This note discusses a pitfall of using the generalized impulse response function (GIRF) in vector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general because it is invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extreme because it yields a set of response functions that are based on extreme identifying assumptions that contradict each other, unless the covariance matrix is diagonal. With an empirical example, the present note demonstrates that the GIRF may yield quite misleading economic inferences

Hyeongwoo, Kim

This note discusses a pitfall of using the generalized impulse response function (GIRF) in vector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general because it is invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extreme because it yields a set of response functions that are based on extreme identifying assumptions that contradict each other, unless the covariance matrix is diagonal. With a help of empirical examples, the present note demonstrates that the GIRF may yield quite misleading economic inferences

Kim, Hyeongwoo

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Generalized Impulse Response Analysis:
General or Extreme?
Hyeongwoo Kim
Auburn University
April 2009
Online at http://mpra.ub.uni-muenchen.de/53951/
MPRA Paper No. 53951, posted 26. February 2014 08:01 UTC
Generalized Impulse Response Analysis: General or Extreme?∗
Hyeongwoo Kim†
Auburn University
July 2012
Abstract
This note discusses a pitfall of using the generalized impulse response function (GIRF) in
vector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general because
it is invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extreme
because it yields a set of response functions that are based on extreme identifying assumptions
that contradict each other, unless the covariance matrix is diagonal. With a help of empirical
examples, the present note demonstrates that the GIRF may yield quite misleading economic
inferences.
Keywords: Generalized Impulse Response Function; Orthogonalized Impulse Response Func-
tion; Vector Autoregressive Models
JEL Classification: C13; C32; C51
∗A thank goes to Michael Stern for helpful comments. Madeline Kim provided excellent research assistance.
†Department of Economics, Auburn University, Auburn, AL 36849. Tel: 334-844-2928. Fax: 334-844-4615. Email:
gmmkim@gmail.com
1
1 Introduction
Notwithstanding its popularity, the orthogonalized impulse response function (OIRF; Sims, 1980)
analysis of structural vector autoregressive (VAR) models is subject to the so-called Wold-ordering
problem.1 That is, when one changes the order of the VAR with an alternative identifying assump-
tion, she may obtain dramatically diﬀerent response functions (Lütkepohl, 1991).
Pesaran and Shin (1998) propose the generalized impulse response function (GIRF), an ordering-
invariant approach, based on the work of Koop et al. (1996). The GIRF has been employed by
many researchers: Boyd et al. (2001), Cheung et al. (2004), and Huang et al. (2008), to name a
few. This note shows, however, that the GIRF may results in misleading inferences caused by its
extreme identification schemes.
The remainder of this note is organized as follows. Section 2 analytically demonstrates why the
GIRF actually can be considered extreme in perspective of its identification method. In Section 3,
I provide two empirical examples that highlights substantial diﬀerences between the GIRF and the
OIRF. Section 4 concludes.
2 A Pitfall of the GIRF
This section provides a simple analytical explanation on a pitfall in using the GIRF, which demon-
strates potentially serious problems of using the GIRF.
Let  () and  () denote the GIRF and the OIRF at time  + , respectively, when
there is one standard error shock at time  to the th variable in an -variate VAR with y =
[1 2 · · · ]0. Pesaran and Shin’s (1998) Proposition 3.1 implies 1() = 1().2 Define
˜ () as the OIRF when  is ordered first in y. By construction, 1() = ˜1(). Now re-order
the vector so that y = [2 1 3 · · · ]0, which yields 2() = ˜2() by the proposition
1The OIRF recursively identifies the structural shocks by using the Choleski decomposition factor of the covariance
matrix, which yields a unique lower triangular matrix. This scheme, therefore, assumes that the variable ordered first
in the VAR is contemporaneously unaﬀected by all other variables.
2That is, the GIRF and the OIRF coincide for the shock to the first variable in y.
2
and because the GIRF is invariant to the ordering of the variables in y. Repeat this procedure
until we get () = ˜(). Collecting these response functions, the GIRF for the entire system
is,
() =
n
˜1() ˜2() · · · ˜()
o
The GIRF, therefore, is not general in eﬀect because it employs extreme identifying assumptions
that each variable is ordered first. More seriously, ˜() and ˜ () are not consistent with each
other when  6=  unless the covariance matrix is diagonal. For instance, ˜() assumes that
 is not contemporaneously aﬀected by all other variables including , while ˜ () needs an
assumption that  is not contemporaneously aﬀected by all other variables including .3 Hence,
the GIRFs conflict each other. This result also trivially applies to vector error correction models.
