Identification of Macroeconomic Shocks V
Identification of Macroeconomic Shocks:
Variations on the IS-LM Model∗
Thomas J. Jordan
Swiss National Bank
Economic Studies Section
P.O. Box
CH–8022 Z¨
urich, Switzerland
E-mail: jordan.thomas@snb.ch
Carlos Lenz
Universit¨at Basel
WWZ
Petersgraben 51
CH–4003 Basel, Switzerland
E-mail: carlos.lenz@unibas.ch
January 1999
Abstract
Applications of structural VARs to the IS-LM model reach different conclusions
about the relative importance of demand and supply shocks for business cycle fluctuations. This paper analyzes why these discrepancies occur. It is shown that the
results depend critically on the stationarity assumptions for certain variables. Because the results of the unit root and cointegration tests are inconclusive, the results
from structural VARs have to be interpreted with caution. It is crucial to check the
sensitivity of the results with respect to the assumed order of integration of those
variables which are on the borderline between I(1) and I(0).
JEL Classification: C32, E32
We thank Benjamin Friedman, Bennett McCallum, Michel Peytrignet, Georg Rich, Axel Weber,
Georg Winckler, and seminar participants at the Universities of Alicante, Basle, Vienna, at the Swiss
National Bank and at the Jahrestagung des Vereins f¨
ur Socialpolitik for very helpful comments. Both
authors gratefully acknowledge financial support provided by the Swiss National Foundation through
Grants No. 8210-040206 (T.J.) and 12-40498.94 (C.L.).
∗
1. Introduction
The debate about which type of disturbances cause business cycle fluctuations stands traditionally at the center of macroeconomic research. Since the seminal paper by Sims
(1980), the basic tool to evaluate different shocks has been the vectorautoregression
(VAR). The critique by Cooley and LeRoy (1985) led to the development of the structural
VAR approach pioneered by Bernanke (1986), Blanchard and Watson (1986), and Sims
(1986).1 Recently, considerable interest has been devoted to structural VARs consisting
of the variables which form the traditional IS-LM model. The restrictions needed to estimate the structural VAR are taken from the long-run or short-run restrictions which
are postulated by the aggregate supply/aggregate demand version of the textbook IS-LM
model.
There are two main motivations for this kind of analysis. One the one hand, the
validity of the predictions of the IS-LM model can be tested empirically. This will answer
the question about the adequacy of the IS-LM model as a description of the economy.2
On the other hand, the relative impact of the structural shocks on the different variables
can be assessed. Of particular interest in this context is the following question: To what
extent are output fluctuations at business cycle frequencies caused by either demand or
supply shocks? This point is critical for a discussion of whether real business cycle theory
or Keynesian macroeconomics is empirically more relevant.
There are five papers which can be regarded as applications of the IS-LM framework
in a structural VAR analysis with postwar U.S. data: Gal´ı (1992), Keating (1992), Bayoumi and Eichengreen (1993)3, Jordan and Lenz (1994), Gerlach and Smets (1995). The
interesting point is that the results of these studies differ considerably, especially with
respect to the importance of demand and supply shocks for output fluctuations. Table 1
summarizes the variance decompositions for output in the different studies.4
The following differences in the setup of the structural VARs can account for the
1
Structural VARs were originally identified by short-run restrictions. Blanchard and Quah (1989) and
Shapiro and Watson (1988) developed a method to identify shocks by long-run restrictions. Gal´ı (1992)
pioneered the simultaneous use of both types of restrictions.
2
See Gal´ı (1992) for a discussion why the empirical relevance of the IS-LM model remains of great
importance.
3
Two other papers by the same authors (Bayoumi and Eichengreen, 1994, 1995) use the same modeling
strategy. Unfortunately, none of their papers reports variance decompositions.
4
All tables and figures are in the Appendix.
2
discrepancies among these papers: (1) the number of variables used to estimate the same
number of structural shocks, (2) the assumed order of integration of the variables and the
assumed cointegrating relations, (3) the restrictions imposed to identify structural shocks,
and (4) the used data sets (definition of the variables and sample periods). Table 2 gives
an overview of the differences between the studies.
The number of variables included in all the analyzes varies between two and six. However, the maximum number of variables entering a VAR is four, additional variables are
sometimes analyzed by using linear combinations of the variables forming the estimated
system. The set of variables considered embraces output, the price level, real money,
nominal money, as well as nominal and real interest rates. The number of variables included in the VAR determines to what extent demand shocks have to be aggregated. The
assumed order of integration of the variables differs, especially in the case of the inflation
rate and the real interest rate. Most studies expect that the inflation rate is stationary,
only Gal´ı (1992) believes that it has a unit root. There is less consensus about the stationarity of the real interest rate: half the papers which analyze this variable presuppose
that it is stationary, the other half that it is integrated of order one. Depending on the
assumed order of integration of the individual variables, cointegration relations among
them are implicit in the choice of variables used for the analysis. For instance, if the
nominal interest rate and the inflation rate are assumed to be integrated of order one and
the real interest rate is taken to be stationary, then this implies that the nominal interest
rate and the inflation rate are cointegrated. All the papers have the identification scheme
for the aggregate supply shocks in common: Only these shocks have a long-run effect on
output. A variety of identification strategies are applied in order to identify the different
demand shocks.
Even though part of the variation between the results of the studies can be attributed
to the use of different data sets it is still striking that empirical models based on the same
philosophy come to very different conclusions. The purpose of this paper is therefore
two-fold: First, by using the same set of data throughout our analysis, we investigate
why the above-mentioned papers reach different conclusions, especially with respect to
the relative impact of demand and supply shocks on output over short horizons. Second,
we try to assess which model specification is the correct one and try to show whether
demand shocks are in fact important for business cycle fluctuations.
We find that the assumed order of integration of the variables and the cointegrating
relations between variables have a critical influence on the variance decompositions. This
3
is particularly true for the relative importance of demand and supply shocks. The assumptions about the long-run properties of the variables also have an influence on the
impulse responses but to a much lesser extent.It is interesting to note that the specific
identification restrictions are less important for the variance decompositions. This is, of
course, related to the fact that all authors use long-run neutrality of demand shocks to
identify aggregate supply shocks. However, the different strategies used to disentangle
the different types of demand shocks have no dramatic consequences on their effects. Furthermore, we find that a reduction in the number of variables implying the aggregation of
demand shocks leaves the main results basically unchanged.Unit root and cointegration
tests give no clear-cut answer about which model specification is the correct one. Therefore, results from structural VARs have to be interpreted with caution, because the result
may be sensitive to the assumed order of integration of those variables which are on the
borderline between I(1) and I(0).
The remainder of this paper is organized as follows: Section 2 presents the model and
discusses its integration properties. In Section 3, we develop a method for identifying
structural shocks in a VAR with both long-run and short-run restrictions. The empirical
results of five different specifications of the model are presented in Section 4, and Section
5 concludes.
2. A version of the IS-LM model
We start with the premise that real output, the nominal interest rate, nominal money, and
the price level are the four variables that can be used to identify aggregate supply, money
supply, money demand and spending shocks in a VAR model. Based on this assumption
we can ask two basic questions: 1. Which system of variables should be used? There
are six variables to choose from, the four variables mentioned above plus some linear
combinations like the real interest rate and real balances. In addition, each variable that
enters the system can be used in levels or in first differences (even second differences could
be considered), which offers quite a wide range of options. 2. Which restrictions should
be applied in order to identify the structural shocks? In the literature, a large variety of
approaches have been proposed to deal with this second question, but it is far from clear
which of these solutions is optimal.
In order to answer these questions, we take both theoretical and empirical aspects
into account. From a theoretical point of view, the choice of variables and their order
4
of integration should be in line with the structure of the model. Otherwise it becomes
an almost impossible task to apply the identifying restrictions suggested by the model.
In addition, an empirical analysis is needed to establish the time series properties of the
variables. In particular, this means that the empirical order of integration and possibly
existing cointegrating relations among the variables should be tested for rather than imposed. The results of the unit root and cointegration tests can then be used to choose
the correct specification of the model.
2.1. The model
In this section we present a log-linear version of the IS-LM model which is useful as a
means of choosing the variables and their order of integration. The IS and LM equations
establish the usual equilibrium conditions in the goods and money markets
i − E∆p+1 = f + β1 y
m − p = d + β2 y + β3 i,
(1)
(2)
where i is the nominal interest rate, p is the price level, f are autonomous fiscal expenditures, y is output, m is money supply, and d is the autonomous part of money demand.
E and ∆ are the expectations and difference operators, respectively, and all variables,
except the nominal interest rate, are in logs. The long-run aggregate supply curve is
vertical. This means that long-run changes in output are governed by aggregate supply
(uas ) shocks
∆y = uas .
(3)
In addition, we assume that the nominal money supply is either
∆m = β4 ∆y + β5 ∆i + β6 umd + ums
(4-a)
∆2 m = β4 ∆y + β5 ∆i + β6 umd + ums,
(4-b)
or
depending on the integration order of nominal money. Money supply depends therefore
on all shocks in the model. Note that we consider equations (1) – (4) to be a long-run
formulation, i.e. the difference operators apply to distant points in time. This is in line
with the comparative-static concept of the IS-LM model, which makes no exact predictions
about adjustment paths.
5
Taking first differences and labeling changes in fiscal expenditures as IS-shocks (uis )
and changes in the autonomous part of money demand as money demand shocks (umd ),
we can write the IS-LM model as follows:
∆(i − E∆p+1 ) = γ1 ∆y + uis
(5)
∆(m − p) = γ2 ∆y + γ3 ∆i + umd
(6)
∆y = uas
(7)
∆p = γ4 ∆y + γ5 ∆i + γ6 umd + ums
(8-a)
∆2 p = γ4 ∆y + γ5 ∆i + γ6 umd + ums,
(8-b)
where the choice of the price equation (8-a) or (8-b) depends on the integration order of
the money supply process. In order to derive equation (8-b), we impose a cointegrating
relation between ∆m and ∆p.
If all variables on the right-hand side of these equations and the inflation rate are
stationary (which implies E∆2 p+1 = 0 or ∆p = ∆p+1 ), we can solve for the variables in
terms of the structural shocks:
∆y
∆(i − ∆p)
∆(m − p)
∆p
1
α1
=
0
0
1
0
α2 α3
1
0 uas
is
0
u
md
0 u
α4 α5 α6 1
ums
(9)
.
This system represents the long-run effects of each structural shock and implies six
long-run neutrality restrictions: only aggregate supply shocks have a permanent effect on
the level of output, monetary shocks have no permanent effect on the level of the real and
the nominal interest rate, and money supply shocks have no permanent effect on the level
of real balances.
