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

07 Mar 2016-Applied Economics Letters (Routledge)-Vol. 23, Iss: 17, pp 1215-1218

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.

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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 LHuillier
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 LHuillier jplhuillier2010@gmail.com EIEF, via Sallustiana 62, 00187 Rome, Italy.
APPLIED ECONOMICS LETTERS, 2016
http://dx.doi.org/10.1080/13504851.2016.1145342
© 2016 Taylor & Francis
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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
t1
þ B
2
y
t2
þ ...þ B
1
y
t1
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 variancecovar-
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 variancecovariance 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 variancecovariance 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
t1
þ
^
B
2
y
t2
þ ...þ
^
B
1
y
t1
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 variancecovariance 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 variancecovariance 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
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Berg, T. O. 2014. Time Varying Fiscal Multipliers in
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Christiano, L., M. Eichenbaum, and S. Rebelo. 2011. When
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APPLIED ECONOMICS LETTERS
3
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Ilzetzki, E., E. G. Mendoza, and C. A. Vegh. 2013. How Big
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4
M. DONADELLI ET AL.
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Citations
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Abstract: This paper provides novel evidence on the time varying impact of government spending shocks on output in Germany over the years 1970 to 2013. In a first step, I use an expectations-augmented vector autoregressive model with time varying parameters (TVP-VAR) to show that fiscal multipliers are not stable over time but exhibit a u-shaped pattern. While multipliers fluctuate around 2 at the beginning and end of the sample, they are much smaller in between. In a second step, I discuss which factors determine the magnitude of German multipliers and hence explain the observed variation. It turns out that fiscal policy is more effective when business uncertainty is high but less in periods of financial market stress, while the state of the business cycle is minor important. Moreover, I find that fiscal sustainability is a crucial determinant of the multipliers and that these are about 1 euro higher since the loss of monetary policy autonomy due to the adoption of the euro. And finally, I conclude that policy recommendations based on average multipliers are misleading.

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"Differences in measures of the fisc..." refers background or methods in this paper

  • ...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 ∑....

    [...]

  • ...Also, sign restrictions (Uhlig 2005) based on a priori thoughts about the sign of the impact of fiscal policy can be considered....

    [...]

  • ...Because the Cholesky decomposition of ∑ is unique, if ∑P1 ≠ ∑P2, the corresponding Cholesky decompositions CP1 and CP2 satisfy CP1 CP2 (3) From Uhlig (2005), we know that α is given by R....

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