Differences in measures of the fiscal multiplier and the reduced-form vector autoregression

TL;DR: In this paper, the authors show that if the parameters of the reduced-form VAR differ, then the dynamic effects of fiscal policy differ as well, generically and for any set of identification assumptions.

Abstract: The literature has recently asked whether the effects of fiscal policy vary with the state of the economy (Christiano, Eichenbaum, and Rebelo 2011; Rendahl 2014; Auerbach and Gorodnichenko 2012). We study this question in the context of vector autoregression (VAR) estimation. We show formally that, if (asymptotically) the parameters of the reduced-form VAR differ, then the dynamic effects of fiscal policy differ as well, generically and for any set of identification assumptions. Thus, in theory, the econometrician can detect these differences (either across time or space) generically just by relying on reduced-form VAR estimation.

Summary

1. Introduction

• Since the first observations of photoconductivity in organic materials[1], the scientific interest in organic solar cells (OSCs) as important candidates for a reliable and economically viable renewable energy technology has increased enormously[2, 3, 4, 5].
• Given the relative energy level alignment of the three components (Fig. 1, right), it is required that the Si-PCPDTBT sensitizer is spatially separated from fullerene phase.
• In the light of these findings, as a first-step towards the modeling of complex ternary blends, it is of primary importance to study comparatively the binary interactions and miscibility of different polymers with fullerenes and to clarify the physical quantities (such as the concentration and density) that control the miscibility and the interfaces.
• PTB7-th and Si-PCPDTBT with the fullerenederivative PC71BM, by calculating the mixing free energy as a function of the relative polymer:fullerene concentration and blend density.the authors.

2. Results and discussion

• The authors calculate the free energy change ∆Gm/Vm upon mixing the polymers with the fullerene acceptor.
• In order to calculate ∆Hm/Vm, for each relative concentration, 10 starting configurations with different distribution of the fullerenes in the polymer matrix are considered, for a total of∼500 structures, and annealed at room temperature and pressure for ∼20 ns.
• Nevertheless, the local minima observed at specific concentrations (e.g. fullerene weight ratio of 50% for PTB7, 65% for PTB7-th and 68% for Si-PCPDTBT) allow to realize kinetically stable BHJs[14].
• The case of Si-PCPDTBT versus PTB7-th is less obvious since the corresponding ∆Hm/Vm curves intersect at several concentrations.
• In order to explain the dependence of ∆Hm/Vm on fullerene weight ratio (x) the authors consider a simple model based on the matching between the average fullerene-fullerene distance d and the polymer period L.

2.1. Effect of non-equilibrium conditions

• The results described in Fig. 2 are obtained in ideal conditions and at equilibrium density, but polymer blends are processed from solution using different solvents and deposition techniques.
• The density of the blend is rapidly increased as the solvent evaporates and, at given density, the blend morphology starts to form.
• Fullerene weight ratio, the enthalpy of mixing depends on the density of the blend, also known as At fixed polymer.
• This is an important result that shows the impact of density on the miscibility of the donor:acceptor pair and, in turn, on the final morphology of the blends obtained during nonequilibrium conditions occurring during layer processing.
• Nevertheless, their analysis at variable density gives useful indications at an affordable computational cost.

2.2. Microscopic analysis of the blends

• In order to provide physical (i.e. microscopic) insight into the values of ∆Hm/Vm reported in Figs. 2 and 4, the authors have separated the different contributions of ∆Hm/Vm (also indicated as potential energy, Ep).
• This can attributed to the stiffness of the polymer; when PTB7-th and fullerenes are closely-packed the π −π interactions are stronger (lowering dispersive energy) but the polymer chains undergo larger distortions (increasing the dihedral energy).
• This is confirmed also by the radial pair distribution functions g(r) (see Fig. S3 in SI) that indicates higher aggregation in PTB7-th.
• The results are reported in Table 1 and show that the highest adhesion to fullerenes is that of PTB7-th, followed by PTB7 and Si-PCPDTBT.
• The energy cost of bending (dihedral term) increases rapidly for the more rigid PTB7-th so that the rather flexible PTB7 polymer is energetically favored and has better miscibility with fullerenes.