If the inflation rate is non-stationary and money and prices are cointegrated, the model
is
∆y
∆(i − ∆p)
∆(m − p)
2
∆p
1
α1
=
0
0
1
0
α2 α3
1
0 uas
is
0
u
md
α4 u
α5 α6 α7 1
ums
(10)
,
so that only five long-run neutrality restrictions exist. Another possibility is the assumption of a stationary real interest rate as suggested in Gal´ı (1992). This means that the
6
nominal interest rate and the inflation rate are cointegrated. In this case, the IS curve is
horizontal so that supply and fiscal shocks do not affect the real interest rate in the long
run. In this case, only three exploitable long-run neutrality restrictions exist.
If the inflation rate is stationary, the model suggests six long-run neutrality restrictions
which can be used to identify the structural shocks. If the inflation rate has a unit root,
the IS-LM model provides less than six long-run restrictions. In that case, short and
long-run restrictions have to be combined in order to identify the structural VAR. The
short-run restrictions come from the less formal and more implicit assumption about
the dynamics of the IS-LM model. For instance, Gal´ı (1992) assumes that output does
not react contemporaneously to money demand and supply shocks. Other short-run
restrictions can be inferred from the assumptions about the money supply process.
2.2. Aggregation of demand shocks
As already mentioned above, some authors also use models consisting of only two or three
variables and it is easy to see that this can be interpreted as an aggregation of demand
shocks. The money supply and money demand shocks can be aggregated to a money
market shock (umm ). By dropping the third line of equation (9), we get
∆y
∆p
1
∆(i − ∆p)
= α1
0
1
0 uas
is
0
u
α2 α3 1
umm
(11)
.
This is the model used in Jordan and Lenz (1994) and comes quite close to the model
used by Gerlach and Smets (1995). Both papers use the nominal interest rate instead of
the real interest rate. Since Gerlach and Smets (1995) assume the real interest rate to
be stationary, however, they can only apply two long-run restrictions and therefore they
need a short-run restriction in order to separate monetary from fiscal shocks.
Going one step further, fiscal and monetary shocks can be aggregated to an aggregate
demand shock. Eliminating the nominal interest rate from equation (11) yields
uas
∆y 1 0
=
ad
u
α1 1
∆p
,
which is the model used by Bayoumi and Eichengreen (1993).
7
(12)
2.3. Variations on the IS-LM model
In order to find out where the discrepancies in the results of the different versions of the
IS-LM model come from, we estimate five empirical models:
Model 1 consists of the same VAR as in Gal´ı (1992), with a stationary real interest
rate. However, we use a different set of short-run identifying restrictions, which allows us
to evaluate the influence of different identifying assumptions on the results. In the short
run, prices do not react to money demand and money supply shocks. Furthermore, we
presume that contemporaneous GNP does not enter the money supply rule.
Model 2 is the VAR formed by the variables of equation (9), where it is assumed that
the inflation rate is stationary and the real interest rate has a unit root. In order to
identify the structural shocks, we can rely on the long-run restrictions implied by the
model. The comparison of Model 1 and Model 2 should shed some light on the impact
of various systems on the results. Comparing Model 2 to the long-run model of Keating
(1992) should also give some insights about the discrepancies arising from using different
data sets.
Model 3 takes a unit root in both the inflation rate and the real interest rate. The
VAR corresponds to equation (10). There are five long-run restrictions implied by the
model. Furthermore, we assume that the inflation rate does not react contemporaneously
to money supply shocks. This exercise should give further insight into the effect of different
systems on the results.
Model 4 corresponds to the three-variable system in equation (11), where we assume a
stationary inflation rate and a non-stationary real interest rate. The shocks are identified
with long-run restrictions. This allows us to draw some conclusions about the use of
different data sets and different types of restrictions as we compare the results to those
in Jordan and Lenz (1994) and Gerlach and Smets (1995). In addition, we can assess the
effects of aggregating the money demand and supply shocks to a single money market
shock.
Model 5 is the VAR consisting of output growth and the inflation rate as in equation
(12) and the shocks are again identified by long-run restrictions. A comparison of the
8
results of Model 5 with those of Bayoumi and Eichengreen (1993) and those of Models 2
and 4 shows the effects of different data sets and the impact of the aggregation of all
demand shocks to a single shock.
3. Identification of the structural shocks
The structural shocks of the IS-LM model can be identified by a combination of longrun and short-run identifying restrictions on the VAR representation of the variables in
question. The idea of using both long-run and short-run restrictions simultaneously was
pioneered by Gal´ı (1992) and can be seen as a natural extension of earlier structural VAR
approaches that proposed to use either only short-run or only long-run restrictions.
The implementation of the restrictions is a straightforward exercise, provided they
have a special kind of recursive structure as we can demonstrate for the general case. We
start with a given vector of stationary variables xt which is assumed to have a structural
vector moving average (VMA) representation
xt = A(L)ut ,
where A(L) denotes a matrix polynomial in the lag-operator L and ut is a vector of unobservable structural shocks. We assume that the structural shocks are mutually uncorrelated and normalize their variance to unity, i.e. E(ut u′t ) = I. Inversion of the structural
VMA representation yields the structural VAR representation
B(L)xt = ut ,
where B(L) = A−1 (L). In order to identify the unobservable structural shocks contained
in ut , a set of restrictions is placed on A(L) and B(L). This allows us to represent the
structural shocks as linear combinations of the disturbances of the reduced form system
in xt .
We first estimate a reduced form VAR in xt
∆xt = D(L)∆xt−1 + ǫt ,
where ǫt is a vector of serially uncorrelated reduced form disturbances with covariance
matrix Ω. Inverting this VAR yields the Wold VMA representation
xt = C(L)ǫt ,
9
where C0 equals the identity matrix. A comparison between the reduced form and structural VMAs immediately shows that the following relation must hold
A0 ut = ǫt ,
that is, once A0 is known, the structural shocks can be computed from the reduced form
disturbances. In order to compute A0 we first note that
Ω = A0 A′0 ,
which places n(n + 1)/2 restrictions on the n2 different elements of A0 . This means that
we need n(n − 1)/2 additional restrictions to identify A0 . The method of Gal´ı (1992)
amounts to imposing restrictions on the immediate responses or the contemporaneous
structural relations between the variables (given by A0 and B0 respectively) and on the
long-run responses to the structural shocks (given by A(1)). The additional restrictions
are imposed on A0 , either directly or via the relationships
A−1
0 = B0
and C(1)A0 = A(1).
˜ and an arbitrary orthonormal matrix
Denoting the Cholesky decomposition of Ω by Ω
by S, we can write:
˜
ΩS
= A0
˜ ′ )−1 S = B ′
(Ω
0
˜
C(1)ΩS
= A(1).
Note that the knowledge of S is equivalent to the knowledge of A0 and, consequently,
that n(n − 1)/2 restrictions are needed to determine S. We show now that it is possible
to determine S explicitly if the restrictions have a ’triangular’ structure: Consider a
˜ (Ω
˜ ′ )−1 , and C(1)Ω.
˜
(n − 1) × n matrix Z which consists of the appropriate rows of Ω,
The ’triangular’ structure of the restrictions ensures that Z can be ordered in such a way
that, if multiplied by the first n − 1 columns of S, the following holds
Z[ S1 . . . Sn−1
0 0 ...
0 0 ...
]=
..
.. ..
.
. .
10
0
·
..
.
0 · ... ·
,
where Si denotes the i-th column of S. Using the equation above, the orthogonality of S
can be exploited to solve recursively for each of its columns:
Mi Si =
0
..
.
0
and Si′ Si = 1 i = 1, . . . , n,
where
zi,1
..
.
M1 = Z
Mi =
zn−1,1
...
...
S1′
..
.
′
Si−1
zi,n
..
.
zn−1,n
i = 2, . . . , n − 1
Mn =
S1′
..
.
′
Sn−1
.
4. Empirical Results
In this section we carry out the estimation under the different setups discussed at the
end of Section 2. We use the same data set as Gal´ı (1992). The sample period runs from
1955:I to 1987:III and the data includes the log of real GNP (y), the yield on three-month
Treasury bills (i), the log of the Consumer Price Index (p), and the log of M1 (m). All
data is from CITIBASE, except for measures of M1 previous to 1959 which are from the
Federal Reserve Bulletin.
We then test for the order of integration of the individual variables and for possible
cointegrating relations among some of the variables. These tests are critical for the following reason: All variables included in the VAR have to be stationary to avoid spurious
regression problems. The estimated system is also misspecified if it includes first differences of cointegrated variables. The empirical results of the unit root and cointegration
tests are therefore a tool to discriminate among the different specifications in order to
find the correct model.
4.1. The Impact of Different Identifying Restrictions
We use the system proposed by Gal´ı (1992) to check whether different identification
restrictions can lead to different impulse responses and variance decompositions. In contrast to Gal´ı, we identify the structural VAR of our Model 1 by a different set of short-run
11
restrictions: MS and MD shocks do not affect the prices within the first quarter and contemporaneous GNP does not enter the money supply rule. The results are presented in
Figure 1 and Table 5.
Considering the impulse responses, the only differences arising from using a new set of
short-run identification restrictions are due to the impact of the MD shocks on ∆m, ∆p
and m−p in the short run. In our setup, nominal money growth first declines and becomes
positive only after 4 quarters. The inflation rate is first increasing and jumps below its
original level only after 8 quarters. Real balances therefore first decline and reach their
old level after about 12 quarters. All the other impulse responses look very similar to
those reported to those in Gal´ı (1992). Concerning the variance decompositions, we find
that the relative importance of IS and MS shocks changes compared to Gal´ı’s results. In
general MS shocks become more important for y, i, m, and ∆p; and less important for
m−p and i−∆p. The opposite is true for IS shocks. However, the impact of supply shocks
on all variables is very similar to Gal´ı (1992). Supply shocks are the dominant source of
output fluctuations even in the very short run. We conclude that different identification
restrictions do not lead to a change in the relative importance of demand and supply
shocks for output variations if the system remains unchanged.
With both possible identification schemes, the system proposed by Gal´ı leads to some
implausible results. This is most evident in the version with the original set of restrictions.