In the next section, I show that such inconsistency may lead to misleading economic inferences.
3 Empirical Examples
This section provides empirical illustration to compare the implications of the GIRF with those of
the OIRF.
I first use a quadvariate VAR model of the US per capita investment (), consumption (),
real GDP (), and the government expenditure share () relative to the real GDP, measured in
logarithms. The data frequency is quarterly and the observations span from 1948Q1 to 2008Q4,
obtained from the Federal Reserve Bank of St. Louis FRED data base.
I report the ordering-free GIRF and the OIRF with an ordering [   ] to a government
expenditure shock in Figure 1.4 Both response functions display a decrease in , which may be
consistent with the crowding-out eﬀect. However, responses of  and  exhibit noticeably diﬀerent
dynamic adjustments. For example, the GIRF implies a significant decrease in  at the 5% level
over a year, while the OIRF implies significantly positive responses of  for about a year. Since 
3 If it is diagonal, there is no gain of using a structural VAR model, because it coincides with a reduced-form VAR,
in other words, equation-by-equation least squares estimations.
495% confidence bands are obtained from 5,000 nonparametric bootstrap simulations.
3
includes not only private but also public sector outputs, it is surprising to see a substantial decreases
in  in the short-run as we seen from the GIRF.5 Similarly, the GIRF implies a significant decrease
in  for about a year, while the OIRF displays slow positive adjustments of  which are insignificant.
Figure 1 about here
I implement another example to better understand why there may be such diﬀerences between
the GIRF and the OIRF. For this, I use a trivariate VAR model of , , and  from the previous
exercise, which was also employed by Pesaran and Shin (1998).
Note that the GIRFs to (one standard error) investment shock (Panel 2-a in Figure 2) coincide
with the OIRFs to an -shock when  is assumed to be contemporaneously unaﬀected by other two
variables,  and  (Panel 2-b). Note also that under this assumption, the OIRFs to a -shock are
very diﬀerent from the corresponding GIRFs. However, the GIRFs to a -shock are identical to the
OIRFs when  is ordered first in the VAR (Panel 2-d) by construction. Again, the other OIRFs
under that assumption are quite diﬀerent from the corresponding GIRFs. Likewise, the GIRFs to
a -shock are identical only to the OIRFs to a -shock when  is ordered first (Panel 2-c).
Based on these findings, it seems important to estimate and report response functions based
on the underlying economic model. For example, if one interprets -shocks as an output (supply)
shock, while -shocks and -shocks are treated as expenditure (demand) shocks, she may employ
an ordering [  ] assuming that  does not contemporaneously respond to demand shocks. Then,
she will report the response functions to an -shock, for instance, that are very diﬀerent from the
GIRFs both quantitatively and qualitatively. If one believes that  is primarily driven by animal
spirit, she may employ [  ] instead and report quite smaller responses of  to a -shock than the
corresponding GIRF.
Figure 2 about here
5For this, private sector activities should decrease by more than an increase in the increases in the public sector.
4
4 Conclusion
This note points out that there is a pitfall in using the GIRF. Economic inferences based on the
GIRF can be misleading because the GIRF employs a set of extreme identifying assumptions that
contradict each other unless the covariance matrix is diagonal. Our empirical example demonstrates
that this is by no means a negligible matter. In such cases, it would be more reasonable to use
identifying assumptions that consistently describes the underlying economic model.
5
References
1. Boyd, Derick, Gugielmo M. Caporale, Ron Smith (2001), "Real Exchange Rate Eﬀects on the
Balance of Trade: Cointegration and the Marshall-Lerner Condition," International Journal
of Finance and Economics 6, 187—200.
2. Cheung, Yin-Wong, Kon S. Lai, and Michael Bergman (2004), "Dissecting the PPP Puzzle:
The Unconventional Roles of Nominal Exchange Rate and Price Adjustments," Journal of
International Economics 64, 135-150.