3. Conclusions

• By using the Flory-Huggins theory, the authors have calculated through atomistic simulations the free energy of mixing of three polymer:fullerene blends as a function of their relative concentration and density.
• Since a hole transfer from the host to the sensitizer is expected, the predicted placement is able to deactivate hole traps and enhance the fill factor of the cell, as observed in experiments.
• The qualitative ∆Hm/Vm dependence on fullerene weight ratio is shown to be related to the matching between the average fullerene-fullerene distance and the polymer period.
• The detailed microscopic understanding of the miscibility requires to break down the enthalpy into different contributions.
• For PTB7-th blends the authors find a competition between the π−π polymer-fullerene interaction and polymer torsional energy, which is not found in PTB7 and Si-PCPDTBT.

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Applied Economics Letters
ISSN: 1350-4851 (Print) 1466-4291 (Online) Journal homepage: http://www.tandfonline.com/loi/rael20
Differences in measures of the fiscal multiplier
and the reduced-form vector autoregression
Michael Donadelli, Adriana Grasso, Jean-Paul L’Huillier & Valentina Milano
To cite this article: Michael Donadelli, Adriana Grasso, Jean-Paul L’Huillier & Valentina
Milano (2016): Differences in measures of the fiscal multiplier and the reduced-form vector
autoregression, Applied Economics Letters, DOI: 10.1080/13504851.2016.1145342
To link to this article: http://dx.doi.org/10.1080/13504851.2016.1145342
Published online: 07 Mar 2016.
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Differences in measures of the fiscal multiplier and the reduced-form vector
autoregression
Michael Donadelli
a
, Adriana Grasso
b,c
, Jean-Paul L’Huillier
c
and Valentina Milano
b,c
a
Research Center SAFE and Goethe University Frankfurt, Frankfurt, Germany;
b
LUISS Guido Carli, Rome, Italy;
c
Einaudi Institute for Economics
and Finance (EIEF), Rome, Italy
ABSTRACT
The literature has recently asked whether the effects of fiscal policy vary with the state of the
economy (Christiano, Eichenbaum, and Rebelo 2011; Rendahl 2014; Auerbach and
Gorodnichenko 2012). We study this question in the context of vector autoregression (VAR)
estimation. We show formally that, if (asymptotically) the parameters of the reduced-form VAR
differ, then the dynamic effects of fiscal policy differ as well, generically and for any set of
identification assumptions. Thus, in theory, the econometrician can detect these differences
(either across time or space) generically just by relying on reduced-form VAR estimation.
KEYWORDS
Fiscal policy;
macroeconomic fluctuations
JEL CLASSIFICATION
E62; E32
I. Introduction
What determines the effects of fiscal policy in actual
economies
is currently a key policy issue. At some
level, it seems natural to expect that there is not one
constant and universal ‘multiplier’, but rather multi-
pliers that depend both on the state of the economy
and its underlying structure. This naturally leads to
questions like: Was the multiplier different in the
1970s than in the 2000s? Is the multiplier in
Germany similar to the multiplier in France?
We show that, in well-behaved cases, different
reduced-form parameters values of a nonsingular
VAR imply different structural dynamic responses
of variables to a fiscal shock. The result is quite
powerful because it applies for any set of identifica-
tion restrictions. Said differently, if the parameters of
the reduced-form are different (say across two dif-
ferent data sets), there is no identification restriction
such that the obtained effects of fiscal policy are the
same. The only exception is zero measure cases. That
is, the statement holds generically over the para-
meter space.
Thus, we note that in principle the researcher
interested in a yes/no answer to the questions above
could skip the identification step of the structural
shocks. The answer can be obtained by studying the
reduced-form directly, and thus independently of
identification assumptions. This is, of course, a meth-
odological advantage. In this letter, we formally estab-
lish the asymptotical basis for this approach. This is
the first step towards the construction of a test that
could potentially be used in practice.
Our result is useful for two reasons. First, in
practice, researchers working on fiscal policy estima-
tion through VARs have to make an identification
decision. In VARs with more than two variables,
multiple possibilities for the ordering of the variables
can be considered, each leading to a different
Cholesky decomposition. Also, sign restrictions