IS shocks have a strong impact on the inflation rate in the long run, whereas MS supply
shocks have a much smaller effect. IS shocks also have a strong impact on the long-run
money growth. Although the IS shock has a positive impact on the inflation rate, it does
not influence the real interest rate. This is incompatible with conventional macroeconomic
models.5
4.2. The Impact of Different Assumptions about the Order of Integration
We now concentrate on the effects of different assumptions about the order of integration
and estimate two new models: Model 2 assumes that the inflation rate is stationary and
that the real interest rate has a unit root; it consists of the variables ∆y, ∆(i − ∆p), ∆p,
and ∆m − ∆p. As discussed in Section 2, this model can be identified only by long-run
restrictions. Model 3 assumes that both the real interest rate and the inflation have a
unit root; it consists of the variables ∆y, ∆(i − ∆p), ∆2 p, and ∆m − ∆p and therefore
5
This is also mentioned in the critique by King (1993) of the Gal´ı (1992) paper.
12
requires that ∆m and ∆p are cointegrated. The set of identification restrictions used in
Model 3 is similar to the one used by Gal´ı (1992).
The results of Model 2 are shown in Table 6 and Figure 2. Comparing the results to
those of Model 1, the impulse responses are different for i, ∆p, and i − ∆p. AS shocks
lower i and i − ∆p in the long run, whereas IS shocks increase i and i − ∆p. MS and
MD shocks have only a short-run impact on the interest rates. None of the shocks has a
long-run impact on ∆p because of the assumed stationarity of ∆p. The impulse responses
for output are similar to the system discussed above. All impulse responses of this model
are exactly what we expect from the theoretical IS-LM model.
With respect to the variance decompositions, we observe a dramatic change. Demand
shocks, mainly IS and MS shocks, are now the dominant source of output fluctuations
at business cycle frequencies. Consequently, different assumptions about the order of
integration of the time series and the cointegration among the variables change the relative
importance of demand and supply shocks in the short run.
To further analyze the impact of a different integration assumptions on the results,
we estimate Model 3. The results are in Table 7 and Figure 3. The impulse responses are
similar to those of Model 2. The main difference is the impulse response of the inflation
rate which is not restricted to be zero in the long run. We find that changes in the inflation
rate are mainly due to money demand and money supply shocks. It is interesting to note
that the contribution of demand shocks to the variance of real output lies between those
of Models 1 and 2. In contrast to Model 1 demand shocks are important for output
fluctuations in the short run, but not as important as in Model 2. All three demand
shocks contribute equally to output fluctuations, this implies that money demand shocks
contribute more than in Model 2.
Keating (1992) uses a system which has the same assumptions about the order of
integration as our Model 2, but he uses the nominal interest rate instead of the real
interest rate and money growth instead of inflation in his VAR setup. He also uses only
long-run restrictions to identify the structural shocks. If we compare his results to those of
our Model 2, we find that they are very similar. However, in our Model 2, money supply
shocks are generally more important than in the VAR used by Keating. Comparing the
results of Keating (1992) to Model 1, we see again the dramatic change in the relative
importance of demand and supply shocks due to a different VAR system.
Considering the results of Models 1 to 3, we conclude that the relative importance of
supply and demand shocks for output fluctuations at business cycle frequencies depends on
13
the assumed order of integration of the variables and, therefore, on whether the variables
enter the VAR in levels or in first differences. The results change completely for different
integration assumptions. The use of different identification assumptions can also change
the impulse responses of some variables, but the changes are less dramatic.
4.3. The Impact of Different Data Sets
Another question of some interest is whether the use of different data sets changes the
results in any significant way if the same system is estimated. The comparison of either
the results of Model 4 to those of Jordan and Lenz (1994) or those of Model 2 to those of
Keating (1992) shows that the impulse responses are not very much influenced by different
data sets. The variance decomposition shows that the relative impact of the different
shocks changes. However, the relative importance of demand and supply shocks for output
fluctuations is approximately the same. Bayoumi and Eichengreen (1993) estimate a twovariable system, which can be compared to the system of equation (12). They do not
present variance decompositions for their model. However, their impulse responses look
very similar to our results. We conclude that the use of different data sets does not affect
the form of the impulse responses very much. However, the relative importance between
the shocks can change. In the cases considered, the relative importance of demand and
supply shocks for output fluctuations remains the same for different data sets.
4.4. The Impact of Aggregation of Demand Shocks
An interesting aspect of the VAR analysis with IS-LM variables is the question whether the
reduction of the four-variable VAR to a three and two-variable system changes the results
of the impulse responses and the variance decompositions. This point thus addresses
the question whether the aggregation of structural demand shocks basically changes the
importance of the demand shocks relative to the supply shocks.
This question can be answered by looking at the results for Models 2, 4, and 5 which
use only long-run restrictions to identify structural shocks. Model 4 aggregates money
demand and supply shocks into a money market (MM) shock. The restrictions are that
MM and IS shocks do not have long-run effects on y and MM shocks do not have long-run
effects on i. In Model 5, all demand shocks are represented by a single aggregate demand
(AD) shock which is assumed to leave output unaltered in the long run. Tables 8 and 9
and Figures 4 and 5 report the results.
14
If we compare the impact of the supply shocks on the output variance, we find a
relatively high correspondence between the three models. The relative importance of
supply and demand shocks varies only slightly between the models. We therefore conclude
that the reduction of the VAR or the aggregation of the shocks does not change the
basic results of the relative importance between demand and supply shocks for output
fluctuations. The aggregation of shocks seems, in any case, less critical than the order of
integration.
4.5. Unit Roots and Cointegration
The order of integration for y, i, m, and m − p is undisputed. The results of our unit
root tests as well as those of other studies show that these variables are I(1).6 Therefore,
we do not present the results of the unit root tests for these variables. The situation is
different for the inflation rate and the real interest rate. Table 3 shows the results of the
Augmented-Dickey-Fuller and Phillips-Perron tests at different lag-lengths.7 The results
of these unit root tests are mixed. For both variables, the Phillips-Perron tests reject the
null hypothesis of a unit root, whereas the Augmented-Dickey-Fuller tests point towards
non-stationarity of the time series.
The studies of Keating (1992), Bayoumi and Eichengreen (1993), Gerlach and Smets
(1995), and Jordan and Lenz (1994) all assume the inflation rate to be stationary. The
unit root tests presented in Gal´ı (1992) also indicate that the inflation rate is stationary.
However, he concludes that the inflation rate must have a unit root because his tests
indicate that the real interest rate is stationary. This conclusion further implies cointegration between the nominal interest rate and the inflation rate. If the variables were not
cointegrated, the real interest rate could not be stationary. A unit root in the inflation
rate also implies that ∆m is I(1) and cointegrated with ∆p if real money growth is to
be stationary. This leads to an additional restriction for the order of integration of the
variables included in the VAR.
In order to shed some more light on this issue, we test for cointegration between i and
∆p as well as ∆m and ∆p by using the Johansen procedure. Table 4 summarizes the
results of this exercise at different lag-lengths.
The tests show that the null hypothesis of no cointegration between i and ∆p is
6
An exception is the study by Gerlach and Smets (1995), where i is assumed to be stationary.
7
The minimal lag-length for both the unit root and cointegration tests was chosen so that the residuals
from the regressions are uncorrelated.
15
never rejected if the maximum eigenvalue statistic is used. Considering the trace statistic
this hypothesis is rejected at the 10 percent level at lag-lengths 3 and 4, but we cannot
reject the null of one cointegrating relation. However, this does not imply that the real
interest rate is stationary. Given that i is I(1), the presence of only one unit root in
the system consisting of i and ∆p can be due to either a stationary inflation rate or
cointegration between i and ∆p, that is, a stationary real interest rate. In order to
discriminate between these cases, we test whether the vectors [0 1] and [1
− 1] are
contained in the cointegrating space at those lag-lengths where a cointegrating relation
between i and ∆p is found. The results are, again, unclear: At three lags, the presence
of [0 1] in the cointegrating space is rejected, whereas the presence of [1 − 1] cannot be
rejected. At four lags both vectors could be contained in the cointegrating space. To sum
up, one can say that these results confirm the findings of the unit root tests: The data
gives no clear answer to the question about the stationarity of ∆p and i − ∆p.
Concerning the cointegration between ∆m and ∆p, the picture is not clear, either.
The tests show that the null of no cointegration can be rejected at three lags in favor of
one cointegrating relation. At four lags, only the maximum eigenvalue test rejects the
absence of cointegration; and at five lags, two unit roots seem to be present in the system
formed by ∆m and ∆p. This contradicts the unit root tests where both real and nominal
money appear to be stationary in first differences. At those lag-lengths where there seems
to be only one unit root in the system, we check whether it is due to the stationarity of
∆m or whether ∆m and ∆p are cointegrated. That is, we test whether the vectors [1 0]
or [1 − 1] are contained in the cointegrating space. The results of these tests show that
neither hypothesis can be rejected.
Given the evidence from the unit root and cointegration tests none of the models
presented in Section 2 can a priori be discarded. Furthermore, the exercise carried out in
this subsection does not provide the basis for a assessment of the models in accordance
with their ability to fit the data. Therefore, the main conclusion we can draw from these
tests is that one must be very cautious in interpreting the results from structural VARs.
In particular, it seems to be crucial to check the sensitivity of the results with respect
to the assumed order of integration of the variables which are on the borderline between
I(1) and I(0).
16
5. Conclusions
We have presented a structural VAR analysis of the IS-LM model under a variety of
identifying assumptions. The main contribution of this paper is to show that the relative
importance of demand and supply shocks for output fluctuations depends critically on
the stationarity assumptions for certain variables. The responses of the variables to
different shocks also depend to some extent on particular identifying restrictions, but
those differences are less important. It is also shown that the aggregation of demand
shocks does not alter their relative importance with respect to supply shocks.
The obtained results show that in structural VAR analysis special care needs to be
taken in determining the order of integration of the variables forming the estimated system. This is particularly true in those cases where unit root tests do not yield clear
results. With respect to the order of integration of the variables in the IS-LM model, the
critical decision has to be made about the stationarity of both the real interest rate and
the inflation rate. If the real interest rate has a unit root and the inflation rate is stationary, we see that demand shocks are very important for output fluctuations at business
cycle frequencies. If the real interest rate is stationary and the inflation rate has a unit
root, then supply shocks are the dominant source of output fluctuations even in the short
run. If both the real interest rate and the inflation rate have a unit root, the results are
in-between the other two cases.
The evidence from the unit root and cointegration tests does not allow to favor one of
the models over another. Therefore, one must be very cautious in interpreting the results
from structural VARs and it is crucial to check the sensitivity of the results with respect
to the assumed order of integration of the variables which are on the borderline between
I(1) and I(0).
