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

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.

Topics: Vector autoregression (59%), Fiscal multiplier (50%)

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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

© 2016 Taylor & Francis

Downloaded by [Johann Christian Senckenberg] at 11:30 07 March 2016

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 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

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.

Downloaded by [Johann Christian Senckenberg] at 11:30 07 March 2016

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):

141–164. doi:10.1257/jep.24.4.141.

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

Is the Government Spending Multiplier Large?” Journal of

Political Economy 119 (1): 78–121. doi:10.1086/659312.

APPLIED ECONOMICS LETTERS

3

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Ilzetzki, E., E. G. Mendoza, and C. A. Vegh. 2013. “How Big

(Small) are Fiscal Multipliers?” Journal of Monetary

Economics 60: 239–254. doi:10.1016/j.jmoneco.2012.10.011.

Kirchner, M., J. Cimadomo, and S. Hauptmeier.

2010. “Transmission of Government Shocks in the Euro

Area: Time Variation and Driving Forces.” ECB Working

Paper 1219.

Primiceri, G. 2005. “Time Varying Structural Vector

Autoregressions and Monetary Policy.” Review of Economic

Studies 72 (3): 821–852. doi:10.1111/roes.2005.72.issue-3.

Rafiq, S. 2014. “UK Fiscal Multipliers in the Postwar Era: Do

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

Review of Economic Studies. Forthcoming.

Uhlig, H. 2005. “What are the Effects of Monetary

Policy on Output? Results from an Agno stic

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jmoneco.2004.05.007.

4

M. DONADELLI ET AL.

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...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 ∑....

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...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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