3. Huang, Ying, Salih N. Neftci, and Feng Guo (2008), "Swap Curve Dynamics Across Markets:
Case of US Dollar versus HK Dollar," Journal of International Financial Markets, Institu-
tions, and Money 18, 79-93.
4. Koop, Gary, M. Hashem Pesaran, and Simon Potter (1996), "Impulse Response Analysis in
Nonlinear Multivariate Models," Journal of Econometrics 74, 119-147.
5. Lütkephol, Helmut (1991), Introduction to Multiple Time Series Analysis, Springer-Verlag,
Berlin, Germany.
6. Pesaran, M. Hashem and Yongcheol Shin (1998), "Generalized Impulse Response Analysis in
Linear Multivariate Models," Economics Letters 58, 17-29.,
7. Sims, Christopher (1980), "Macroeconomics and Reality," Econometrica 48, 1-48.
6
Firgue 1. Response Functions to a Government Share Shock
Note: Solid lines are point estimates. Dashed lines are 95% nonparametric confidence bands from 5,000
bootstrap simulations.
7
Firgue 2. Further Comparisons of Impulse Response Functions
8


Generalized Impulse Response Analysis: General or Extreme?

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveGeneralized Impulse Response Analysis:General or Extreme?Hyeongwoo KimAuburn UniversityApril 2009Online at http://mpra.ub.uni-muenchen.de/53951/MPRA Paper No. 53951, posted 26. February 2014 08:01 UTCGeneralized Impulse Response Analysis: General or Extreme?∗Hyeongwoo Kim†Auburn UniversityJuly 2012AbstractThis note discusses a pitfall of using the generalized impulse response function (GIRF) invector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general becauseit is invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extremebecause it yields a set of response functions that are based on extreme identifying assumptionsthat contradict each other, unless the covariance matrix is diagonal. With a help of empiricalexamples, the present note demonstrates that the GIRF may yield quite misleading economicinferences.Keywords: Generalized Impulse Response Function; Orthogonalized Impulse Response Func-tion; Vector Autoregressive ModelsJEL Classification: C13; C32; C51∗A thank goes to Michael Stern for helpful comments. Madeline Kim provided excellent research assistance.†Department of Economics, Auburn University, Auburn, AL 36849. Tel: 334-844-2928. Fax: 334-844-4615. Email:gmmkim@gmail.com11 IntroductionNotwithstanding its popularity, the orthogonalized impulse response function (OIRF; Sims, 1980)analysis of structural vector autoregressive (VAR) models is subject to the so-called Wold-orderingproblem.1 That is, when one changes the order of the VAR with an alternative identifying assump-tion, she may obtain dramatically different response functions (Lütkepohl, 1991).Pesaran and Shin (1998) propose the generalized impulse response function (GIRF), an ordering-invariant approach, based on the work of Koop et al. (1996). The GIRF has been employed bymany researchers: Boyd et al. (2001), Cheung et al. (2004), and Huang et al. (2008), to name afew. This note shows, however, that the GIRF may results in misleading inferences caused by itsextreme identification schemes.The remainder of this note is organized as follows. Section 2 analytically demonstrates why theGIRF actually can be considered extreme in perspective of its identification method. In Section 3,I provide two empirical examples that highlights substantial differences between the GIRF and theOIRF. Section 4 concludes.2 A Pitfall of the GIRFThis section provides a simple analytical explanation on a pitfall in using the GIRF, which demon-strates potentially serious problems of using the GIRF.Let  () and  () denote the GIRF and the OIRF at time  + , respectively, whenthere is one standard error shock at time  to the th variable in an -variate VAR with y =[1 2 · · · ]0. Pesaran and Shin’s (1998) Proposition 3.1 implies 1() = 1().2 Define˜ () as the OIRF when  is ordered first in y. By construction, 1() = ˜1(). Now re-orderthe vector so that y = [2 1 3 · · · ]0, which yields 2() = ˜2() by the proposition1The OIRF recursively identifies the structural shocks by using the Choleski decomposition factor of the covariancematrix, which yields a unique lower triangular matrix. This scheme, therefore, assumes that the variable ordered firstin the VAR is contemporaneously unaffected by all other variables.2That is, the GIRF and the OIRF coincide for the shock to the first variable in y.2and because the GIRF is invariant to the ordering of the variables in y. Repeat this procedureuntil we get () = ˜(). Collecting these response functions, the GIRF for the entire