17
References
Bayoumi, T. and B. Eichengreen (1993) “Shocking aspects of european monetary
unification.” In F. Torres and F. Giavazzi (eds.), Adjustment and Growth in the European Monetary Union. Cambridge University Press, Cambridge (UK).
Bayoumi, T. and B. Eichengreen (1994) “Macroeconomic adjustment under BrettonWoods and the post Bretton-Woods float: An impulse response analysis.” The Economic Journal, 104:813–827.
Bayoumi, T. and B. Eichengreen (1995) “Is there a conflict between EC enlargement
and European monetary unification?” Greek Economic Review.
Bernanke, B. S. (1986) “Alternative explanations of the money-income correlation.”
Carnegie-Rochester Conference Series on Public Policy, 25:49–100.
Blanchard, O. J. and D. Quah (1989) “The dynamic effects of aggregate demand
and supply disturbances.” The American Economic Review, 79:655–673.
Blanchard, O. J. and M. Watson (1986) “Are business cycles all alike?” In R. Gordon (ed.), The American Business Cycle, pp. 123–156. NBER and University of Chicago
Press.
Cooley, T. F. and S. F. LeRoy (1985) “Atheoretical macroeconomics: A critique.”
Journal of Monetary Economics, 16:283–308.
Gal´ı, J. (1992) “How well does the IS-LM model fit postwar U.S. data?” The Quarterly
Journal of Economics, 107:709–738.
Gerlach, S. and F. Smets (1995) “The monetary transmission mechanism: Evidence
from the G-7 countries.” Discussion Paper 1219, Center for European Policy Research.
Johansen, S. and K. Juselius (1990) “The full information maximum likelihood procedure for inference on cointegration — with applications to the demand for money.”
Oxford Bulletin of Economics and Statistics, 52:169–210.
Jordan, T. J. and C. Lenz (1994) “Demand and supply shocks in the IS-LM model:
Empirical findings for five countries.” Working Paper 94-8, Universit¨
at Bern.
18
Keating, J. W. (1992) “Structural approaches to vector autoregressions.” Federal Reserve Bank of St. Louis, Review, 74(5):37–57.
King, R. G. (1993) “Will the New Keynesian macroeconomics resurrect the IS-LM
model?” Journal of Economic Perspectives, 7:67–82.
Osterwald-Lenum, M. (1992) “A note with quantiles of the asymptotic distribution
of the maximum likelihood cointegration rank test statistics.” Oxford Bulletin of Economics and Statistics, 54:461–472.
Runkle, D. E. (1987) “Vector autoregressions and reality.” Journal of Business &
Economic Statistics, 5:437–442.
Shapiro, M. D. and M. W. Watson (1988) “Sources of business cycle fluctuations.”
In S. Fischer (ed.), NBER Macroeconomics Annual 1988. The MIT Press, Cambridge
(Mass.) and London.
Sims, C. A. (1980) “Macroeconomics and reality.” Econometrica, 48:1–48.
Sims, C. A. (1986) “Are forecasting models usable for policy analysis?” Federal Reserve
Bank of Minneapolis, Quarterly Review, pp. 2–16.
Appendix: Tables and Figures
Table 1
Contribution of Supply Shocks to Fluctuations in U.S. GNP:
Results from four Studies
Output Variance explained
Paper
Gal´ı (1992)
Keating (1992)
Jordan and Lenz (1994)
Gerlach and Smets (1995)
Horizon
by AS Shocks (in %)
1
69
5
67
10
72
1
17
4
24
8
45
1
13
4
28
8
49
1
46
4
71
20
90
Table 2
VAR Setups in Different Studies
Included
Order of
Cointegrating
Type of Identifying
Variables
Integration
Relations
Restrictions
Bayoumi and Ei-
∆y
y ∼ I(1)
none
only long-run
chengreen (1993)
∆p
p ∼ I(1)
Gal´ı (1992)
∆y
y ∼ I(1)
∆m − ∆p
long and short-run
∆i
i ∼ I(1)
i − ∆p
i − ∆p
∆p ∼ I(1)
∆(m − p)
∆m ∼ I(1)
∆y
y ∼ I(1)
∆i
i ∼ I(1)
∆m
m ∼ I(1)
∆(m − p)
p ∼ I(1)
Jordan and
∆y
y ∼ I(1)
Lenz (1994)
∆i
i ∼ I(1)
∆p
p ∼ I(1)
Gerlach and
∆y
y ∼ I(1)
Smets (1995)
∆p
p ∼ I(1)
i
i ∼ I(0)
Paper
Keating (1992)
none
only long-run
none
only long-run
none
long and short-run
Table 3
Unit Root Tests
T (ˆ
ρ − 1)
(ˆ
ρ − 1)/ˆ
σρˆ
Variable
k
ADF
PP
ADF
PP
∆p
2
-11.8
-34.0
-2.46
-4.67
3
-12.6
-38.5
-2.49
-4.89
4
-14.3
-41.7
-2.53
-5.04
2
-17.8
-61.8
-2.86
-6.46
3
-14.2
-69.1
-2.51
-6.70
4
-12.9
-76.0
-2.34
-6.94
i − ∆p
5% Crit. Val.
-13.7
-2.89
T (ˆ
ρ − 1) and (ˆ
ρ − 1)/ˆ
σρˆ denote the modified bias and t-statistics
for the Augmented Dickey-Fuller (ADF) and the Phillips-Perron
(PP) tests respectively. The tests were carried out with correction for serial correlation at k lags. Both regressions include a
constant but no trend.
Table 4
Cointegration Tests
∆xt =
k−1
i=1
Π = α β′
Γi ∆xt−i + Πxt−1 + ǫt
2×2
2×r r×2
x′t = [i ∆pt ]
k
H0
λmax
3
r=0
r=1
4
r=0
r=1
5
r=0
r=1
13.72
–
11.96
–
10.92
–
Trace
β ′ = [0 1]
β ′ = [1 − 1]
–
–
19.61∗
5.97
5.14∗∗
18.22∗
6.35
–
1.46
15.92
0.14
–
1.64
–
–
–
–
x′t = [∆mt
∆pt ]
–
k
H0
λmax
Trace
β ′ = [1 0]
β ′ = [1 − 1]
3
r=0
16.62∗∗
22.10∗∗∗
–
–
r=1
5.48
5.48
r=0
14.16∗
18.51
r=1
4.35
r=0
10.44
4
5
r=1
, ∗, and
∗ ∗
∗
–
–
13.88
–
2.25
–
2.19
1.89
–
1.78
–
–
–
–
∗ ∗ denote significance at the 10, 5, and 1 percent levels, respectively.
The columns labelled λmax and Trace contain the corresponding statistics for the
cointegration rank in the terminology of Johansen and Juselius (1990). Inference is
based on the critical values tabulated in Osterwald-Lenum (1992).
The last two columns contain the test statistics for the corresponding hypotheses
about the cointegration vectors β. In the present case they are χ21 distributed.
All regressions contain a constant which is restricted to the cointegration space.
Table 5
Variance Decomposition: Model 1
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
65 (6)
19
(5)
11
(6)
6
(4)
4
64 (7)
14
(5)
3
(4)
19
(4)
10
71 (6)
10
(4)
6
(3)
14
(4)
20
82 (3)
6
(2)
4
(2)
8
(3)
1
9
(7)
0
(2)
79 (21)
12
(23)
4
3
(3)
7
(3)
70 (15)
20
(12)
10
2
(3)
20
(6)
39 (15)
40
(13)
20
1
(2)
26
(7)
19 (13)
54
(13)
1
0
(3)
0
(1)
0
(19)
100
(21)
4
2
(3)
0
(1)
11 (15)
87
(17)
10
3
(3)
1
(1)
12 (14)
85
(16)
20
2
(3)
5
(2)
13 (13)
79
(14)
1
27 (7)
73
(7)
0
(0)
0
(0)
4
20 (5)
49
(6)
6
(8)
25
(9)
10
10 (3)
42
(6)
3
(9)
45
(10)
20
7
(3)
38
(7)
2
(10)
53
(12)
1
6
(7)
17
(7)
0
(15)
77
(18)
4
18 (8)
20
(7)
17 (13)
46
(17)
10
19 (7)
52
(7)
15
(6)
14
(6)
20
20 (5)
65
(6)
5
(3)
10
(3)
1
1
(4)
3
(3)
7
(22)
89
(22)
4
15 (5)
23
(6)
12 (16)
51
(15)
10
13 (6)
22
(6)
18 (18)
47
(16)
20
13 (6)
21
(6)
21 (19)
46
(17)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 6
Variance Decomposition: Model 2
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
6
(5)
66 (6)
1
(1)
27
(5)
4
12
(7)
48 (8)
1
(1)
39
(6)
10
39
(7)
31 (7)
1
(1)
29
(4)
20
69
(5)
16 (4)
0
(0)
15
(2)
1
19
(8)
30 (7)
0
(1)
50
(8)
4
23
(7)
59 (7)
3
(1)
15
(2)
10
17
(7)
73 (7)
2
(0)
8
(1)
20
15
(7)
79 (7)
1
(0)
5
(1)
1
4
(6)
1
(2)
67
(6)
28
(4)
4
11
(4)
8
(2)
52
(5)
30
(4)
10
11
(3)
11 (2)
50
(5)
28
(4)
20
13
(4)
10 (2)
48
(5)
29
(4)
1
77
(4)
0
(1)
15
(3)
7
(1)
4
73
(3)
5
(2)
13
(2)
8
(1)
10
60
(5)
12 (3)
9
(2)
18
(3)
20
58
(5)
13 (4)
8
(1)
20
(3)
1
6
(6)
1
(3)
81
(6)
11
(2)
4
40
(8)
7
(3)
40
(6)
13
(2)
10
50
(7)
23 (5)
23
(4)
3
(1)
20
48
(8)
32 (6)
19
(3)
1
(0)
1
16 (10)
1
(3)
34
(7)
49
(7)
4
35
(7)
14 (4)
14
(3)
38
(5)
10
23
(6)
40 (5)
9
(2)
28
(4)
20
12
(3)
69 (4)
4
(1)
15
(3)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 7
Variance Decomposition: Model 3
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
34
(9)
33 (6)
20
(6)
12
(2)
4
54
(8)
14 (4)
16
(5)
15
(3)
10
78
(5)
6
(2)
9
(3)
7
(2)
20
88
(3)
3
(1)
5
(2)
4
(1)
1
56
(8)
42 (8)
0
(0)
2
(0)
4
19
(7)
33 (6)
11
(4)
36
(5)
10
7
(4)
35 (6)
19
(6)
40
(6)
20
4
(3)
35 (7)
22
(6)
39
(6)
1
7
(5)
13 (6)
0
(1)
80
(6)
4
21
(7)
19 (5)
3
(1)
58
(6)
10
19
(7)
25 (5)
3
(1)
53
(6)
20
16
(6)
23 (5)
7
(2)
54
(6)
1
19
(6)
8
(4)
73
(6)
0
(0)
4
15
(5)
5
(3)
51
(6)
29
(5)
10
9
(3)
3
(2)
48
(6)
40
(6)
20
7
(2)
4
(2)
45
(7)
44
(6)
1
19 (19)
3
(3)
17
(7)
61
(9)
4
48
(9)
12 (4)
25
(6)
15
(4)
10
34
(8)
19 (5)
43
(8)
4
(1)
20
22
(8)
21 (6)
52
(8)
4
(1)
1
8
(6)
33 (7)
5
(3)
55
(7)
4
10
(3)
44 (6)
20
(4)
26
(4)
10
8
(4)
64 (5)
14
(3)
14
(2)
20
6
(4)
80 (4)
7
(2)
7
(1)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 8
Variance Decomposition: Model 4
Type of Shock
Variable
Horizon
y
1
5
(4)
69 (5)
26
(4)
4
18 (7)
54 (7)
32
(5)
10
47 (7)
31 (6)
22
(3)
20
75 (4)
14 (3)
10
(2)
1
32 (9)
35 (7)
33
(6)
4
23 (8)
66 (8)
11
(2)
10
17 (8)
77 (7)
5
(1)
20
14 (7)
83 (7)
3
(1)
1
83 (3)
0
(0)
17
(3)
4
77 (4)
9
(2)
14
(2)
10
64 (6)
18 (4)
18
(3)
20
62 (7)
19 (5)
19
(3)
1
11 (7)
0
(2)
89
(7)
4
44 (6)
13 (3)
43
(6)
10
29 (7)
37 (6)
34
(5)
20
15 (4)
67 (5)
18
(3)
i
∆p
i − ∆p
AS
IS
MM
Standard errors are given in parentheses. They were calculated
using Runkle’s (1987) bootstrapping method based on 200 replications.
