trivial DSGE, adapted from Perotti
(Jun2011xlviii
,) where a representative household maximize an utility function of
consumption C , Et ∑
j=t
+∞
βj
lnC j subject to the constraint Ct+Kt =Zt Kt−1
α
where the level of output Yt comes out from the production function Yt =Zt Kt
α
which is function of the level of technology Zt – such that
ln Zt=(1−ρ)ln Z+ρln Zt −1+εz ,t−1 – and the level of capital Kt (which
depreciates at rate 100% at each period).
This optimization problem solved by the household leads to the following FOCs:
page 29 out of 158

1
Ct
=αβEt
(1
Ct+1
Zt+1 Kt
α−1
)
Ct+Kt =Zt Kt−1
α
ln Zt=(1−ρ)ln Z+ρln Zt −1+εt−1
In steady state, the variables become:
z=Z
k=(zαβ)
−
1
α−1 K
c=z k
α
−k
Log-linearising the variables around their steady state we have:
−θ1 ct=Et (−ct+1)+zt+(α−1)kt where θ1=(αβ z k
α−1
)
−1
cct+k kt=z k
α
zt+α z k
α
kt−1
zt=ρ zt−1+εz ,t−1
A procedure equivalent to that described in Malley (the Blanchard Kahn method) is to
rewrite the model in expected value terms, such that Et
̃X t+1=K ̃X t where
̃X t=
[xt
st
] , xt are the variables decided by agents at each time t and st is the
state vector including all information agents need in order to decide. We rewrite
(with θ2=(−θ1 ck
−1
), θ3=k
−1
(α z k
−1
−c(α−1)) , θ4=k−1
(z k ρ−c) ):
Et
[
ct+1
k t+1
zt+1
ιt +1
]=
[
θ1 α−1 1 0
θ2 θ3 θ4 z k(α−1)
0 0 ρ 1
0 0 0 0
][
ct
kt
zt
ιt
]where ιt =εz ,t−1 .
Decomposing K as K=C DC−1
(where C is the matrix of eigenvectors and D
page 30 out of 158

the matrix of eigenvalues) one can rewrite C−1
Et
̃X t+1=C DC−1
̃X t . If the length
of xt is the same as the number of eigenvalues outside the unit circle, then the
solution of the optimization problem can be found imposing that Et
̃X t+1 does not
explode as t →+∞ .
Let us arbitrarily calibrate the parameters, so that we can start working numerically.
We will set β=0.9 , α=0.5 , ρ=0.8 , z=5 , σz
2
=0.25 so that K becomes:
K =
[
1 −0.5 1 0
−1.2222 0.7087 2.7778 2.2222
0 0 0.8 1
0 0 0 0
]
The spectral decomposition of K into K=C DC−1
is:
– C=
[
0.6100 0.4693 0.5668 0.4772
−0.7924 0.8831 0.7766 0.8774
0 0 0.2749 −0.0385
0 0 0 0.0308
]such that C*
=C−1
exists;
– D=
[
1.6495 0 0 0
0 0.0591 0 0
0 0 0.8 0
0 0 0 0
].
Since the number of degrees of freedom for the household is equal to the number of
eigenvalues of K outside the unit circle, the optimization problem has a solution.
In particular, since ∣D1,1∣>1 , in order to find a stable solution, we have to impose
C1,1
*
ct+C1,2
*
kt+C1,3
*
zt+C1,4
*
ιt =0 .
Imposing the condition above into the system, we obtain a state space
representation
st+1=Πst+W εz ,t+1
f t=U st
where st=[kt , zt ,ιt ]' and f t =ct .
page 31 out of 158

In particular, Π=
[(K 2,2 – K2,1
C1,2
*
C1,1
*
) (K2,3 – K2,1
C1,3
*
C1,1
*
) (K2,4 – K2,1
C1,4
*
C1,1
*
)
0 K3,3 K3,4
0 0 0
],
W =
[
0
0
1]and U=
[(K1,2 – K1,1
C1,2
*
C1,1
*
) (K1,3 – K1,1
C1,3
*
C1,1
*
) (– K1,1
C1,4
*
C1,1
*
)].
This state space form can easily be rewritten into a VAR system:
X t=
[Π [0]
U Π [0]]X t−1+[W
U W ]εt where X t=
[st
f t
].
Numerically, given our calibration, we have:
[
kt+1
zt +1
ιt +1
ct+1
]=
[
0.0591 2.0927 0.9319 0
0 0.8 1 0
0 0 0 0
0.0019 1.3142 1.5898 0
][
kt
zt
ιt
ct
]+
[
0
0
1
1.0557
]εz ,t
It is interesting to notice that consumption, immediately reacts to a technological
shock, even if the shock have not already affected nor kt neither zt , thus not
having already had any immediate impact on income Yt .
Assuming that kt and zt are observed with measurement error, ε1 and ε2 (of
variance σ1
2
=σ2
2
=0.0001 ,) one can write:
[
kt
zt
ιt
ct
]=
[
0.0591 2.0927 0.9319 0
0 0.8 1 0
0 0 0 0
0.0019 1.3142 1.5898 0
][
kt−1
zt−1
ιt−1
ct−1
]+
[
1 0 0
0 1 0
0 0 1
0 0 1.0557
][
ε1,t
ε2,t
εz ,t
].
Here, it is easy to see prove that – given our calibration – if, for example, ιt is not
included in the system, it is not possible to rewrite the model into a VAR form.
page 32 out of 158

Indeed, considering only [kt , zt ,ct ]' , we can rewrite the system as:
[
1−0.0591 L −2.0927 L 0
0 1−0.8 L 0
−0.0019L 1.3142 L 1][
kt
zt
ct
]=
[
1 0 0.9319 L
0 1 L
0 0 1.0557+1.5898 L][
ε1,t
ε2,t
εz ,t
]
In particular det
([
1 0 0.9319 z
0 1 z
0 0 1.0557+1.5898z])=z vanish for z=−0.6640 , which
is inside the unit circle, thus making it impossible to recover the fundamental shocks
by estimating the innovation of a VAR system including only [kt , zt ,ct ]'
1.6.1.2 A second example: posticipated implementation of taxes
A second example by Canova is an economy where there is a lag between the
programming, the legislation and the implementation of income tax rates.
Similar examples are shown by Favero and Giavazzi (2010xlix
) and Perotti (2011l
.)
We will use a simple DSGE model similar to that proposed in the previous paragraph.
In particular, the representative household maximize Et ∑
j=t
+∞
β j
lnC j subject to the
constraint Ct+Kt +Gt =Z Kt−1
α
where the level of output Yt comes out from the
production function Yt =Z Kt
α
which is function of a fixed level of technology Z ,
and the level of capital Kt (which depreciates at rate 100% at each period).
We added a government spending Gt , which is financed period-by-period by a
lump-sum taxation Tt=Gt ruled by lnTt =(1−ρ)lnT+ρ lnTt−1+ετ ,t−1 , where
ετ ,t is the govern announcement, made at time t , about the fiscal policy to be
implemented at time t+1 .
page 33 out of 158