systemis,() =n˜1() ˜2() · · · ˜()oThe GIRF, therefore, is not general in effect because it employs extreme identifying assumptionsthat each variable is ordered first. More seriously, ˜() and ˜ () are not consistent with eachother when  6=  unless the covariance matrix is diagonal. For instance, ˜() assumes that is not contemporaneously affected by all other variables including , while ˜ () needs anassumption that  is not contemporaneously affected by all other variables including .3 Hence,the GIRFs conflict each other. This result also trivially applies to vector error correction models.In the next section, I show that such inconsistency may lead to misleading economic inferences.3 Empirical ExamplesThis section provides empirical illustration to compare the implications of the GIRF with those ofthe OIRF.I first use a quadvariate VAR model of the US per capita investment (), consumption (),real GDP (), and the government expenditure share () relative to the real GDP, measured inlogarithms. The data frequency is quarterly and the observations span from 1948Q1 to 2008Q4,obtained from the Federal Reserve Bank of St. Louis FRED data base.I report the ordering-free GIRF and the OIRF with an ordering [   ] to a governmentexpenditure shock in Figure 1.4 Both response functions display a decrease in , which may beconsistent with the crowding-out effect. However, responses of  and  exhibit noticeably differentdynamic adjustments. For example, the GIRF implies a significant decrease in  at the 5% levelover a year, while the OIRF implies significantly positive responses of  for about a year. Since 3 If it is diagonal, there is no gain of using a structural VAR model, because it coincides with a reduced-form VAR,in other words, equation-by-equation least squares estimations.495% confidence bands are obtained from 5,000 nonparametric bootstrap simulations.3includes not only private but also public sector outputs, it is surprising to see a substantial decreasesin  in the short-run as we seen from the GIRF.5 Similarly, the GIRF implies a significant decreasein  for about a year, while the OIRF displays slow positive adjustments of  which are insignificant.Figure 1 about hereI implement another example to better understand why there may be such differences betweenthe GIRF and the OIRF. For this, I use a trivariate VAR model of , , and  from the previousexercise, which was also employed by Pesaran and Shin (1998).Note that the GIRFs to (one standard error) investment shock (Panel 2-a in Figure 2) coincidewith the OIRFs to an -shock when  is assumed to be contemporaneously unaffected by other twovariables,  and  (Panel 2-b). Note also that under this assumption, the OIRFs to a -shock arevery different from the corresponding GIRFs. However, the GIRFs to a -shock are identical to theOIRFs when  is ordered first in the VAR (Panel 2-d) by construction. Again, the other OIRFsunder that assumption are quite different from the corresponding GIRFs. Likewise, the GIRFs toa -shock are identical only to the OIRFs to a -shock when  is ordered first (Panel 2-c).Based on these findings, it seems important to estimate and report response functions basedon the underlying economic model. For example, if one interprets -shocks as an output (supply)shock, while -shocks and -shocks are treated as expenditure (demand) shocks, she may employan ordering [  ] assuming that  does not contemporaneously respond to demand shocks. Then,she will report the response functions to an -shock, for instance, that are very different from theGIRFs both quantitatively and qualitatively. If one believes that  is primarily driven by animalspirit, she may employ [  ] instead and report quite smaller responses of  to a -shock than thecorresponding GIRF.Figure 2 about here5For this, private sector activities should decrease by more than an increase in the increases in the public sector.44 ConclusionThis note points out that there is a pitfall in using the GIRF. Economic inferences based on theGIRF can be misleading because the GIRF employs a set of extreme identifying assumptions thatcontradict each other unless the covariance matrix is diagonal. Our empirical example demonstratesthat this is by no means a negligible matter. In such cases, it would be more reasonable to useidentifying assumptions that consistently describes the underlying economic model.5References1. Boyd, Derick, Gugielmo M. Caporale, Ron Smith (2001), "Real Exchange Rate Effects on theBalance of Trade: Cointegration and the Marshall-Lerner Condition," International