Table 9
Variance Decomposition: Model 5
Type of Shock
Variable
Horizon
y
1
10
(6)
90
(6)
4
24
(7)
76
(7)
10
50
(5)
50
(5)
20
75
(3)
25
(3)
1
92
(2)
8
(2)
4
84
(3)
16
(3)
10
67
(5)
33
(5)
20
64
(6)
36
(6)
∆p
AS
AD
Standard errors are given in parentheses. They
were calculated using Runkle’s (1987) bootstrapping method based on 200 replications.
Figure 1
Model 1
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 2
Model 2
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 3
Model 3
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 4
Model 4
Responses to AS-Shock
Responses to IS-Shock
Responses to MM-Shock
Figur
Variations on the IS-LM Model∗
Thomas J. Jordan
Swiss National Bank
Economic Studies Section
P.O. Box
CH–8022 Z¨
urich, Switzerland
E-mail: jordan.thomas@snb.ch
Carlos Lenz
Universit¨at Basel
WWZ
Petersgraben 51
CH–4003 Basel, Switzerland
E-mail: carlos.lenz@unibas.ch
January 1999
Abstract
Applications of structural VARs to the IS-LM model reach different conclusions
about the relative importance of demand and supply shocks for business cycle fluctuations. This paper analyzes why these discrepancies occur. It is shown that the
results depend critically on the stationarity assumptions for certain variables. Because the results of the unit root and cointegration tests are inconclusive, the results
from structural VARs have to be interpreted with caution. It is crucial to check the
sensitivity of the results with respect to the assumed order of integration of those
variables which are on the borderline between I(1) and I(0).
JEL Classification: C32, E32
We thank Benjamin Friedman, Bennett McCallum, Michel Peytrignet, Georg Rich, Axel Weber,
Georg Winckler, and seminar participants at the Universities of Alicante, Basle, Vienna, at the Swiss
National Bank and at the Jahrestagung des Vereins f¨
ur Socialpolitik for very helpful comments. Both
authors gratefully acknowledge financial support provided by the Swiss National Foundation through
Grants No. 8210-040206 (T.J.) and 12-40498.94 (C.L.).
∗
1. Introduction
The debate about which type of disturbances cause business cycle fluctuations stands traditionally at the center of macroeconomic research. Since the seminal paper by Sims
(1980), the basic tool to evaluate different shocks has been the vectorautoregression
(VAR). The critique by Cooley and LeRoy (1985) led to the development of the structural
VAR approach pioneered by Bernanke (1986), Blanchard and Watson (1986), and Sims
(1986).1 Recently, considerable interest has been devoted to structural VARs consisting
of the variables which form the traditional IS-LM model. The restrictions needed to estimate the structural VAR are taken from the long-run or short-run restrictions which
are postulated by the aggregate supply/aggregate demand version of the textbook IS-LM
model.
There are two main motivations for this kind of analysis. One the one hand, the
validity of the predictions of the IS-LM model can be tested empirically. This will answer
the question about the adequacy of the IS-LM model as a description of the economy.2
On the other hand, the relative impact of the structural shocks on the different variables
can be assessed. Of particular interest in this context is the following question: To what
extent are output fluctuations at business cycle frequencies caused by either demand or
supply shocks? This point is critical for a discussion of whether real business cycle theory
or Keynesian macroeconomics is empirically more relevant.
There are five papers which can be regarded as applications of the IS-LM framework
in a structural VAR analysis with postwar U.S. data: Gal´ı (1992), Keating (1992), Bayoumi and Eichengreen (1993)3, Jordan and Lenz (1994), Gerlach and Smets (1995). The
interesting point is that the results of these studies differ considerably, especially with
respect to the importance of demand and supply shocks for output fluctuations. Table 1
summarizes the variance decompositions for output in the different studies.4
The following differences in the setup of the structural VARs can account for the
1
Structural VARs were originally identified by short-run restrictions. Blanchard and Quah (1989) and
Shapiro and Watson (1988) developed a method to identify shocks by long-run restrictions. Gal´ı (1992)
pioneered the simultaneous use of both types of restrictions.
2
See Gal´ı (1992) for a discussion why the empirical relevance of the IS-LM model remains of great
importance.
3
Two other papers by the same authors (Bayoumi and Eichengreen, 1994, 1995) use the same modeling
strategy. Unfortunately, none of their papers reports variance decompositions.
4
All tables and figures are in the Appendix.
2
discrepancies among these papers: (1) the number of variables used to estimate the same
number of structural shocks, (2) the assumed order of integration of the variables and the
assumed cointegrating relations, (3) the restrictions imposed to identify structural shocks,
and (4) the used data sets (definition of the variables and sample periods). Table 2 gives
an overview of the differences between the studies.
The number of variables included in all the analyzes varies between two and six. However, the maximum number of variables entering a VAR is four, additional variables are
sometimes analyzed by using linear combinations of the variables forming the estimated
system. The set of variables considered embraces output, the price level, real money,
nominal money, as well as nominal and real interest rates. The number of variables included in the VAR determines to what extent demand shocks have to be aggregated. The
assumed order of integration of the variables differs, especially in the case of the inflation
rate and the real interest rate. Most studies expect that the inflation rate is stationary,
only Gal´ı (1992) believes that it has a unit root. There is less consensus about the stationarity of the real interest rate: half the papers which analyze this variable presuppose
that it is stationary, the other half that it is integrated of order one. Depending on the
assumed order of integration of the individual variables, cointegration relations among
them are implicit in the choice of variables used for the analysis. For instance, if the
nominal interest rate and the inflation rate are assumed to be integrated of order one and
the real interest rate is taken to be stationary, then this implies that the nominal interest
rate and the inflation rate are cointegrated. All the papers have the identification scheme
for the aggregate supply shocks in common: Only these shocks have a long-run effect on
output. A variety of identification strategies are applied in order to identify the different
demand shocks.
Even though part of the variation between the results of the studies can be attributed
to the use of different data sets it is still striking that empirical models based on the same
philosophy come to very different conclusions. The purpose of this paper is therefore
two-fold: First, by using the same set of data throughout our analysis, we investigate
why the above-mentioned papers reach different conclusions, especially with respect to
the relative impact of demand and supply shocks on output over short horizons. Second,
we try to assess which model specification is the correct one and try to show whether
demand shocks are in fact important for business cycle fluctuations.
We find that the assumed order of integration of the variables and the cointegrating
relations between variables have a critical influence on the variance decompositions. This
3
is particularly true for the relative importance of demand and supply shocks. The assumptions about the long-run properties of the variables also have an influence on the
impulse responses but to a much lesser extent.It is interesting to note that the specific
identification restrictions are less important for the variance decompositions. This is, of
course, related to the fact that all authors use long-run neutrality of demand shocks to
identify aggregate supply shocks. However, the different strategies used to disentangle
the different types of demand shocks have no dramatic consequences on their effects. Furthermore, we find that a reduction in the number of variables implying the aggregation of
demand shocks leaves the main results basically unchanged.Unit root and cointegration
tests give no clear-cut answer about which model specification is the correct one. Therefore, results from structural VARs have to be interpreted with caution, because the result
may be sensitive to the assumed order of integration of those variables which are on the
borderline between I(1) and I(0).
The remainder of this paper is organized as follows: Section 2 presents the model and
discusses its integration properties. In Section 3, we develop a method for identifying
structural shocks in a VAR with both long-run and short-run restrictions. The empirical
results of five different specifications of the model are presented in Section 4, and Section
5 concludes.
2. A version of the IS-LM model
We start with the premise that real output, the nominal interest rate, nominal money, and
the price level are the four variables that can be used to identify aggregate supply, money
supply, money demand and spending shocks in a VAR model. Based on this assumption
we can ask two basic questions: 1. Which system of variables should be used? There
are six variables to choose from, the four variables mentioned above plus some linear
combinations like the real interest rate and real balances. In addition, each variable that
enters the system can be used in levels or in first differences (even second differences could
be considered), which offers quite a wide range of options. 2. Which restrictions should
be applied in order to identify the structural shocks? In the literature, a large variety of
approaches have been proposed to deal with this second question, but it is far from clear
which of these solutions is optimal.
In order to answer these questions, we take both theoretical and empirical aspects
into account. From a theoretical point of view, the choice of variables and their order
4
of integration should be in line with the structure of the model. Otherwise it becomes
an almost impossible task to apply the identifying restrictions suggested by the model.