This optimization problem solved by the household leads to the following FOCs:
1
Ct
=αβ Et
(1
Ct+1
Z Kt
α−1
)
Ct+Kt +Tt=Z Kt−1
α
lnTt =(1−ρ)lnT+ρ lnTt−1+ετ ,t−1
The values of the variables in steady state are:
t=T
k=(αβ)
−
1
α−1
c=z k
α
−k−t
Log-linearising the variables around their steady state, we have:
−θ1 ct=Et (−ct+1)+(α−1)kt where θ1=(αβ z k
α−1
)
−1
cct+k kt+t tt=α z k
α
kt−1
tt=ρtt−1+ετ ,t−1
The system of equations can be rewritten in matrix form θ2=(−θ1 ck
−1
) and
θ3=k−1
(α z kα
−c(α−1)) ) as:
Et
[
ct+1
kt+1
tt+1
ιt +1
]=
[
θ1 α−1 0 0
θ2 θ3 −ρt k−1
k−1
t
0 0 ρ 1
0 0 0 0
][
ct
kt
tt
ιt
]where ιt =ετ ,t−1 .
As before, we calibrate the model, and set β=0.9 , α=0.5 , ρ=0.8 , z=5 , t=1 ,
σz
2
=0.25 so that K becomes K =
[
1 −0.5 1 0
−1.2222 0.7087 −0.1580 0.1975
0 0 0.8 1
0 0 0 0
]where the spectral decomposition of K into K=C DC−1
is:
page 34 out of 158

– C=
[
0.6100 0.4693 −0.1244 −0.4216
−0.7924 0.8831 −0.0498 −0.8433
0 0 0.9910 −0.2603
0 0 0 0.2082
]such that C*
=C−1
exists;
– D=
[
1.6495 0 0 0
0 0.0591 0 0
0 0 0.8 0
0 0 0 0
].
Imposing C1,1
*
ct+C1,2
*
kt+C1,3
*
zt+C1,4
*
ιt =0 , one can obtain:
[
kt+1
zt +1
ιt +1
ct+1
]=
[
0.0591 −0.0372 0.1930 0
0 0.8 1 0
0 0 0 0
0.0019 −0.0803 −0.0928 0
][
kt
zt
ιt
ct
]+
[
0
0
1
0.0037
]εz ,t
Let us allow for two measurement error, ε1 and ε2 (of variance σ1
2
=σ2
2
=0.0001 )
on the observable kt and zt . Therefore, we will have:
[
kt
zt
ιt
ct
]=
[
0.0591 −0.0372 0.1930 0
0 0.8 1 0
0 0 0 0
0.0019 −0.0803 −0.0928 0
][
kt−1
zt−1
ιt−1
ct−1
]+
[
1 0 0
0 0 0
0 0 1
0 1 0.0037
][
ε1,t
ε2,t
εz ,t
].
The representation only considering [kt , zt ,ct ]' can be rewritten as:
[
1−0.0591 L +0.0372 L 0
0 1−0.8 L −L
−0.0019L 1.3142 L 1 ][
kt
zt
ct
]=
[
1 0 0.1930 L
0 1 L
0 0 0.0037−0.0928 L][
ε1,t
ε2,t
εz ,t
]
In particular det
([
1 0 0.1930z
0 1 z
0 0 0.0037−0.0928 z])=z vanish for z=0.04 , which is
inside the unit circle, thus making it impossible to recover the fundamental shocks by
estimating the innovation of a VAR system including only [kt ,tt ,ct ]' .
page 35 out of 158

1.7 How to cope with non-fundamentalness in our theoretical models
In the previous paragraph we have shown how the lack of relevant information in
rational expectations models can lead to the non-fundamentalness problem.
As suggested by Perotti (2011li
,) the most intuitive solution for econometricians is to
implemented an EVAR (Expectation Augmented VAR,) which includes in the VAR
system the missing information (the series ιt , in our theoretical models.)
A similar solution is implemented, for example, by Favero and Giavazzi (2010lii
) who
included in a fiscal VAR the Romer&Romer (2010liii
) series of fiscal shocks identified
with the narrative approach, reclassified by Mertens and Ravn (2009liv
) into
anticipated and non-anticipated shocks.
We propose, instead, a different theoretical solution. Let us go back to the model
with technological innovation slowly spreading and assume we can ask the
representative household about her expectation about the future level (say at t+1 )
of production yt=zt +αk t .
If this information is known without error, we have:
Ψt =Et ( yt+1)=Et (zt+αkt+1)=θ2 ct+θ3 kt +(θ4+ρ)zt +(z kα−1
+1)εz ,t−1 .
Moreover, from the computations in the previous paragraphs, we know that:
– Et
[
ct+1
kt+1
zt+1
]=
[
θ1 α−1 1
θ2 θ3 θ4
0 0 ρ ][
ct
kt
zt
]+
[
0
z kα−1
1 ]εz ,t−1 ;
– C1,1
*
ct+C1,2
*
kt+C1,3
*
zt+C1,4
*
εz ,t−1=0 .
Putting together the equations above, the system can be rewritten in a VAR form:
page 36 out of 158

Et
[
ct+1
kt+1
zt+1
Ψt+1
]=
[
θ1 α−1 1 0
θ2−
δC1,1
*
C1,4
*
θ3−
δC1,2
*
C1,4
*
θ4−
δC1,3
*
C1,4
*
0
−λ−1
θ2 −λ−1
θ3 ρ−λ−1
(θ4+ρ) λ−1
γ1 γ2 γ3 γ4
][
ct
kt
zt
Ψt
]+
[
0
0
0
λ
]εz ,t
with λ=(z kα−1
+1) and δ z kα−1
and γ1 , γ2 , γ3 and γ4 are the coefficients in
Ψt+1=θ2(θ1ct +(α−1)kt+zt )+
+θ3
((θ2−
δ C1,1
*
C1,4
*
)ct+
(θ3−
δC1,1
*
C1,4
*
)kt+
(θ4−
δC1,1
*
C1,4
*
)zt
)+
+(θ4+ρ)(−λ−1
θ2 ct −λ−1
θ3 kt+(ρ−λ−1
(θ4+ρ))zt −λ−1
Ψt)
From this simple theoretical model, we have shown how including in the VAR system
forward looking variables containing information about agent’s expectations for the
future can help in coping with the problem of non-fundamentalness.
page 37 out of 158

1.8 Why might confidence matter?
As stated above, the lack of information which may affect small VARs making it
impossible to recover the real structural shocks can be corrected by including in the
system some forward looking variables. Previous work by Bachmann & Sims (2010lv
)
and Barsky & Sims (2010lvi
and 2011lvii
) suggests that a relevant forward looking
variable that should be included in the system is consumer confidence.
One possibility is that confidence is nothing but a proxy of macroeconomic
fundamentals, that sum up some information contained in the agents’ information
set. This view does not lead us to give confidence an important role itself, but show
how confidence might be useful for econometricians in order to cope with the lack of
information. We can call this the “informational view” of confidence, coherent with
the possibility that consumer confidence can incorporate news about the future state
of economy (i.e. anticipated taxes or anticipated technology innovations,) which
cannot already be found in the variables in the econometricians’ information set.
Bachmann & Sims (2010lviii
,) examined possible reasons why confidence might matter
itself in the transmission of monetary and fiscal policy. Among others possibilities,
they proposed some hypothesis that, although coherent with the informational view,
stressed the fact that consumer confidence is only a subjective (though aggregate)
perception of fundamentals, and therefore it can also reflect a subjective component
not entirely related (or even totally independent) from fundamentals.
Indeed, it is reasonable – even if, as noticed by Bachmann & Sims (2010lix
,) this
hypothesis lacks of a coherent theoretical structure in literature of DSGE models that
allows confidence of having an important and independent role and, as shown by
Barsky & Sims (2010lx
,) met with limited empirical success – a shift in the perception
of fundamentals by consumers (even if only subjective) can determine changes in
the consumers’ choices, thus determining changes in aggregate consumption, output
and employment. This view, which Bachmann & Sims (2010lxi
) refer to as “animal
page 38 out of 158