(Uhlig 2005) based on a priori thoughts about the
sign of the impact of fiscal policy can be considered.
Second, it is not ex-ante obvious that the result
holds. It could well be that there exist two different
sets of identification assumptions that could some-
how lead to the same structural responses for some
values of reduced-form parameters. Our result
ensures that, generically, this is impossible.
Our article adds to a rapidly growing literature
aiming to compare the fiscal multiplier over time
and across countries. Auerbach, Gale, and Harris
(2010) review the debate regarding the size of the
fiscal multiplier and provide fresh and provocative
thoughts regarding how details of the fiscal imple-
mentation and the state of the economy may influ-
ence the effectiveness of policy. Auerbach and
CONTACT Jean-Paul L’Huillier jplhuillier2010@gmail.com EIEF, via Sallustiana 62, 00187 Rome, Italy.
APPLIED ECONOMICS LETTERS, 2016
http://dx.doi.org/10.1080/13504851.2016.1145342
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Gorodnichenko (2012) provide explicit estimates of
fiscal multipliers when the economy is in recession
and compare them to the estimates when the econ-
omy is not. Ilzetzki, Mendoza, and Vegh (2013)
show that the effects of the policy depend on key
country characteristics, such as the level of develop-
ment, the exchange rate regime and public indebt-
edness, among others.
Pioneering work by Primiceri (2005) used a VAR
with time varying parameters to study the effects of
monetary policy. In the context of fiscal policy, a
natural step to take is to analyse following a similar
approach. Some papers have already taken that
route, see for instance, Kirchner, Cimadomo, and
Hauptmeier (2010), Rafiq (2014), or Berg (2014).
For the reasons explained above, our results below
naturally fit this research agenda.
II. Theory
Consider the reduced-form VAR model
y
t
¼ B
1
y
t1
þ B
2
y
t2
þ ...þ B
1
y
t1
þ u
t
(1)
where y
t
is a N × 1 vector, of which one of the
variables is the measure of fiscal policy of interest.
The variables in y
t
need not have any particular
order. We denote by V(u
t
)=∑ the variance–covar-
iance matrix of the innovations u
t
.
Once model (1) has been estimated by ordinary
least squares (OLS), we assume that identification of
structural shocks can be achieved
1
via a nonsingular
A matrix such that
u
t
¼ Aε
t
where ԑ
t
is the vector of structural shocks and it is
such that
Vðε
t
Þ¼I
The mapping A implies
AA
0
¼
X
(2)
Definition 1 A set of identifying restrictions R is
given by N(N – 1)/2 equations such that A is uniquely
pinned down by R and the equations in (2).
The following proposition establishes t hat if the
estimated variance–covariance matrix differs across
data sets, then the estimated impulse response func-
tions (IRFs) differ as well. This proposition is useful
because it suggests that the econometrician can rely
on the reduced-form parameters to gauge differ-
ences in the effect of fiscal pol icy, no matter his
stand in terms of identification.
Proposition 1 Suppose that, for two different popu-
lations, the reduced-form variance–covariance matrix
∑ is given by ∑
P1
and ∑
P2
, ∑
P1
≠∑
P2
. Then, given two
data sets, each from one of these populations, the
estimated IRFs to an (exogenous) impulse to govern-
ment spending differ across data sets, asymptotically
and generically, under any set of identifying restric-
tions R.
Proof. For a given sample {y
1
,...,y
T
}, write the
estimated VAR
y
t
¼
^
B
1
y
t1
þ
^
B
2
y
t2
þ ...þ
^
B
1
y
t1
þ u
t
and estimated ∑ by
^
P
. Because OLS is consistent,
estimates of matrix coefficients
b
B
1
,...,
b
B
1
of con-
verge in probability to B
1
,...,B
l
asymptotically (for
large T):
b
B
1
!
p
B
1
.
.
.
b
B
1
!
p
B
1
Similarly,
d
X
!
p
X
Thus, estimates for each of the data sets
d
X
P1
!
p
X
P1
and
d
X
P2
!
p
X
P2
Without loss of generality, assume that the coeffi-
cients in both populations are the same and given by
B
1
,...,B
l
.
1
It is well known that this assumption is restrictive because of fiscal foresight. Recently, applied researchers have argued that the inclusion of measures of
fiscal expectations is a useful way to alleviate such concerns (see, for instance, the discussion in Auerbach and Gorodnichenko 2012, p. 3. See also Berg
2014.)
2
M. DONADELLI ET AL.
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We are interested in the impulse vector a corre-
sponding to the government shock. This vector is
the column of A that corresponds to the government
impulse in ԑ
t
. Following the terminology in Uhlig
(2005) and using Proposition A.1 therein, this vector
can be obtained as
a ¼ Cα
where α is an N dimensional vector of unit length