In addition, an empirical analysis is needed to establish the time series properties of the
variables. In particular, this means that the empirical order of integration and possibly
existing cointegrating relations among the variables should be tested for rather than imposed. The results of the unit root and cointegration tests can then be used to choose
the correct specification of the model.
2.1. The model
In this section we present a log-linear version of the IS-LM model which is useful as a
means of choosing the variables and their order of integration. The IS and LM equations
establish the usual equilibrium conditions in the goods and money markets
i − E∆p+1 = f + β1 y
m − p = d + β2 y + β3 i,
(1)
(2)
where i is the nominal interest rate, p is the price level, f are autonomous fiscal expenditures, y is output, m is money supply, and d is the autonomous part of money demand.
E and ∆ are the expectations and difference operators, respectively, and all variables,
except the nominal interest rate, are in logs. The long-run aggregate supply curve is
vertical. This means that long-run changes in output are governed by aggregate supply
(uas ) shocks
∆y = uas .
(3)
In addition, we assume that the nominal money supply is either
∆m = β4 ∆y + β5 ∆i + β6 umd + ums
(4-a)
∆2 m = β4 ∆y + β5 ∆i + β6 umd + ums,
(4-b)
or
depending on the integration order of nominal money. Money supply depends therefore
on all shocks in the model. Note that we consider equations (1) – (4) to be a long-run
formulation, i.e. the difference operators apply to distant points in time. This is in line
with the comparative-static concept of the IS-LM model, which makes no exact predictions
about adjustment paths.
5
Taking first differences and labeling changes in fiscal expenditures as IS-shocks (uis )
and changes in the autonomous part of money demand as money demand shocks (umd ),
we can write the IS-LM model as follows:
∆(i − E∆p+1 ) = γ1 ∆y + uis
(5)
∆(m − p) = γ2 ∆y + γ3 ∆i + umd
(6)
∆y = uas
(7)
∆p = γ4 ∆y + γ5 ∆i + γ6 umd + ums
(8-a)
∆2 p = γ4 ∆y + γ5 ∆i + γ6 umd + ums,
(8-b)
where the choice of the price equation (8-a) or (8-b) depends on the integration order of
the money supply process. In order to derive equation (8-b), we impose a cointegrating
relation between ∆m and ∆p.
If all variables on the right-hand side of these equations and the inflation rate are
stationary (which implies E∆2 p+1 = 0 or ∆p = ∆p+1 ), we can solve for the variables in
terms of the structural shocks:
∆y
∆(i − ∆p)
∆(m − p)
∆p
1
α1
=
0
0
1
0
α2 α3
1
0 uas
is
0
u
md
0 u
α4 α5 α6 1
ums
(9)
.
This system represents the long-run effects of each structural shock and implies six
long-run neutrality restrictions: only aggregate supply shocks have a permanent effect on
the level of output, monetary shocks have no permanent effect on the level of the real and
the nominal interest rate, and money supply shocks have no permanent effect on the level
of real balances.
If the inflation rate is non-stationary and money and prices are cointegrated, the model
is
∆y
∆(i − ∆p)
∆(m − p)
2
∆p
1
α1
=
0
0
1
0
α2 α3
1
0 uas
is
0
u
md
α4 u
α5 α6 α7 1
ums
(10)
,
so that only five long-run neutrality restrictions exist. Another possibility is the assumption of a stationary real interest rate as suggested in Gal´ı (1992). This means that the
6
nominal interest rate and the inflation rate are cointegrated. In this case, the IS curve is
horizontal so that supply and fiscal shocks do not affect the real interest rate in the long
run. In this case, only three exploitable long-run neutrality restrictions exist.
If the inflation rate is stationary, the model suggests six long-run neutrality restrictions
which can be used to identify the structural shocks. If the inflation rate has a unit root,
the IS-LM model provides less than six long-run restrictions. In that case, short and
long-run restrictions have to be combined in order to identify the structural VAR. The
short-run restrictions come from the less formal and more implicit assumption about
the dynamics of the IS-LM model. For instance, Gal´ı (1992) assumes that output does
not react contemporaneously to money demand and supply shocks. Other short-run
restrictions can be inferred from the assumptions about the money supply process.
2.2. Aggregation of demand shocks
As already mentioned above, some authors also use models consisting of only two or three
variables and it is easy to see that this can be interpreted as an aggregation of demand
shocks. The money supply and money demand shocks can be aggregated to a money
market shock (umm ). By dropping the third line of equation (9), we get
∆y
∆p
1
∆(i − ∆p)
= α1
0
1
0 uas
is
0
u
α2 α3 1
umm
(11)
.
This is the model used in Jordan and Lenz (1994) and comes quite close to the model
used by Gerlach and Smets (1995). Both papers use the nominal interest rate instead of
the real interest rate. Since Gerlach and Smets (1995) assume the real interest rate to
be stationary, however, they can only apply two long-run restrictions and therefore they
need a short-run restriction in order to separate monetary from fiscal shocks.
Going one step further, fiscal and monetary shocks can be aggregated to an aggregate
demand shock. Eliminating the nominal interest rate from equation (11) yields
uas
∆y 1 0
=
ad
u
α1 1
∆p
,
which is the model used by Bayoumi and Eichengreen (1993).
7
(12)
2.3. Variations on the IS-LM model
In order to find out where the discrepancies in the results of the different versions of the
IS-LM model come from, we estimate five empirical models:
Model 1 consists of the same VAR as in Gal´ı (1992), with a stationary real interest
rate. However, we use a different set of short-run identifying restrictions, which allows us
to evaluate the influence of different identifying assumptions on the results. In the short
run, prices do not react to money demand and money supply shocks. Furthermore, we
presume that contemporaneous GNP does not enter the money supply rule.
Model 2 is the VAR formed by the variables of equation (9), where it is assumed that
the inflation rate is stationary and the real interest rate has a unit root. In order to
identify the structural shocks, we can rely on the long-run restrictions implied by the
model. The comparison of Model 1 and Model 2 should shed some light on the impact
of various systems on the results. Comparing Model 2 to the long-run model of Keating
(1992) should also give some insights about the discrepancies arising from using different
data sets.
Model 3 takes a unit root in both the inflation rate and the real interest rate. The
VAR corresponds to equation (10). There are five long-run restrictions implied by the
model. Furthermore, we assume that the inflation rate does not react contemporaneously
to money supply shocks. This exercise should give further insight into the effect of different
systems on the results.
Model 4 corresponds to the three-variable system in equation (11), where we assume a
stationary inflation rate and a non-stationary real interest rate. The shocks are identified
with long-run restrictions. This allows us to draw some conclusions about the use of
different data sets and different types of restrictions as we compare the results to those
in Jordan and Lenz (1994) and Gerlach and Smets (1995). In addition, we can assess the
effects of aggregating the money demand and supply shocks to a single money market
shock.
Model 5 is the VAR consisting of output growth and the inflation rate as in equation
(12) and the shocks are again identified by long-run restrictions. A comparison of the
8
results of Model 5 with those of Bayoumi and Eichengreen (1993) and those of Models 2
and 4 shows the effects of different data sets and the impact of the aggregation of all
demand shocks to a single shock.
3. Identification of the structural shocks
The structural shocks of the IS-LM model can be identified by a combination of longrun and short-run identifying restrictions on the VAR representation of the variables in
question. The idea of using both long-run and short-run restrictions simultaneously was
pioneered by Gal´ı (1992) and can be seen as a natural extension of earlier structural VAR
approaches that proposed to use either only short-run or only long-run restrictions.
The implementation of the restrictions is a straightforward exercise, provided they
have a special kind of recursive structure as we can demonstrate for the general case. We
start with a given vector of stationary variables xt which is assumed to have a structural
vector moving average (VMA) representation
xt = A(L)ut ,
where A(L) denotes a matrix polynomial in the lag-operator L and ut is a vector of unobservable structural shocks. We assume that the structural shocks are mutually uncorrelated and normalize their variance to unity, i.e. E(ut u′t ) = I. Inversion of the structural
VMA representation yields the structural VAR representation
B(L)xt = ut ,
where B(L) = A−1 (L). In order to identify the unobservable structural shocks contained
in ut , a set of restrictions is placed on A(L) and B(L). This allows us to represent the
structural shocks as linear combinations of the disturbances of the reduced form system
in xt .
We first estimate a reduced form VAR in xt
∆xt = D(L)∆xt−1 + ǫt ,
where ǫt is a vector of serially uncorrelated reduced form disturbances with covariance
matrix Ω. Inverting this VAR yields the Wold VMA representation
xt = C(L)ǫt ,
9
where C0 equals the identity matrix. A comparison between the reduced form and structural VMAs immediately shows that the following relation must hold
A0 ut = ǫt ,
that is, once A0 is known, the structural shocks can be computed from the reduced form
disturbances. In order to compute A0 we first note that
Ω = A0 A′0 ,
which places n(n + 1)/2 restrictions on the n2 different elements of A0 . This means that
we need n(n − 1)/2 additional restrictions to identify A0 . The method of Gal´ı (1992)
amounts to imposing restrictions on the immediate responses or the contemporaneous
structural relations between the variables (given by A0 and B0 respectively) and on the
long-run responses to the structural shocks (given by A(1)). The additional restrictions
are imposed on A0 , either directly or via the relationships
A−1
0 = B0
and C(1)A0 = A(1).
˜ and an arbitrary orthonormal matrix
Denoting the Cholesky decomposition of Ω by Ω
by S, we can write:
˜
ΩS
= A0
˜ ′ )−1 S = B ′
(Ω
0
˜
C(1)ΩS
= A(1).
Note that the knowledge of S is equivalent to the knowledge of A0 and, consequently,
that n(n − 1)/2 restrictions are needed to determine S. We show now that it is possible
to determine S explicitly if the restrictions have a ’triangular’ structure: Consider a
˜ (Ω
˜ ′ )−1 , and C(1)Ω.
˜
(n − 1) × n matrix Z which consists of the appropriate rows of Ω,
The ’triangular’ structure of the restrictions ensures that Z can be ordered in such a way
that, if multiplied by the first n − 1 columns of S, the following holds
Z[ S1 . . . Sn−1
0 0 ...
0 0 ...
]=
..
.. ..
.
. .
10
0
·
..
.
0 · ... ·
,
where Si denotes the i-th column of S. Using the equation above, the orthogonality of S
can be exploited to solve recursively for each of its columns:
Mi Si =
0
..
.
0
and Si′ Si = 1 i = 1, . . . , n,
where
zi,1
..
.
M1 = Z
Mi =
zn−1,1
...
...
S1′
..