spirit view,” is intriguing, however we have to admit that it does not results easy to
create trivial models which allows for these results: Bachmann & Sims modelled
consumer confidence as the error in agent’s prediction of the future level of
technology; if consumers are convinced, for example, that the future level of income
will be higher they will consume more, thus stimulating economy.
Another interesting hypothesis proposed by Barsky & Sims (2010lxii
) about the role of
consumer confidence is that it reflects changes in consumer’s discount factor. In
particular, we believe that a relevant unobserved component of confidence as to the
discount factor might be related to consumer’s perception of risk in terms of
uncertainty about the future: if consumers perceive an higher uncertainty about the
future, they will give an higher value to earlier than to future consumption – thus
increasing their immediate consumption and stimulating economy. This “wait-and-
see” perspective of confidence is studied by Bachmann, Elstner, Sims (2010lxiii
) who
analysed the relation between business uncertainty, economic activity and business
confidence using business survey data at different frequencies: we will adopt to the
analysis of the relation between uncertainty and consumer confidence an approach
similar to that used by Bachmann, Elstner, Sims.
In the next section we will try to impose the minimum theory possible about the role
of confidence, and only try to measure if its inclusion in the classical VARs has an
important impact in correctly estimating the structural shocks. Finally, we will try to
discriminate, still remaining in the SVAR approach framework – which allow us not to
model confidence as did by Barsky & Sims (2010lxiv
) but keep using the minimum
theory and “letting the data speak” – about the two main different views about the
role of confidence described above (the “informational view” and the “animal spirit
view”).
page 39 out of 158

2 EVALUATING THE IMPORTANCE OF CONFIDENCE
2.1 The empirical methodology
As shown in the previous section, in presence of rational expectations, if some
structural shock is anticipated by economical agents, impulse responses estimated
through a structural VAR might be biased.
An interesting approach to cope with this anticipation problem, used for example by
Perotti (Jun2011lxv
), is to augment the VAR with a forward looking variable which
contains information about future values of the variables. This approach is called by
Perotti Expectation VAR (EVAR.)
Our basic idea is that we could use consumer confidence as a forward looking
variable, including it in the VAR system, in order to cope with the non-
fundamentalness problem.
In particular, we will analyse three different framework in which non-
fundamentalness problem may arise. In particular, we will refer to four classical
works which uses the VAR technique to identify the response of economy to shocks:
– Blanchard and Perotti (2002lxvi
, henceforth BP2002) for the identification of
fiscal shocks;
– Christiano, Eichenbaum, Evans (1999lxvii
, henceforth CEE1999) for the
identification of monetary shocks;
– Galì (1999lxviii
, henceforth GAL1999) and Christiano, Eichenbaum, Vigfusson
(2004 lxix
, henceforth CEV2004) for the identification of technology shocks.
Our methodology will be the following:
– we will replicate their results within the original time sample, and extend their
work to the sample 1960Q1-2010Q2, using U.S. data;
– following an approach similar to Barsky and Sims (2010lxx
) and Bachmann and
Sims (2010lxxi
,) we will augment these VARs with a measure of consumer
page 40 out of 158

confidence and trying with different methods if the inclusion of confidence can
play an important role; in particular we will:
– make a first naïf comparison by plotting the impulse responses and
their confidence bands estimated in both the original VAR systems
and in the confidence-augmented ones;
– make a Granger-causality test, similarly to that implemented by
Ramey (2011lxxii
,) in order to check if consumer confidence Granger-
causes the other variables in the structural VARs;
– implement a “brute-force” bootstrap method in order to estimate if
the inclusion of consumer confidence in the VARs’ sysytem
significantly changes the estimated impulse responses to estimated
structural shocks;
– implement an adaption of the fundamentalness tests proposed by Forni,
Gambetti, Sala (2010lxxiii
) and Forni & Gambetti (May2011lxxiv
and Jun2011lxxv
)
in order to test if the inclusion of consumer confidence has made the
estimated shocks fundamental (more precisely we will check if this inclusion
can make us reject the hypothesis of non-fundamentalness for the identified
shocks.)
2.2 Our measure of consumer confidence
Barsky and Sims (2010lxxvi
) as a measure of consumer confidence used EY5: EY5 is
an index computed using the Michigan Survey of Consumers lxxvii
about the
expectations of consumers about their income in 5 years. Bachmann and Sims
(2011lxxviii
,) used instead ICS (the Michigan Index of Consumer Sentiment,) computed
as an average with equal weights of different indexes computed from the Michigan
Survey of Consumers.
We will propose a different index, computed similarly to ICS but – in our opinion –
less subject to the arbitrary choices of the weights: our measure for consumer
confidence will be CPC – confidence principal component – computed as the principal
page 41 out of 158

component of different relevant indexes of consumer taken from Michigan Survey of
Consumers. CPC is basically a weighted sum of the different indexes used for ICS but
the weights are chosen so that CPC explains the most part of variance (and thus
information) disposable.
In particular, our CPC index has been created following the procedure below:
– we first standardized the series so that Zi ,t=
Xi ,t –̂μ(X i)
̂σ(X i)
– since the ADF test (with a constant and 4 lags) gives strong evidence contrary
to the unit root hypothesis for the variables used, we do not need to make
further transformations in order to apply principal component analysis hold;
– we, then, created CPC as the principal component of the standardized series
Z , so that the principal component (as stated above) is the linear
combination of the variables considered that explains the most part of their
variance (in particular ̃CPC=PC(:,1) , with PC=(C D−1/2
)' where C and
D are respectively the matrices of eigenvalues and eigenvalues of Z Z ' /T );
– CPC has been normalized further on so that its standard error is equal to 1.
As we will see, the Granger-causality analysis shows how CPC results to be more
statistically significant than EY5 and ICS in forecasting the future value of the
macroeconomic variables used in the classical VARs.
Further information about the Michigan Survay are given in Appendix2.
page 42 out of 158