Journalof Finance and Economics 6, 187—200.2. Cheung, Yin-Wong, Kon S. Lai, and Michael Bergman (2004), "Dissecting the PPP Puzzle:The Unconventional Roles of Nominal Exchange Rate and Price Adjustments," Journal ofInternational Economics 64, 135-150.3. Huang, Ying, Salih N. Neftci, and Feng Guo (2008), "Swap Curve Dynamics Across Markets:Case of US Dollar versus HK Dollar," Journal of International Financial Markets, Institu-tions, and Money 18, 79-93.4. Koop, Gary, M. Hashem Pesaran, and Simon Potter (1996), "Impulse Response Analysis inNonlinear Multivariate Models," Journal of Econometrics 74, 119-147.5. Lütkephol, Helmut (1991), Introduction to Multiple Time Series Analysis, Springer-Verlag,Berlin, Germany.6. Pesaran, M. Hashem and Yongcheol Shin (1998), "Generalized Impulse Response Analysis inLinear Multivariate Models," Economics Letters 58, 17-29.,7. Sims, Christopher (1980), "Macroeconomics and Reality," Econometrica 48, 1-48.6Firgue 1. Response Functions to a Government Share ShockNote: Solid lines are point estimates. Dashed lines are 95% nonparametric confidence bands from 5,000bootstrap simulations.7Firgue 2. Further Comparisons of Impulse Response Functions8

MPRAMunich Personal RePEc ArchiveGeneralized Impulse Response Analysis:General or Extreme?Kim HyeongwooAuburn UniversityApril 2009Online at http://mpra.ub.uni-muenchen.de/17014/MPRA Paper No. 17014, posted 31. August 2009 14:32 UTCGeneralized Impulse Response Analysis: General or Extreme?∗Hyeongwoo Kim†Auburn UniversityApril 2009AbstractThis note discusses a pitfall of using the generalized impulse response function (GIRF) invector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general because itis invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extreme becauseit yields a set of response functions that are based on extreme identifying assumptions thatcontradict each other, unless the covariance matrix is diagonal. With an empirical example, thepresent note demonstrates that the GIRF may yield quite misleading economic inferences.Keywords: Generalized Impulse Response Function, Orthogonalized Impulse Response Func-tion, Vector Autoregressive ModelsJEL Classification: C13, C32, C51∗I am grateful to Michael Stern for helpful comments.†Department of Economics, Auburn University, 415 W. Magnolia Ave., Auburn, AL 36849. Tel: 334-844-2928.Fax: 334-844-4615. Email: gmmkim@gmail.com11 A Pitfall of the GIRFNotwithstanding its popularity, the orthogonalized impulse response function (OIRF; Sims, 1980)analysis of structural vector autoregressive (VAR) models is subject to the so-called Wold-orderingproblem.1 That is, when one changes the order of the VAR with an alternative identifying assump-tion, she may obtain dramatically different response functions (Lu¨tkepohl, 1991). Pesaran and Shin(1998) propose an ordering-invariant approach, the generalized impulse response function (GIRF),based on the work of Koop et al. (1996). The GIRF has been employed by many researches, toname a few, Boyd et al. (2001), Cheung et al. (2004), and Huang et al. (2008).However, it is important to recognize that there is a pitfall of using the GIRF. Let ψgyj (n) andψoyj (n) denote the GIRF and the OIRF at time t+n, respectively, when there is one standard errorshock at time t to the jth variable in an m-variate VAR with yt = [y1,t y2,t · · · ym,t]′. Pesaran andShin’s (1998) Proposition 3.1 implies ψgy1(n) = ψoy1(n).2 Define ψ˜oyj (n) as the OIRF when yj,t isordered first in yt. Then, ψgy1(n) = ψ˜oy1(n). Now re-order the vector so that yt = [y2,t y1,t · · · ym,t]′,which yields ψgy2(n) = ψ˜oy2(n) by the proposition and because the GIRF is invariant to the orderingof the variables in yt. Repeat this procedure until we get ψgym(n) = ψ˜oym(n). Collecting theseresponse functions, then, the GIRF for the entire system is,ψg(n) ={ψ˜oy1(n) ψ˜oy2(n) · · · ψ˜oym(n)}The GIRF, therefore, is not general in effect because it employs extreme identifying assumptionsthat each variable is ordered first. More seriously, ψ˜oyi(n) and ψ˜oyj (n) are not consistent with eachother when i 6= j if the covariance matrix is non-diagonal.3,4 Hence, the GIRFs conflict each other.This result trivially applies to vector error correction models also. In next section, I show that suchinconsistency may lead to misleading economic inferences.1The OIRF recursively identifies the structural shocks by using the Choleski decomposition of the covariancematrix, which yields a unique lower triangular matrix. This scheme, therefore, assumes that