.
′
Si−1
zi,n
..
.
zn−1,n
i = 2, . . . , n − 1
Mn =
S1′
..
.
′
Sn−1
.
4. Empirical Results
In this section we carry out the estimation under the different setups discussed at the
end of Section 2. We use the same data set as Gal´ı (1992). The sample period runs from
1955:I to 1987:III and the data includes the log of real GNP (y), the yield on three-month
Treasury bills (i), the log of the Consumer Price Index (p), and the log of M1 (m). All
data is from CITIBASE, except for measures of M1 previous to 1959 which are from the
Federal Reserve Bulletin.
We then test for the order of integration of the individual variables and for possible
cointegrating relations among some of the variables. These tests are critical for the following reason: All variables included in the VAR have to be stationary to avoid spurious
regression problems. The estimated system is also misspecified if it includes first differences of cointegrated variables. The empirical results of the unit root and cointegration
tests are therefore a tool to discriminate among the different specifications in order to
find the correct model.
4.1. The Impact of Different Identifying Restrictions
We use the system proposed by Gal´ı (1992) to check whether different identification
restrictions can lead to different impulse responses and variance decompositions. In contrast to Gal´ı, we identify the structural VAR of our Model 1 by a different set of short-run
11
restrictions: MS and MD shocks do not affect the prices within the first quarter and contemporaneous GNP does not enter the money supply rule. The results are presented in
Figure 1 and Table 5.
Considering the impulse responses, the only differences arising from using a new set of
short-run identification restrictions are due to the impact of the MD shocks on ∆m, ∆p
and m−p in the short run. In our setup, nominal money growth first declines and becomes
positive only after 4 quarters. The inflation rate is first increasing and jumps below its
original level only after 8 quarters. Real balances therefore first decline and reach their
old level after about 12 quarters. All the other impulse responses look very similar to
those reported to those in Gal´ı (1992). Concerning the variance decompositions, we find
that the relative importance of IS and MS shocks changes compared to Gal´ı’s results. In
general MS shocks become more important for y, i, m, and ∆p; and less important for
m−p and i−∆p. The opposite is true for IS shocks. However, the impact of supply shocks
on all variables is very similar to Gal´ı (1992). Supply shocks are the dominant source of
output fluctuations even in the very short run. We conclude that different identification
restrictions do not lead to a change in the relative importance of demand and supply
shocks for output variations if the system remains unchanged.
With both possible identification schemes, the system proposed by Gal´ı leads to some
implausible results. This is most evident in the version with the original set of restrictions.
IS shocks have a strong impact on the inflation rate in the long run, whereas MS supply
shocks have a much smaller effect. IS shocks also have a strong impact on the long-run
money growth. Although the IS shock has a positive impact on the inflation rate, it does
not influence the real interest rate. This is incompatible with conventional macroeconomic
models.5
4.2. The Impact of Different Assumptions about the Order of Integration
We now concentrate on the effects of different assumptions about the order of integration
and estimate two new models: Model 2 assumes that the inflation rate is stationary and
that the real interest rate has a unit root; it consists of the variables ∆y, ∆(i − ∆p), ∆p,
and ∆m − ∆p. As discussed in Section 2, this model can be identified only by long-run
restrictions. Model 3 assumes that both the real interest rate and the inflation have a
unit root; it consists of the variables ∆y, ∆(i − ∆p), ∆2 p, and ∆m − ∆p and therefore
5
This is also mentioned in the critique by King (1993) of the Gal´ı (1992) paper.
12
requires that ∆m and ∆p are cointegrated. The set of identification restrictions used in
Model 3 is similar to the one used by Gal´ı (1992).
The results of Model 2 are shown in Table 6 and Figure 2. Comparing the results to
those of Model 1, the impulse responses are different for i, ∆p, and i − ∆p. AS shocks
lower i and i − ∆p in the long run, whereas IS shocks increase i and i − ∆p. MS and
MD shocks have only a short-run impact on the interest rates. None of the shocks has a
long-run impact on ∆p because of the assumed stationarity of ∆p. The impulse responses
for output are similar to the system discussed above. All impulse responses of this model
are exactly what we expect from the theoretical IS-LM model.
With respect to the variance decompositions, we observe a dramatic change. Demand
shocks, mainly IS and MS shocks, are now the dominant source of output fluctuations
at business cycle frequencies. Consequently, different assumptions about the order of
integration of the time series and the cointegration among the variables change the relative
importance of demand and supply shocks in the short run.
To further analyze the impact of a different integration assumptions on the results,
we estimate Model 3. The results are in Table 7 and Figure 3. The impulse responses are
similar to those of Model 2. The main difference is the impulse response of the inflation
rate which is not restricted to be zero in the long run. We find that changes in the inflation
rate are mainly due to money demand and money supply shocks. It is interesting to note
that the contribution of demand shocks to the variance of real output lies between those
of Models 1 and 2. In contrast to Model 1 demand shocks are important for output
fluctuations in the short run, but not as important as in Model 2. All three demand
shocks contribute equally to output fluctuations, this implies that money demand shocks
contribute more than in Model 2.
Keating (1992) uses a system which has the same assumptions about the order of
integration as our Model 2, but he uses the nominal interest rate instead of the real
interest rate and money growth instead of inflation in his VAR setup. He also uses only
long-run restrictions to identify the structural shocks. If we compare his results to those of
our Model 2, we find that they are very similar. However, in our Model 2, money supply
shocks are generally more important than in the VAR used by Keating. Comparing the
results of Keating (1992) to Model 1, we see again the dramatic change in the relative
importance of demand and supply shocks due to a different VAR system.
Considering the results of Models 1 to 3, we conclude that the relative importance of
supply and demand shocks for output fluctuations at business cycle frequencies depends on
13
the assumed order of integration of the variables and, therefore, on whether the variables
enter the VAR in levels or in first differences. The results change completely for different
integration assumptions. The use of different identification assumptions can also change
the impulse responses of some variables, but the changes are less dramatic.
4.3. The Impact of Different Data Sets
Another question of some interest is whether the use of different data sets changes the
results in any significant way if the same system is estimated. The comparison of either
the results of Model 4 to those of Jordan and Lenz (1994) or those of Model 2 to those of
Keating (1992) shows that the impulse responses are not very much influenced by different
data sets. The variance decomposition shows that the relative impact of the different
shocks changes. However, the relative importance of demand and supply shocks for output
fluctuations is approximately the same. Bayoumi and Eichengreen (1993) estimate a twovariable system, which can be compared to the system of equation (12). They do not
present variance decompositions for their model. However, their impulse responses look
very similar to our results. We conclude that the use of different data sets does not affect
the form of the impulse responses very much. However, the relative importance between
the shocks can change. In the cases considered, the relative importance of demand and
supply shocks for output fluctuations remains the same for different data sets.
4.4. The Impact of Aggregation of Demand Shocks
An interesting aspect of the VAR analysis with IS-LM variables is the question whether the
reduction of the four-variable VAR to a three and two-variable system changes the results
of the impulse responses and the variance decompositions. This point thus addresses
the question whether the aggregation of structural demand shocks basically changes the
importance of the demand shocks relative to the supply shocks.
This question can be answered by looking at the results for Models 2, 4, and 5 which
use only long-run restrictions to identify structural shocks. Model 4 aggregates money
demand and supply shocks into a money market (MM) shock. The restrictions are that
MM and IS shocks do not have long-run effects on y and MM shocks do not have long-run
effects on i. In Model 5, all demand shocks are represented by a single aggregate demand
(AD) shock which is assumed to leave output unaltered in the long run. Tables 8 and 9
and Figures 4 and 5 report the results.
14
If we compare the impact of the supply shocks on the output variance, we find a
relatively high correspondence between the three models. The relative importance of
supply and demand shocks varies only slightly between the models. We therefore conclude
that the reduction of the VAR or the aggregation of the shocks does not change the
basic results of the relative importance between demand and supply shocks for output
fluctuations. The aggregation of shocks seems, in any case, less critical than the order of
integration.
4.5. Unit Roots and Cointegration
The order of integration for y, i, m, and m − p is undisputed. The results of our unit
root tests as well as those of other studies show that these variables are I(1).6 Therefore,
we do not present the results of the unit root tests for these variables. The situation is
different for the inflation rate and the real interest rate. Table 3 shows the results of the
Augmented-Dickey-Fuller and Phillips-Perron tests at different lag-lengths.7 The results
of these unit root tests are mixed. For both variables, the Phillips-Perron tests reject the
null hypothesis of a unit root, whereas the Augmented-Dickey-Fuller tests point towards
non-stationarity of the time series.
The studies of Keating (1992), Bayoumi and Eichengreen (1993), Gerlach and Smets
(1995), and Jordan and Lenz (1994) all assume the inflation rate to be stationary. The
unit root tests presented in Gal´ı (1992) also indicate that the inflation rate is stationary.
However, he concludes that the inflation rate must have a unit root because his tests
indicate that the real interest rate is stationary. This conclusion further implies cointegration between the nominal interest rate and the inflation rate. If the variables were not
cointegrated, the real interest rate could not be stationary. A unit root in the inflation
rate also implies that ∆m is I(1) and cointegrated with ∆p if real money growth is to
be stationary. This leads to an additional restriction for the order of integration of the
variables included in the VAR.
In order to shed some more light on this issue, we test for cointegration between i and
∆p as well as ∆m and ∆p by using the Johansen procedure. Table 4 summarizes the
results of this exercise at different lag-lengths.
The tests show that the null hypothesis of no cointegration between i and ∆p is
6
An exception is the study by Gerlach and Smets (1995), where i is assumed to be stationary.
7
The minimal lag-length for both the unit root and cointegration tests was chosen so that the residuals
from the regressions are uncorrelated.
15
never rejected if the maximum eigenvalue statistic is used. Considering the trace statistic
this hypothesis is rejected at the 10 percent level at lag-lengths 3 and 4, but we cannot
reject the null of one cointegrating relation. However, this does not imply that the real
interest rate is stationary. Given that i is I(1), the presence of only one unit root in
the system consisting of i and ∆p can be due to either a stationary inflation rate or
cointegration between i and ∆p, that is, a stationary real interest rate. In order to
discriminate between these cases, we test whether the vectors [0 1] and [1
− 1] are
contained in the cointegrating space at those lag-lengths where a cointegrating relation
between i and ∆p is found. The results are, again, unclear: At three lags, the presence
of [0 1] in the cointegrating space is rejected, whereas the presence of [1 − 1] cannot be
rejected. At four lags both vectors could be contained in the cointegrating space. To sum
up, one can say that these results confirm the findings of the unit root tests: The data
gives no clear answer to the question about the stationarity of ∆p and i − ∆p.