2.3 Confidence and the identification of fiscal shocks
2.3.1 Replicating BP2002
In order to identify fiscal shocks, we will first replicate an adaptation of the
deterministic trend version of the VAR in BP2002, similar to that implemented by
Swisher (2010lxxix
,) which has the advantage of offering an easy procedure to extend
BP2002 results to a different sample (in particular Swisher proposed an easy
procedure to estimate the immediate elasticity of Tt to Yt shocks). Then, as
suggested by Perotti (2011lxxx
,) we will try to obtain the same results using a simple
Choleski identification. How our database is constructed is explained in Appendix1.
Following the approach adapted from Swisher (2010), we will firstly estimate the VAR
model X t= A(L)X t+C Zt+ut through OLS, where X t=[Tt ,Gt ,Yt ]
'
and Zt is a
matrix of exogenous controls including:
– a vector of ones (for the estimation of the constant):
– a linear trend t ;
– a dummy variable for 1975Q2 (which is an <<isolated temporary tax cut
which was reversed after one quarter>> identified by BP2002lxxxi
as an
exogenous change in taxes which cannot be said to be a normal reaction to
economy);
– 4 lags of the 1975Q2 dummy;
– 4 quarterly dummy variables.
The quadratic trend t2
has been dropped from the original content of Z , leaving
the results qualitatively almost unchanged.
Data is taken quarterly and four lags of A(L) are estimated: as explained in
BP2002, the use of quarterly data (common in literature of VARs,) is central in the
identification of fiscal shocks, because it is reasonable to assume that it takes more
than a quarter to fiscal policy authority to learn about GDP shocks, decide the fiscal
page 43 out of 158

measures to take in response and to implement them: indeed, (as stated in the title
of an important work of Valery Rameylxxxii
) “it is all in the timing!!”
Four lags are estimated: the economical reason, explained in BP2002, is that some
taxes – such as indirect taxes or income taxes when withheld at the source – are
paid within the same quarter of the time of the transaction, while others – such as
corporate income taxes —are paid with some delays but generally in one year (since
are due to transaction generated in the year.)
In BP2002, the effects ut on macroeconomic variables are assumed to be linearly
linked to the orthogonal structural shocks εt , through the relation Bu ut=Bεεt
where BU =
[
1 0 −a1
0 1 −b1
−c1 −c2 1 ], Bε=
[
1 a2 0
b2 1 0
0 0 1], ut=
[
uT ,t
uG ,t
uY ,t
]and εt=
[
εG ,t
εT ,t
εY ,t
].
This way of modelling the relation between shocks and effects is equivalent to
assume that:
– unexpected movements in taxes and spending within a quarter can be due
– the response to unexpected movements of GDP;
– the response to structural shocks to spending;
– the response to structural shocks to taxes;
– unexpected movements in output can be due to
– unexpected movements in taxes;
– unexpected movements in spending;
– other unexpected shocks.
The coefficient a1 represents the immediate elasticity of taxes to unexpected
changes in GDP, and it is estimated by Swisher (2010lxxxiii
) through an OLS on the
model uT ,t =a1 uGDP ,t+ξt using the residuals from the previous VAR (our estimation
page 44 out of 158

gives ̂a1=2.36 for the sample 1960-2010Q2 and ̂a1=1.77 for the sample 1960-
1997Q4, over which in BP2002 (using a slightly different method) a value of 2.09
for this coefficient is found.
Moreover, we will assume (as in Swisher 2010) that government spending do not
immediately react to GDP shocks so that b1=0 (while in BP2002 this result is
estimated from data.)
BP 2002, as well as Swisher 2010, analyses both cases in which one assumes a2=0
or b2=0 . In the first case one is assuming that Tt does not immediately react to
Gt shocks, while in the second one is assuming that Gt does not immediately react
to Tt shocks. Their analysis show that the two assumptions gives almost the same
results. Therefore we arbitrarily chose for b2=0 .
In order to compute an “average impulse response” of how economy reacts to
shocks without taking into account the quarter in which the shock happens, we
estimate then a model X t=̃A(L)X t+ ̃C ̃Zt+ ̃ut with ̃Z not including the quarterly
dummy variables.
Once this model is estimated, one can identify the shocks considering that
BU
−1
Bε=V =Et (uu' ) : this matrix equation is equivalent to a system of 3 equations
with 6 unknown variables [a1 ,a2 ,b1 ,b2 ,c1 ,c2], therefore we have to include 3
additional constraints so that the solution is unique. Using the estimate
̃V =̂̃u ̂̃u' /(T−l k) , and the constraints on the coefficients (we have already
imposed b1=b2=0 and externally estimated a1 ), we can finally estimate the linear
relation between shocks and effects and therefore compute the impulse responses to
the three different structural shocks.
page 45 out of 158

As suggested by Perotti (Jun2011lxxxiv
,) almost identical results are obtained using a
simple Choleski identification with the ordering X t=[Gt ,Tt ,Y t ]
'
.
A Choleski identification is equivalent to assume the following model structure:
[
Gt
Tt
Y t
]=A (L)
[
Gt−1
Tt−1
Yt−1
]+C Z+
[
d1,1 0 0
d2,1 d2,2 0
d3,1 d3,2 d3,3
][
εG ,t
εT ,t
εY ,t
].
In particular, we will impose d1,1=d2,2=d3,3=1 and leave free to move the
variances of the orthogonal shocks σG
2
, σT
2
, σY
2
.
Assuming that in the linear relation ut=Dεt the elements above the diagonal are
zeros, is equivalent to state the following temporal assumptions on the shocks:
– shocks on T and Y different from εG do not affect G within the quarter;
– shocks on Y different from εG do not affect T within the quarter.
Again we estimate D from ̂D ̂Σ ̂D'= ̂V =̂u ̂u' /(T −l k) , subject to the zero
restrictions on the elements above the diagonal. The matrix Z of the controls
includes a vector of ones, the linear trend, the dummy for 1975Q2 and its 4 lags.
In both the identification methods, we find that the impulse responses to fiscal taxes
slightly changes if we use the original time sample or we extend the results to 1960-
2010Q2. In particular, we see that:
– for the sample 1960-1997Q4,
– a 1% GOV shock implies
– a slight increase of GDP (circa 0.15% after 20 quarters) which
become significant only after 15 quarters;
– a less significant positive movement of TAX, of 0.5% after 20
quarters, positive at 68% significance after 10 quarters;
page 46 out of 158

– a 1% TAX shock implies:
– a negative impact on GDP in a 20 quarters horizon (around -0.1%),
which becomes significantly negative only after 11 quarters (at least
at a 68% degree of significance);
– a negative impact on G, with a negative pick of -0.3% after 10
quarters;
– extending the sample to 1960-2010Q2, shocks in GOV and TAX on other
economical variables are flattened, and less significant.
Since both the identification methods lead us to qualitatively identical results, we will
only present results obtained with the Choleski identification.
In particular, a first estimation on real data will be used to generate 1000 series,
created from the estimated model X t=̂A(L)X t+̂C Zt+ut
*
where X 0 is the initial
observed state of X , and u*
is a random re-sample from the series ̂u of the
estimated errors from the linear regression. Confidence bands (at 95% and 68%
confidence levels) are estimated using percentiles from the estimated responses from
the bootstrapped series. Instead plotting the impulse responses from the first
estimation, for illustrative purposes, we have plotted the median obtained from the
bootstrap: qualitative results, however, are very similar.
page 47 out of 158

BP2002 (1960-1997Q4) with Choleski (GTY): responses to a εG shock.
BP2002 (1960-1997Q4) with Choleski (GTY): responses to a εT shock.
page 48 out of 158
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
im p u ls e re s p o n s e o f
G
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
1
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
= [ 1 0 0 ] ', e s tim a t e d w ith a V A R w ith # la g s = 4
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
G
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y

BP2002 (1960-2010Q2) with Choleski (GTY): responses to a εG shock.
BP2002 (1960-2010Q2) with Choleski (GTY): responses to a εT shock.
page 49 out of 158
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
G
0 5 1 0 1 5 2 0
-2
-1
0
1
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-0 .6
-0 .4
-0 .2
0
0 .2
G
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
T
0 5 1 0 1 5 2 0
-0 .2
0
0 .2
0 .4
0 .6
Y