the variable ordered firstin the VAR is contemporaneously unaffected by all other variables.2The GIRF and the OIRF coincide for the first variable in yt.3ψ˜oyi(n) assumes yi,t is not contemporaneously affected by all other variables including yj,t, while ψ˜oyj (n) needs anassumption that yj,t is not contemporaneously affected by all other variables including yi,t.4If it is diagonal, there is no gain of using a structural VAR model, because it coincides with a reduced-form VAR,in other words, equation-by-equation least squares estimations.22 An Empirical IllustrationThis section provides an empirical illustration to compare the implications of the GIRF with thoseof the OIRF. I use a trivariate VAR model of the US per capita investment (i), consumption (c),and real GDP (y), measured in logarithms, as Pesaran and Shin (1998) did. The data frequencyis quarterly and observations span from 1947Q1 to 2008Q4, obtained from the Federal Reserve ofSt. Louis FRED data bank.Note that the GIRFs to (one standard error) investment shock (Panel 1-a in Figure 1) coincidewith the OIRFs to an i-shock when i is assumed to be contemporaneously unaffected by other twovariables, c and y (Panel 1-b). Note also that under this assumption, the OIRFs to a y-shock arevery different from the corresponding GIRFs. However, the GIRFs to a y-shock are identical to theOIRFs when y is ordered first in the VAR (Panel 1-d) by construction. Again, the other OIRFsunder that assumption are quite different from the corresponding GIRFs. Likewise, the GIRFs toa c-shock are identical only to the OIRFs to a c-shock when c is ordered first (Panel 1-c).What I claim here is that one has to estimate and report response functions based on hereconomic model. For example, if one interprets y-shocks as an output (supply) shock, while i-shocks and c-shocks are treated as expenditure (demand) shocks, she may employ an ordering[y i c] assuming that y does not contemporaneously respond to demand shocks. Then, she willreport the response functions to an i-shock, for instance, that are very different from the GIRFsboth quantitatively and qualitatively. If one believes that i is primarily driven by animal spirit,she may employ [i y c] instead and reports quite smaller responses of i to a y-shock than thecorresponding GIRF. The responses of i to a c-shock are again a lot different in Panels 1-b and 1-das i exhibits delayed overshooting for a year, while the GIRF produces bigger responses of i to ac-shock and delayed overshooting persists only for a half year.I am not claiming that the OIRF is better than the GIRF because there are many otheralternative options available.5 It seems more reasonable to me to use an identifying assumptionthat consistently describes the underlying economic models rather than to use the GIRF with acombination of extreme assumptions that conflict with each other.5For example, one may employ over-identified or partially identified systems. A just-identified non-recursive systemcan also be considered.33 ConclusionThis note points out that there is a pitfall of using the GIRF. Economic inferences based on theGIRF can be misleading because the GIRF employs a set of extreme identifying assumptions thatcontradict each other unless the covariance matrix is diagonal. Our empirical example demonstratesthat this is by no means a negligible matter.4ReferencesBoyd, D., G. M. Caporale, R. Smith, 2001. Real Exchange Rate Effects on the Balance of Trade:Cointegration and the Marshall-Lerner Condition. International Journal of Finance and Eco-nomics 6, 187–200.Cheung, Y.-W., K. S. Lai, M. Bergman, 2004. Dissecting the PPP Puzzle: The UnconventionalRoles of Nominal Exchange Rate and Price Adjustments. Journal of International Economics 64,135–150.Huang, Y., S. N. Neftci, F. Guo, 2008. Swap Curve Dynamics Across Markets: Case of US Dollarversus HK Dollar. Journal of International Financial Markets, Institutions, and Money 18, 79–93.Koop, G., M. H. Pesaran, S. Potter, 1996. Impulse Response Analysis in Nonlinear MultivariateModels. Journal of Econometrics 74.Lu¨tkephol, H., 1991. Introduction to Multiple Time Series Analysis. Springer-Verlag, Berlin, Ger-many.Pesaran, M. H., Y. Shin, 1998. Generalized Impulse Response Analysis in Linear MultivariateModels. Economics Letters 58, 17–29.Sims, C., 1980. Macroeconomics and Reality. Econometrica 48, 1–48.5Firgue 1. Impulse Response Function to One Standard Error Shock6

English

MPRA
Munich Personal RePEc Archive
Generalized Impulse Response Analysis:
General or Extreme?