and C is the Cholesky decomposition of ∑. Because
the Cholesky decomposition of ∑ is unique, if ∑
P1
≠
∑
P2
, the corresponding Cholesky decompositions C
P1
and C
P2
satisfy
C
P
1
Þ C
P
2
(3)
From Uhlig (2005), we know that α is given by R.
Our goal is to show that
a
P1
Þ a
P2
Either a
P1
Þ a
P2
directly from (3), or in the knife-
edge case that a
P1
= a
P2
we need to prove that the
claim holds generically. To do so, consider an arbi-
trarily small perturbation ζ 2 IR
þ
that implies a
perturbed a
0
1
given by
a
0
P1
¼ C
P1
þ ζIðÞα
and a
0
P1
Þ a
P2
. Because the Cholesky decomposi-
tion is unique, the perturbation ζdefines uniquely a
corresponding variance–covariance matrix
P
0
P1
with
Cholesky equal to C
P1
þ ζI, given by
X
0
P1
¼ C
P1
þ ζIðÞC
P1
þ ζIðÞ0
The expression shows that
P
0
P1
can b e made
arbitrarily close to
P
P1
by taking ζ small. To con-
clude, generically the IRFs to a government spend-
ing shock differ, as claimed. This completes the
proof.
For reasons of space, we have focused on the case
where parameters of the variance–covariance matrix
differ. A similar argument can be given for differ-
ences on parameters in
b
B
1
,...,
b
B
1
. To see this, notice
that for large T, the IRFs are
y
t
¼ Aε
t
; y
tþ1
¼ B
1
Aε
t;
y
tþ2
¼ B
2
1
Aε
t
þ B
2
Aε
t
; :::
Thus, a difference in
b
B
1
,...,
b
B
1
.between popula-
tions will also imply, generically, a difference in the
IRFs.
A remark is important regarding the precise
meaning of Proposition 1 holding generically. As it
is clear in the proof, this means that the result
applies to open and dense sets of parameters. As
such, it is possible to find examples in which the
result does not hold. But all those examples have to
be knife-edge cases, and thus in some sense mislead-
ing. This is the claim in the Proposition.
III. Conclusion
We have proven a theorem relating the p ara-
meters
of a reduced-form VAR and the IRFs to
an identified fiscal policy shock. The theorem
shows that, independently of identification
assumptions, diff erent reduced-form parameters
induce (asymptotically and generically) different
IRFs to a fiscal policy shock. Based on this result,
a researcher solely interested in differences
between fisca l policy multipliers across time,
states or countries can skip the identification
step as a first pass to the data. A formal deriva-
tion of a statistical test based on this idea is left
for future work.
Acknowledgements
We thank Pierpaolo Benigno, Daniele Terlizzese and LUISS
Guido Carli seminar participants for useful comments.
Grasso and Milano thank the Research Center SAFE for
hospitality and support during the preparation of this article.
Disclosure statement
No potential conflict of interest was reported by the authors.
References
Auerbach, A., W. G. Gale, and B. H. Harris. 2010. “Activist
Fiscal Policy.” Journal of Economic Perspectives 24 (4):
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Auerbach, A., and Y. Gorodnichenko. 2012. “Measuring the
Output Responses to Fiscal Policy.” American Economic
Journal-Economic Policy 4 (2): 1–27. doi:10.1257/pol.4.2.1.
Berg, T. O. 2014. “Time Varying Fiscal Multipliers in
Germany.” MPRA Working Paper 57223.
Christiano, L., M. Eichenbaum, and S. Rebelo. 2011. “When
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3
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Ilzetzki, E., E. G. Mendoza, and C. A. Vegh. 2013. “How Big
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State Dependent Shocks Matter?” CESifo Economic Studies
60 (1): 213–245. doi:10.1093/cesifo/ift011.
Rendahl, P. 2016. Fiscal Policy in an Unemployment Crisis
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M. DONADELLI ET AL.
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Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Differences in measures of the fiscal multiplier and the reduced-form vector autoregression" ?

The authors study this question in the context of vector autoregression ( VAR ) estimation. The authors show formally that, if ( asymptotically ) the parameters of the reduced-form VAR differ, then the dynamic effects of fiscal policy differ as well, generically and for any set of identification assumptions. 

Q2. What are the future works mentioned in the paper "Differences in measures of the fiscal multiplier and the reduced-form vector autoregression" ?

A formal derivation of a statistical test based on this idea is left for future work. 

Q3. What is the proof of the theorem?

The theorem shows that, independently of identification assumptions, different reduced-form parameters induce (asymptotically and generically) different IRFs to a fiscal policy shock.