2.3.2 Augmenting BP2002 with confidence
Since the two specifications lead to similar qualitative results, we will present only
the augmentation of the VAR extending the the Choleski identification.
Moreover, since similar qualitative results are obtained using EY5 and CPC as
augmenting variables, we only present results obtained with CPC, since it has slightly
narrower confidence bands.
Given the previous ordering X t=[Gt ,Tt ,Yt ]' , we have to chose in which place to
insert the confidence variable Ψt (CPC or EY5).
Let us speak about the implication of the different Choleski ordering. We will denote
Xti as the new vector containing X t augmented with Ψt in the i -th place.
In the following arguments, we will assume that the null hypothesis that confidence
has an important role in the business cycle is true.
In the case of i=4 , we will have X t4 =[Gt ,Tt ,Y t ,Ψt ]' : the short run
constraints imposed by the Choleski decomposition of V =Et (uu') with this
ordering imply that every shocks affect consumer confidence within a quarter (in
other words, every shock on G , T and Y is immediately perceived by consumers
and affect their confidence lever), on the other hand any exogenous shock affecting
consumer confidence do not affect G , T and Y within the quarter: the fact that
other shocks having an immediate impact on Ψ do not affect G and T within the
quarter is reasonable if we see consumer confidence as an anticipating variable for
fiscal policy shocks (which therefore should move before G and T in presence of
such shocks), on the other hand it is more questionable the fact that such a shock
does not immediately affect Y , since a shock affecting consumer confidence – even
if generated from news on future state of macroeconomic variables – might have an
immediate effect on consumption, and therefore on GDP.
page 50 out of 158

Imposing i=3 , we will have X t3 =[Gt ,Tt , Ψt ,Y t ]' : this ordering implies once
againt that fiscal shocks are able to affect consumer confidence within the quarter,
but G and T are not immediately affected by shocks on Ψ such as news shock or
sentiment shock; on the other hand, Y is affected by these shocks within the
quarter. Another implicit assumption of this ordering is that Ψ is not affected by
other shocks affecting Y , therefore limiting the “perceptive capacity” of consumers
and allowing for other shocks affecting Y within the quarter which are not
recognized by consumers.
In the previous two identifications, consumer confidence is allowed to assume an
anticipative role of policy shocks, since with this ordering one could identify the
anticipated policy shock as a shock immediately affecting Ψ which takes at least a
quarter to affect G and T : for example, if an unexpected new tax is announced
one quarter before its implementation, such a shock would immediately affect
consumer confidence, but would take at least 1 quarter to be observed in the taxes
series by econometricians; the channel from which we would see such a shock
affecting taxes would be the linear relation expressed by A(L) that links Ψt to
Tt+1 and Gt+1 .
Finally, ordering confidence first, that is i=1 Xt1 =[Ψt ,Gt ,Tt ,Yt ]' , we are
virtually eliminating the possibility of an anticipative role of Ψ within the quarter,
since one cannot identify the confidence shock as a shock due to anticipate news
about policy.
Another possible ordering, which instead we will not take into account. is i=2 (
Xt2 =[Gt ,Ψt ,Tt ,Yt ]' ), which would still conserve the possibility of an anticipative
role of Ψt about G is still conserved, but would eliminate every short-run
predictive role on T .
page 51 out of 158

In Bachman & Sims 2010 lxxxv
the importance of confidence is highlighted – following
an approach previously adopted by Sims & Zha 2006lxxxvi
– by plotting the impulse
responses (with their confidence bands) from the augmented VAR, where shocks are
identifies with a Choleski decomposition of the variance-convariance matrix with
confidence ordered last, together with the “counter-factual” impulse responses.
In particular, the counter-factual responses are generated by shocking the other
shocks and keeping fixed consumer confidence, by artificially adding a series for ε4
such that the level of confidence is held constant.
Their approach makes them claim that <<the inclusion of confidence in the system
does little to alter either the qualitative or quantitative dynamics of the response of
economic activity to policy shocks>>, on the other hand it does not makes easy to
evaluate the statistical significance of this “little”. On the other hand, the evaluation
of this significance is central: the more the impact is statistically significant, the more
evidence we have collected in favour of the importance of confidence in correctly
identifying structural shocks. Moreover, by imposing confidence to be ordered last,
one is implicitly assuming questionable assumptions about the role of confidence.
Therefore we will start with a first naïf analysis of the statistical difference between
the original VAR and the augmented one, under the different assumptions i=1,3,4
: in particular, we will plot together the median impulses with their 95% and 68%
confidence bands from the bootstrapped series generated with both the original
estimated VAR model (with X t=[Gt ,Tt ,Y t ]' ) and the augmented one (with Xti ).
Dark lines refer to the impulse responses estimated through the augmented VAR,
while the light ones are the responses estimated through the original non-nested
VAR. We will presents the results obtained through the sample 1960-2010Q2.
Moreover, the only impulse responses shown for consumer confidence are those
estimated with the confidence-augmented VAR systems.
page 52 out of 158

Choleski BP2002 augmented with i=4 : responses to a εG shock
Choleski BP2002 augmented with i=4 : responses to a εT shock
page 53 out of 158
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
G
0 5 1 0 1 5 2 0
-2
-1
0
1
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-1 0
-5
0
5
1 0
C P C
= [1 0 0 0 ], e s t im a te d w it h a V A R w it h # la g s = 4
0 5 1 0 1 5 2 0
-0 .6
-0 .4
-0 .2
0
0 .2
G
0 5 1 0 1 5 2 0
-1
0
1
2
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-5
0
5
C P C

page 54 out of 158
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
G
0 5 1 0 1 5 2 0
-2
-1
0
1
T
0 5 1 0 1 5 2 0
-1 0
-5
0
5
1 0
C P C
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-0 .6
-0 .4
-0 .2
0
0 .2
G
0 5 1 0 1 5 2 0
-1
0
1
2
T
0 5 1 0 1 5 2 0
-5
0
5
C P C
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y

page 55 out of 158
0 5 1 0 1 5 2 0
-1 0
-5
0
5
1 0
C P C
0 5 1 0 1 5 2 0
-0 .5
0
0 .5
1
1 .5
G
0 5 1 0 1 5 2 0
-2
-1
0
1
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-1 0
-5
0
5
C P C
0 5 1 0 1 5 2 0
-0 .6
-0 .4
-0 .2
0
0 .2
G
0 5 1 0 1 5 2 0
-1
0
1
2
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y

2.3.3 Evaluating the importance of confidence in the estimation of fiscal shocks
Even if very naïf, our approach makes us notice that in all 3 cases considered (
i=1,3,4 ), in more than 50% of the cases, the impulses of T and Y in response
to a εT shock estimated with the original non-augmented VAR are outside the 95%
bands estimated for the augmented model.
This suggest us that the estimated impulse responses are significantly different. On
the other hand, the fact that the bands estimated through the two different models
overlap, makes it a little more complicated to evaluate the significance of this
difference.
2.3.4 Doing it with a Granger-causality analysis
The first solution in order to make it clearer if confidence has an effective impact in
the estimation of the impulse responses to structural shocks is to carry out a
Granger-causality analysis.
In a bivariate context, we say that a variable x Granger-cause another variable y
variable if it helps in predicting the future value of y . More precisely lxxxvii
, we say
that x fails to linearly Granger-cause y if we have:
MSE ( yt +s | yt , yt−1 ,..)=MSE ( yt +s | yt , yt−1 ,.. , xt , xt−1 ,..), ∀s>0 .
In our notation, we will write x →
gc
y to indicate that x linearly Granger-cause y ,
and x →
gc
y to indicate that x fails to linearly Granger-cause y .
In the bi-variate context, in order to test Granger-causality one should test the null
hypotesis β1=β2=..=βp=0 against βi≠0 , ∀i=1,2,.. lxxxviii
in the linear relation
yt=c0+c1t+ A(L) yt+B(L) xt (where A(L)=∑
i=1
q
αi L
i
and B(L)=∑
i=1
p
βi L
i
)
estimated with OLS. To test this, we use the statistic a standard F-statistic
page 56 out of 158