Kim Hyeongwoo
Auburn University
April 2009
Online at http://mpra.ub.uni-muenchen.de/17014/
MPRA Paper No. 17014, posted 31. August 2009 14:32 UTC
Generalized Impulse Response Analysis: General or Extreme?∗
Hyeongwoo Kim†
Auburn University
April 2009
Abstract
This note discusses a pitfall of using the generalized impulse response function (GIRF) in
vector autoregressive (VAR) models (Pesaran and Shin, 1998). The GIRF is general because it
is invariant to the ordering of the variables in the VAR. The GIRF, in fact, is extreme because
it yields a set of response functions that are based on extreme identifying assumptions that
contradict each other, unless the covariance matrix is diagonal. With an empirical example, the
present note demonstrates that the GIRF may yield quite misleading economic inferences.
Keywords: Generalized Impulse Response Function, Orthogonalized Impulse Response Func-
tion, Vector Autoregressive Models
JEL Classification: C13, C32, C51
∗I am grateful to Michael Stern for helpful comments.
†Department of Economics, Auburn University, 415 W. Magnolia Ave., Auburn, AL 36849. Tel: 334-844-2928.
Fax: 334-844-4615. Email: gmmkim@gmail.com
1
1 A Pitfall of the GIRF
Notwithstanding its popularity, the orthogonalized impulse response function (OIRF; Sims, 1980)
analysis of structural vector autoregressive (VAR) models is subject to the so-called Wold-ordering
problem.1 That is, when one changes the order of the VAR with an alternative identifying assump-
tion, she may obtain dramatically different response functions (Lu¨tkepohl, 1991). Pesaran and Shin
(1998) propose an ordering-invariant approach, the generalized impulse response function (GIRF),
based on the work of Koop et al. (1996). The GIRF has been employed by many researches, to
name a few, Boyd et al. (2001), Cheung et al. (2004), and Huang et al. (2008).
However, it is important to recognize that there is a pitfall of using the GIRF. Let ψgyj (n) and
ψoyj (n) denote the GIRF and the OIRF at time t+n, respectively, when there is one standard error
shock at time t to the jth variable in an m-variate VAR with yt = [y1,t y2,t · · · ym,t]′. Pesaran and
Shin’s (1998) Proposition 3.1 implies ψgy1(n) = ψoy1(n).
2 Define ψ˜oyj (n) as the OIRF when yj,t is
ordered first in yt. Then, ψ
g
y1(n) = ψ˜oy1(n). Now re-order the vector so that yt = [y2,t y1,t · · · ym,t]′,
which yields ψgy2(n) = ψ˜oy2(n) by the proposition and because the GIRF is invariant to the ordering
of the variables in yt. Repeat this procedure until we get ψ
g
ym(n) = ψ˜oym(n). Collecting these
response functions, then, the GIRF for the entire system is,
ψg(n) =
{
ψ˜oy1(n) ψ˜
o
y2(n) · · · ψ˜oym(n)
}
The GIRF, therefore, is not general in effect because it employs extreme identifying assumptions
that each variable is ordered first. More seriously, ψ˜oyi(n) and ψ˜
o
yj (n) are not consistent with each
other when i 6= j if the covariance matrix is non-diagonal.3,4 Hence, the GIRFs conflict each other.
This result trivially applies to vector error correction models also. In next section, I show that such
inconsistency may lead to misleading economic inferences.
1The OIRF recursively identifies the structural shocks by using the Choleski decomposition of the covariance
matrix, which yields a unique lower triangular matrix. This scheme, therefore, assumes that the variable ordered first
in the VAR is contemporaneously unaffected by all other variables.
2The GIRF and the OIRF coincide for the first variable in yt.