F=
(I p
̂β)' (R(X ' X )
−1
R' )
−1
(I p
̂β)
p ̂σ
, where X contains a vector of ones, the
linear trend, q lags of y and p lags of x , ̂β is the vector of estimated coefficients
βi , I p is an identity matrix of dimension p× p , and R=[[0]p×(k− p) I p ].
Unfortunately, since some of the variables in our VARs are integrated of order 1 (we
cannot refuse the unit root hypothesis with the ADF test with the constant, a linear
trend and 4 lags for G , while we can refuse it at 5% significance level for T and
10% significance level for Y – and, in any case, there is much debate about the
stationarity of these variables in literaturelxxxix
) things may be a bit more complicated,
since when in a VAR in levels we operate with integrated variables the F-statistics
may not have a standard F-distribution xc
.
Fortunately, when the variable x is an I (0) stationary process, we know that p
times the F-statistic used for testing the hypothesis that x does not Granger-cause
a unit root process has a standard limiting χp
2
distributionxci
. Indeed, we feel
confident enough that confidence is a non-stationary process, since the ADF test on
confidence variables gives us:
– a p-value less than 0.001 in favour of the unit-root hypothesis for CPC (with
the ADF-statistic computed with 4 lags, and no constant – since by
construction CPC is zero-mean);
– a p-value of 0.0045 in favour of the unit-root hypothesis for EY5 (computed
with 4 lags and the constant).
In our multivariate VAR model Xti =A(L) X ti +ut , Granger-causality is a bit more
complex than in the bi-variate case. For simplicity, we will make only one-by-one test
for testing the hypothesis that consumer confidence Granger-cause the different
variables in the VAR system. Moreover, without complicating things about the limiting
page 57 out of 158

distribution of the F-statistic, we will add, as a set of control variables, the lags of the
other variables contained in the VAR: we will therefore test the coefficients of B(L)
in the equation yt=c0+c1t+ A(L) yt+B(L) xt +C(L)Zt .
The fact that one cannot refuse the hypothesis that confidence Granger-causes a
variable represents a piece of evidence that confidence provides useful information in
order to forecasts future values of this variable.
In particular, we have found (approximating the distribution of p times the F-statistic
to a χp
2
) that:
– using 4 lags of EY5 (and of the other variables in the VAR system of BP2002)
– p(χ p
2
> p F)=0.5806 against Ψ →
gc
G ;
– p(χ p
2
gc
T ;
– p(χ p
2
> p F)<0.0001 against Ψ →
gc
Y ;
– using 4 lags of CPC (and of the other variables in the VAR system of BP2002)
– p(χ p
2
gc
G ;
– p(χ p
2
gc
T ;
– p(χ p
2
> p F)<0.0001 against Ψ →
gc
Y .
Confidence seems then to help in predicting T and Y , but to have low predictive
power about G .
In order to better verify this, we have also computed the statistics for testing
Granger-causality in the simple bivariate relation yt=c0+c1t+ A4(L) yt+CPCt −1
with 4 lags of the considered variable (without any other control variable) and 1 lag
of confidence. Here we present the results:
– p(χ p
2
> p F)=0.7684 for CPC →
gc
G ;
page 58 out of 158

– p(χ p
2
> p F)<0.0001 for CPC →
gc
T ;
– p(χ p
2
> p F)<0.0001 for CPC →
gc
Y ;
On the other hand, we have not investigated with an high degree of formality about
the reverse Granger-causality relation (because of the unit-root problem cited
above): we simply noticed that in a regression of
CPCt= A4(L)CPCt +B4(L)xt −1 the F-statistics computed would not have been
sufficient to refuse the hypothesis that G and T do not Granger-cause CPC in the
case G and T were not unit root.
From this analysis we have collected some evidence in favour of the fact that
consumer confidence is important in correctly estimating the impact of structural
fiscal shocks in BP2002. In particular, we see that it is important for well predicting
the response of T and Y (and, therefore, for estimating their dynamic relation).
Incidentally, we also find that the inclusion of confidence in the system of the VAR
also increase significance of estimated coefficients (thus helping in better estimating
the dynamics between the variables). Indeed in a VAR with 4 lags, computing the
p-values values as if the – for simplicity, and only for illustrative purposes – 4 times
the F-statistics were distributed as χ4
2
(that is something highly improbable in this
case, since some of the regressors may be unit root processes), we see that in
general p-values are smaller.
2.3.5 Doing it brute-force
The previous findings brought us some evidence in favour of the hypothesis that
confidence might be important. In particular, since the objects we are interested in
are the impulse responses to structural shocks, we want to quantify the importance
of consumer confidence, and understand if its inclusion in the VAR system has a
statistically significant impact in the estimation.
page 59 out of 158

In order to do so, we have developed a simple “brute-force” method to evaluate the
significance of the difference between the impulse responses estimated with the VAR
system augmented with consumer confidence and these estimated with the non-
augmented original VAR:
– the first step is to estimate the augmented VAR and to store the prediction
errors;
– then we will generate 1000 series from this model, by randomly re-sampling
the estimated errors (as we did before for computing the median andh the
confidence bands of the impulse responses);
– for each generated series we will estimate both the augmented and the non-
augmented model;
– for each series, after running OLS for the estimation of the coefficients of the
two model, we have to identify the structural shocks from the estimated
variance-covariance matrix of the innovations and compute their impulse
responses on the variables
– for the original non-augmented model, we will identify the structural
shocks using the original identification;
– for the confidence-augmented model, we will identify the structural
shocks such that their main features are maintained, so that the
impulse responses are somehow comparable with the original ones;
– finally, for each series, we will compute the difference between the computed
impulse responses – augmented and non-augmented – in a 20 quarters
horizon.
This approach allow us to compute the confidence bands for the difference between
the impulse responses estimated including confidence and those estimated with the
original specification, without studying the distribution of the statistical object
“difference between the impulse responses.”
For example, applying the method explained above to BP2002, we are interested to
page 60 out of 158

compute the difference between the different impact of fiscal shocks, εT and εG :
for instance, if we find that the 95% confidence bands for the difference between
two responses to the same shock estimated under the two different specification are
above the x-axis, we can say that for at least 95% of the series the inclusion of
consumer confidence in the VAR system has lead to estimate an impulse response
which is above that computed with the non-augmented VAR.
This approach has two basic weaknesses:
– firstly, the results are highly subject to the identification restrictions used in
the augmented VAR; indeed, one basic assumption underlying this approach is
that the shocks identified in the two different specification are comparable,
while in fact it is possible that the additional restriction needed on the
augmented VAR make us estimate shocks that have a “different meaning”
from these estimated in the original specification;
– secondly, the method does not take into account second type errors (that
confidence has no role in helping the identification of the shocks) while
generating the bootstrapped series; on the other hand, in our work, this
analysis comes after having already collected evidence in favour of the null
hypothesis under which the method works.
The first weakness cited above does not result to be particularly relevant in the
estimation of the fiscal shocks, while it will be more important in the identification of
the technology shock, and will require some more reasoning.
We will now plot the differences between the impulse responses computed with the
augmented VAR system and the original responses under the different assumptions
i=4,3,1 . As stated above, if the difference results to be positive at some j -th
horizon for some k -th variable after an initial shock of the l -th structural shock, it
means that the considered augmented specification predicts an higher value of
X k , j k periods after the εl ,0 shock.
page 61 out of 158