3ψ˜oyi(n) assumes yi,t is not contemporaneously affected by all other variables including yj,t, while ψ˜
o
yj (n) needs an
assumption that yj,t is not contemporaneously affected by all other variables including yi,t.
4If it is diagonal, there is no gain of using a structural VAR model, because it coincides with a reduced-form VAR,
in other words, equation-by-equation least squares estimations.
2
2 An Empirical Illustration
This section provides an empirical illustration to compare the implications of the GIRF with those
of the OIRF. I use a trivariate VAR model of the US per capita investment (i), consumption (c),
and real GDP (y), measured in logarithms, as Pesaran and Shin (1998) did. The data frequency
is quarterly and observations span from 1947Q1 to 2008Q4, obtained from the Federal Reserve of
St. Louis FRED data bank.
Note that the GIRFs to (one standard error) investment shock (Panel 1-a in Figure 1) coincide
with the OIRFs to an i-shock when i is assumed to be contemporaneously unaffected by other two
variables, c and y (Panel 1-b). Note also that under this assumption, the OIRFs to a y-shock are
very different from the corresponding GIRFs. However, the GIRFs to a y-shock are identical to the
OIRFs when y is ordered first in the VAR (Panel 1-d) by construction. Again, the other OIRFs
under that assumption are quite different from the corresponding GIRFs. Likewise, the GIRFs to
a c-shock are identical only to the OIRFs to a c-shock when c is ordered first (Panel 1-c).
What I claim here is that one has to estimate and report response functions based on her
economic model. For example, if one interprets y-shocks as an output (supply) shock, while i-
shocks and c-shocks are treated as expenditure (demand) shocks, she may employ an ordering
[y i c] assuming that y does not contemporaneously respond to demand shocks. Then, she will
report the response functions to an i-shock, for instance, that are very different from the GIRFs
both quantitatively and qualitatively. If one believes that i is primarily driven by animal spirit,
she may employ [i y c] instead and reports quite smaller responses of i to a y-shock than the
corresponding GIRF. The responses of i to a c-shock are again a lot different in Panels 1-b and 1-d
as i exhibits delayed overshooting for a year, while the GIRF produces bigger responses of i to a
c-shock and delayed overshooting persists only for a half year.
I am not claiming that the OIRF is better than the GIRF because there are many other
alternative options available.5 It seems more reasonable to me to use an identifying assumption
that consistently describes the underlying economic models rather than to use the GIRF with a
combination of extreme assumptions that conflict with each other.
5For example, one may employ over-identified or partially identified systems. A just-identified non-recursive system
can also be considered.
3
3 Conclusion
This note points out that there is a pitfall of using the GIRF. Economic inferences based on the
GIRF can be misleading because the GIRF employs a set of extreme identifying assumptions that
contradict each other unless the covariance matrix is diagonal. Our empirical example demonstrates
that this is by no means a negligible matter.
4
References
Boyd, D., G. M. Caporale, R. Smith, 2001. Real Exchange Rate Effects on the Balance of Trade:
Cointegration and the Marshall-Lerner Condition. International Journal of Finance and Eco-
nomics 6, 187–200.
Cheung, Y.-W., K. S. Lai, M. Bergman, 2004. Dissecting the PPP Puzzle: The Unconventional
Roles of Nominal Exchange Rate and Price Adjustments. Journal of International Economics 64,
135–150.
Huang, Y., S. N. Neftci, F. Guo, 2008. Swap Curve Dynamics Across Markets: Case of US Dollar
versus HK Dollar. Journal of International Financial Markets, Institutions, and Money 18, 79–93.
Koop, G., M. H. Pesaran, S. Potter, 1996. Impulse Response Analysis in Nonlinear Multivariate
Models. Journal of Econometrics 74.
Lu¨tkephol, H., 1991. Introduction to Multiple Time Series Analysis. Springer-Verlag, Berlin, Ger-
many.
Pesaran, M. H., Y. Shin, 1998. Generalized Impulse Response Analysis in Linear Multivariate
Models. Economics Letters 58, 17–29.
Sims, C., 1980. Macroeconomics and Reality. Econometrica 48, 1–48.
5
Firgue 1. Impulse Response Function to One Standard Error Shock
6


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