For illustrative purposes, the line shown as the difference of impulse response for
CPC is equal to the response estimated in the augmented specification. Another
possibility was, for instance, to show the difference between the response estimated
in the augmented specification and that estimated with a VAR system where the
coefficients are constrained so that consumer confidence has no effect at any time
on the other variables.
page 62 out of 158

BP2002 (1960-2010Q2): difference in responses to a εG shock (GTY vs i=4 )
BP2002 (1960-2010Q2): difference in responses to a εT shock (GTY vs i=4 )
page 63 out of 158
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
d iff in e s tim a t e d irfs o f
G
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
d iff in e s tim a te d irfs o f
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-2
0
2
4
6
C P C
d iffe re n c e in irfs t o ε0
= [1 0 0 0 ], a u g m e n t e d V A R - o rig in a l V A R , e s t im a te d w ith # la g s = 4
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
G
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y
0 5 1 0 1 5 2 0
-1 0
-5
0
5
C P C

page 64 out of 158
0 5 1 0 1 5 2 0
-0 .3
-0 .2
-0 .1
0
0 .1
G
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
1
T
0 5 1 0 1 5 2 0
-4
-2
0
2
4
C P C
0 5 1 0 1 5 2 0
-0 .2
-0 .1
0
0 .1
0 .2
Y
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
G
0 5 1 0 1 5 2 0
-1 .5
-1
-0 .5
0
0 .5
T
0 5 1 0 1 5 2 0
-1
0
1
2
3
C P C
0 5 1 0 1 5 2 0
-0 .3
-0 .2
-0 .1
0
0 .1
Y

page 65 out of 158
0 5 1 0 1 5 2 0
-2
-1
0
1
C P C
0 5 1 0 1 5 2 0
-0 .3
-0 .2
-0 .1
0
0 .1
G
0 5 1 0 1 5 2 0
-1
-0 .5
0
0 .5
1
T
0 5 1 0 1 5 2 0
-0 .2
-0 .1
0
0 .1
0 .2
Y
0 5 1 0 1 5 2 0
-2
-1
0
1
C P C
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
G
0 5 1 0 1 5 2 0
-1 .5
-1
-0 .5
0
0 .5
T
0 5 1 0 1 5 2 0
-0 .4
-0 .2
0
0 .2
0 .4
Y

From the graphs above we can state that, while the inclusion of CPC in the system
does not significantly change the estimated impulse responses of the economic
variables when a εG shock affects economy, on the other hand – under all
assumptions i=1,3,4 on the ordering of confidence in the system – the estimated
responses of T and Y to a tax shock εT are significantly different when CPC is
included in the estimated VAR or it is not (at least until the 10th
quarter at 95% level
of significance, and often beyond the 15th
quarter at 68% level of significance).
In particular, we see that positive εT shocks leads to a negative impacts in all cases,
which however results to be less flattened when CPC is included.
A possible explication of this result is that confidence partly solves the “anticipation
problem” (already presented in the first section of this paper) treated in Perotti
(Feb2011xcii
:) if a tax is announced before the implementation, the variables will show
lower responses on macroeconomic variables contemporary to the effective
implementation. Indeed agents, if liquidity constraints are not too tight to prevent
changes in their choices and Barro-Ricardo equivalencexciii
does not hold, might have
already started to react to the shock at the time of the announcement.
We will test this hypothesis in the third section of this work.
2.3.6 Doing it with a large factor model
The theoretical reason why we choose to include consumer confidence in the VAR
system, as explained in the first section of this work, is that we believe that this
variable may help solving the non-fundamentalness problem affecting small VARs.
Forni, Gambetti, Sala (2010xciv
) and Forni & Gambetti (May2011xcv
and Jun2011xcvi
)
propose an interesting approach to test if the shocks recovered from a VAR system
are fundamental.
page 66 out of 158

In order to do this, they appealed to the hypothesis that macroeconomic data are the
observables from a large factor model xcvii
X t=χt+ξt with χt =A Ft where:
– the matrix A of the loadings represents the linear relation between common
component χt of the variables and the r principal factors Ft underlying the
set of economic variables;
– ξt represents the matrix containing the idiosyncratic component of each
variable, and can be interpreted – as in Forni & Gambetti (Jun2011xcviii
) – as a
measurement error.
Moreover, the dynamics of the model – which is observable to econometricians
trough X t only – concerns the principal factors through the MA process
Ft=N (L)εt , where εt is the vector containing the l structural fundamental
shocks animating the economy.
Let us imagine that only a subset ̃X t= ̃χt+ ̃ξt of k variables contained in X t is
observable. If ̃ξt is small, one one could directly test that if the subsystem
̃χt = ̃A N (L)εt satisfies the condition for fundamentalness of the system, that
rank ( ̃A N (L))=l , ∀ z∈ℂs.t.∣z∣<1 .
Indeed, for ̃ξt small enough, rejecting the hypothesis that εt is ̃χt -fundamental
would also represent a strong evidence against the null hypothesis that the shocks
εt are ̃X t -fundamental.
The testing procedure proposed by Forni, Gambetti, Sala (2010xcix
) and Forni &
Gambetti (May2011c
) can be implemented as follows:
– the first step is the estimation of the large factor model X t= A Ft+ξt ,
where the factors are estimated as the first r principal components of a
dataset including a large number of macroeconomic series (more details about
page 67 out of 158

the data are given in Appendix1;)
– a large number (say 1000) of series of factors are generated randomly re-
sampling the factors taken in blocks of τ contiguous records, where τ is
chosen sufficiently large in order to <<retain relevant lagged auto- and cross-
covariancesci
;>>
– for each j -th series Fj one have to estimate N (L) , and this can be done
by inverting an estimated VAR process G(L) Ftj =ut and identifying εt as
the first l principal components of ut ;
– if the system is square (that is k =l ), the p-value against the null hypothesis
of fundamentalness can be easily computed as the ratio between the number
of times the highest root of the equation det ( ̃A N (L))=0 is outside the unit
circle and the number of series generated.
It is important to notice that this test does not depend on the identification chosen
for the shocks. Indeed, if the factor model exactly contains l structural shocks, the
l principal components ̃εt of the innovations of the factors should contain all the
information provided by εt (in fact it should hold that ̃εt=Γεt , so that if χt lies in
the space generated by present and past values of ̃εt , it also lies on the space
generated by present and past values of εt .)
However, this testing procedure has some drawbacks.
Firstly, to be practically implemented, it requires that the system to be tested is a
square system; therefore, given a VAR system, it is necessary to assume that the
number of shocks in the economy is equal to the number of variables considered, or
at least that information contained in the first k shocks is the same contained in the
k shocks we want to identify.
Moreover, let us assume that we are considering a VAR such that k=l where l is
the real number of shocks in the economy: it is possible that the VAR system does
page 68 out of 158

not contain useful variables to recover all the l shock in the economy, but is
sufficient to correctly recover, say, l−1 shocks: the test described above will reject
the hypothesis of fundamentalness since the χt does not lie on the space generated
by present and past values of εt : indeed, this test is created such that the
hypothesis of fundamentalness is rejected when χt does not contain enough
information to recover all the l structural shocks.
As a consequence of the previous issue, let us imagine that a bivariate VAR contains
enough information to only recover the first structural shock, and the inclusion of
consumer confidence in the system provides the further information needed such
that also the second structural shock can be recovered, but not the third. The testing
procedure proposed above is much likely to reject again the hypothesis of
fundamentalness, even if we can now recover the first two shocks we were
interested in.
An interesting alternative method to test fundamentalness is provided by Forni &
Gambetti (Jun2011cii
). The basic idea underlying the following testing procedure is
that a fundamental shock (or, equivalently, a combination of fundamental shocks)
should be unpredictable by using the information provided in the information set.
Moreover, as a proxy of the information set, we can use the first r principal factors
estimated using the procedure described above.
More precisely, one can state that if Ft Granger-causes ̂εj ,t (where ̂εj is one of
the orthogonal shocks somehow identified using the structural VAR approach), if
follows that ̂εj cannot be a structural shock ciii
.
This procedure has a lot of advantages.
For first, one can test the fundamentalness of a single identified shock of interest,
without having to check that the entire VAR system is a fundamental representation:
page 69 out of 158

this testing procedure is therefore very useful in our case, since it can can be used to
state if the inclusion of consumer confidence in the VAR system helps in rejecting the
Granger-causality hypothesis that F t →
gc
̂ε j ,t
for some j -th somehow identified
shock.
Moreover, since we identify the shocks ̂ε such that they are orthogonal to each
other, testing bivariately if every single shock ̂εj is Granger-caused by F t is
equivalent to test if the whole vector ̂ε (or even a subset of it) is Granger-caused by
Ft : for example, rejecting that Ft →
gc
̂ε1,t
at some α confidence level and that
Ft →
gc
̂ε2,t
at the same α confidence level should be equivalent to reject the
hypothesis that Ft →
gc
[̂ε1,t ,̂ε2,t ]' at a degree of confidence of α2
−2α .
Our Granger-causality tests will be implemented using the standard F -statistic and
computing the p-values using its standard asymptotic distribution, since the principal
factors estimated in our model are stationary by construction. Moreover, since the
inclusion of too many factors as well as too many lags could weaken the power of
the test adding useless regressors, we chose to implement for each ̂εj shock a
series of Granger-causality tests using:
– 1, 2, 4 lags;
– the first principal factor, the first two principal factors, the first three principal
factor... and so on until the last r -th principal factor.
We choose r=13 , as in the baseline specification of Forni, Gambetti, Sala (2010civ
.)
As a rule of decision, we choose to consider as the p-value p*
for Ft →
gc
̂ε j ,t
, the
minimum among the p-values p(χp
2
> p F) found in the different tests. This
procedure results “prudential” in our sense, since it is constructed in order to
page 70 out of 158

rejecting the hypothesis that Ft →
gc
̂ε j ,t
even if only one test implemented would
reject it.
In particular we found that for the original BP2002 we have:
– p*
=0.6007 against F →
gc
εG
;
– p*
<0.0001 against F →
gc
εT
.
We have then collected some evidence in favour of the fact that a εG shock (as
identified in BP2002) is fundamental, while we are highly convinced that εT shock is
not: this help us in understanding why the inclusion of Ψ in the VAR system does
not change significantly the impulse responses to a εG shock, indeed we seem to
already have enough information to recover εG even in the non-augmented VAR.
However, even if we include Ψ in the VAR system, under every of the different
i=1,3,4 augmented identifications, we still find that p*
<0.0001 for F →
gc
εT
.
On the other hand, since we have already proved the importance of consumer
confidence in the identification of the tax shock, we conclude that even if confidence
brings important information, we are still far from recovering the real shock εT .
Instead of using p*
for testing Ft →
gc
̂εj ,t
, more precise reasoning can be based on
the analysis of the entire table of p-values with 1, 2, 4 lags and the principal factors
from the 1st
to the r -th. We report the tables of the p-values of every test on
Ft →
gc
̂εj ,t
made in our work in Appendix3.
page 71 out of 158

2.4 Confidence and the identification of monetary shocks
2.4.1 Replicating CEE1999
As to the identification of monetary shocks – as stated by Christiano, Eichenbaum
and Evans (1999cv
) – in literature there is a certain agreement about their qualitative
effects on macroeconomic variables.
In particular, in order to identify monetary shocks, we will first replicate and extend
the results obtained by Christiano, Eichenbaum and Evans (1999cvi
, henceforth
CEE1999.) under the specification replicated by Bachman & Sims (Aug2010cvii
.)
Similarly to what we have done for BP2002, we will estimate with OLS a VAR model
X t=α+ A(L)X t+ut with four lags, where (as defined in Appendix1)
X t=[Yt ,Ydeflt , Pcommt , FFRt , M2t ]
'
, and α is a vector of constants.
Once again, the relation between effects and shocks is modelled as a linear relation
ut=Dεt , where D is a lower triangular matrix: in this framework, keeping this
ordering in the Choleski identification is equivalent to state that the monetary policy
authority notices and reacts to unexpected shocks in macroeconomic variables within
the quarter. In particular, the monetary shock εFFR is identified as the 4th
shock in
the ε vector.
We will first run the estimation on the sample 1965Q3-1995Q2, then extend it to
1960Q1-2010Q2. As we will see, the results obtained extending the sample are
qualitatively identical to that obtained from the original dates. In particular, a εFFR
positive monetary shock implies a statistically significant negative movement of GDP.
page 72 out of 158

CEE1999 (1965Q3-1995Q2) with Choleski: responses to a εFFR shock
CEE1999 (1960-2010Q2) with Choleski: responses to a εFFR shock
page 73 out of 158
0 1 0 2 0
-1 0
-5
0
5
x 1 0
-3
Y
0 1 0 2 0
-0 .4
-0 .2
0
0 .2
0 .4
d e fl
0 1 0 2 0
-2
-1
0
1
2
P C O M
0 1 0 2 0
-0 .5
0
0 .5
1
1 .5
F F R
0 1 0 2 0
-0 .0 2
-0 .0 1
0
0 .0 1
M 2
= [0 0 0 1 0 ]', e s tim a te d w ith a V A R w ith # la g s = 4
0 1 0 2 0
-1 0
-5
0
5
x 1 0
-3
Y
0 1 0 2 0
-0 .4
-0 .2
0
0 .2
0 .4
d e fl
0 1 0 2 0
-2
-1
0
1
P C O M
0 1 0 2 0
-1
-0 .5
0
0 .5
1
F F R
0 1 0 2 0
-1 5
-1 0
-5
0
5
x 1 0
-3
M 2
= [0 0 0 1 0 ] ', e s t im a te d w ith a V A R w ith # la g s = 4

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