Concerning the cointegration between ∆m and ∆p, the picture is not clear, either.
The tests show that the null of no cointegration can be rejected at three lags in favor of
one cointegrating relation. At four lags, only the maximum eigenvalue test rejects the
absence of cointegration; and at five lags, two unit roots seem to be present in the system
formed by ∆m and ∆p. This contradicts the unit root tests where both real and nominal
money appear to be stationary in first differences. At those lag-lengths where there seems
to be only one unit root in the system, we check whether it is due to the stationarity of
∆m or whether ∆m and ∆p are cointegrated. That is, we test whether the vectors [1 0]
or [1 − 1] are contained in the cointegrating space. The results of these tests show that
neither hypothesis can be rejected.
Given the evidence from the unit root and cointegration tests none of the models
presented in Section 2 can a priori be discarded. Furthermore, the exercise carried out in
this subsection does not provide the basis for a assessment of the models in accordance
with their ability to fit the data. Therefore, the main conclusion we can draw from these
tests is that one must be very cautious in interpreting the results from structural VARs.
In particular, it seems to be crucial to check the sensitivity of the results with respect
to the assumed order of integration of the variables which are on the borderline between
I(1) and I(0).
16
5. Conclusions
We have presented a structural VAR analysis of the IS-LM model under a variety of
identifying assumptions. The main contribution of this paper is to show that the relative
importance of demand and supply shocks for output fluctuations depends critically on
the stationarity assumptions for certain variables. The responses of the variables to
different shocks also depend to some extent on particular identifying restrictions, but
those differences are less important. It is also shown that the aggregation of demand
shocks does not alter their relative importance with respect to supply shocks.
The obtained results show that in structural VAR analysis special care needs to be
taken in determining the order of integration of the variables forming the estimated system. This is particularly true in those cases where unit root tests do not yield clear
results. With respect to the order of integration of the variables in the IS-LM model, the
critical decision has to be made about the stationarity of both the real interest rate and
the inflation rate. If the real interest rate has a unit root and the inflation rate is stationary, we see that demand shocks are very important for output fluctuations at business
cycle frequencies. If the real interest rate is stationary and the inflation rate has a unit
root, then supply shocks are the dominant source of output fluctuations even in the short
run. If both the real interest rate and the inflation rate have a unit root, the results are
in-between the other two cases.
The evidence from the unit root and cointegration tests does not allow to favor one of
the models over another. Therefore, one must be very cautious in interpreting the results
from structural VARs and it is crucial to check the sensitivity of the results with respect
to the assumed order of integration of the variables which are on the borderline between
I(1) and I(0).
17
References
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Bayoumi, T. and B. Eichengreen (1994) “Macroeconomic adjustment under BrettonWoods and the post Bretton-Woods float: An impulse response analysis.” The Economic Journal, 104:813–827.
Bayoumi, T. and B. Eichengreen (1995) “Is there a conflict between EC enlargement
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Cooley, T. F. and S. F. LeRoy (1985) “Atheoretical macroeconomics: A critique.”
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Oxford Bulletin of Economics and Statistics, 52:169–210.
Jordan, T. J. and C. Lenz (1994) “Demand and supply shocks in the IS-LM model:
Empirical findings for five countries.” Working Paper 94-8, Universit¨
at Bern.
18
Keating, J. W. (1992) “Structural approaches to vector autoregressions.” Federal Reserve Bank of St. Louis, Review, 74(5):37–57.
King, R. G. (1993) “Will the New Keynesian macroeconomics resurrect the IS-LM
model?” Journal of Economic Perspectives, 7:67–82.
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of the maximum likelihood cointegration rank test statistics.” Oxford Bulletin of Economics and Statistics, 54:461–472.
Runkle, D. E. (1987) “Vector autoregressions and reality.” Journal of Business &
Economic Statistics, 5:437–442.
Shapiro, M. D. and M. W. Watson (1988) “Sources of business cycle fluctuations.”
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Bank of Minneapolis, Quarterly Review, pp. 2–16.
Appendix: Tables and Figures
Table 1
Contribution of Supply Shocks to Fluctuations in U.S. GNP:
Results from four Studies
Output Variance explained
Paper
Gal´ı (1992)
Keating (1992)
Jordan and Lenz (1994)
Gerlach and Smets (1995)
Horizon
by AS Shocks (in %)
1
69
5
67
10
72
1
17
4
24
8
45
1
13
4
28
8
49
1
46
4
71
20
90
Table 2
VAR Setups in Different Studies
Included
Order of
Cointegrating
Type of Identifying
Variables
Integration
Relations
Restrictions
Bayoumi and Ei-
∆y
y ∼ I(1)
none
only long-run
chengreen (1993)
∆p
p ∼ I(1)
Gal´ı (1992)
∆y
y ∼ I(1)
∆m − ∆p
long and short-run
∆i
i ∼ I(1)
i − ∆p
i − ∆p
∆p ∼ I(1)
∆(m − p)
∆m ∼ I(1)
∆y
y ∼ I(1)
∆i
i ∼ I(1)
∆m
m ∼ I(1)
∆(m − p)
p ∼ I(1)
Jordan and
∆y
y ∼ I(1)
Lenz (1994)
∆i
i ∼ I(1)
∆p
p ∼ I(1)
Gerlach and
∆y
y ∼ I(1)
Smets (1995)
∆p
p ∼ I(1)
i
i ∼ I(0)
Paper
Keating (1992)
none
only long-run
none
only long-run
none
long and short-run
Table 3
Unit Root Tests
T (ˆ
ρ − 1)
(ˆ
ρ − 1)/ˆ
σρˆ
Variable
k
ADF
PP
ADF
PP
∆p
2
-11.8
-34.0
-2.46
-4.67
3
-12.6
-38.5
-2.49
-4.89
4
-14.3
-41.7
-2.53
-5.04
2
-17.8
-61.8
-2.86
-6.46
3
-14.2
-69.1
-2.51
-6.70
4
-12.9
-76.0
-2.34
-6.94
i − ∆p
5% Crit. Val.
-13.7
-2.89
T (ˆ
ρ − 1) and (ˆ
ρ − 1)/ˆ
σρˆ denote the modified bias and t-statistics
for the Augmented Dickey-Fuller (ADF) and the Phillips-Perron
(PP) tests respectively. The tests were carried out with correction for serial correlation at k lags. Both regressions include a
constant but no trend.
Table 4
Cointegration Tests
∆xt =
k−1
i=1
Π = α β′
Γi ∆xt−i + Πxt−1 + ǫt
2×2
2×r r×2
x′t = [i ∆pt ]
k
H0
λmax
3
r=0
r=1
4
r=0
r=1
5
r=0
r=1
13.72
–
11.96
–
10.92
–
Trace
β ′ = [0 1]
β ′ = [1 − 1]
–
–
19.61∗
5.97
5.14∗∗
18.22∗
6.35
–
1.46
15.92
0.14
–
1.64
–
–
–
–
x′t = [∆mt
∆pt ]
–
k
H0
λmax
Trace
β ′ = [1 0]
β ′ = [1 − 1]
3
r=0
16.62∗∗
22.10∗∗∗
–
–
r=1
5.48
5.48
r=0
14.16∗
18.51
r=1
4.35
r=0
10.44
4
5
r=1
, ∗, and
∗ ∗
∗
–
–
13.88
–
2.25
–
2.19
1.89
–
1.78
–
–
–
–
∗ ∗ denote significance at the 10, 5, and 1 percent levels, respectively.
The columns labelled λmax and Trace contain the corresponding statistics for the
cointegration rank in the terminology of Johansen and Juselius (1990). Inference is
based on the critical values tabulated in Osterwald-Lenum (1992).
The last two columns contain the test statistics for the corresponding hypotheses
about the cointegration vectors β. In the present case they are χ21 distributed.
All regressions contain a constant which is restricted to the cointegration space.
Table 5
Variance Decomposition: Model 1
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
65 (6)
19
(5)
11
(6)
6
(4)
4
64 (7)
14
(5)
3
(4)
19
(4)
10
71 (6)
10
(4)
6
(3)
14
(4)
20
82 (3)
6
(2)
4
(2)
8
(3)
1
9
(7)
0
(2)
79 (21)
12
(23)
4
3
(3)
7
(3)
70 (15)
20
(12)
10
2
(3)
20
(6)
39 (15)
40
(13)
20
1
(2)
26
(7)
19 (13)
54
(13)
1
0
(3)
0
(1)
0
(19)
100
(21)
4
2
(3)
0
(1)
11 (15)
87
(17)
10
3
(3)
1
(1)
12 (14)
85
(16)
20
2
(3)
5
(2)
13 (13)
79
(14)
1
27 (7)
73
(7)
0
(0)
0
(0)
4
20 (5)
49
(6)
6
(8)
25
(9)
10
10 (3)
42
(6)
3
(9)
45
(10)
20
7
(3)
38
(7)
2
(10)
53
(12)
1
6
(7)
17
(7)
0
(15)
77
(18)
4
18 (8)
20
(7)
17 (13)
46
(17)
10
19 (7)
52
(7)
15
(6)
14
(6)
20
20 (5)
65
(6)
5
(3)
10
(3)
1
1
(4)
3
(3)
7
(22)
89
(22)
4
15 (5)
23
(6)
12 (16)
51
(15)
10
13 (6)
22
(6)
18 (18)
47
(16)
20
13 (6)
21
(6)
21 (19)
46
(17)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 6
Variance Decomposition: Model 2
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
6
(5)
66 (6)
1
(1)
27
(5)
4
12
(7)
48 (8)
1
(1)
39
(6)
10
39
(7)
31 (7)
1
(1)
29
(4)
20
69
(5)
16 (4)
0
(0)
15
(2)
1
19
(8)
30 (7)
0
(1)
50
(8)
4
23
(7)
59 (7)
3
(1)
15
(2)
10
17
(7)
73 (7)
2
(0)
8
(1)
20
15
(7)
79 (7)
1
(0)
5
(1)
1
4
(6)
1
(2)
67
(6)
28
(4)
4
11
(4)
8
(2)
52
(5)
30
(4)
10
11
(3)
11 (2)
50
(5)
28
(4)
20
13
(4)
10 (2)
48
(5)
29
(4)
1
77
(4)
0
(1)
15
(3)
7
(1)
4
73
(3)
5
(2)
13
(2)
8
(1)
10
60
(5)
12 (3)
9
(2)
18
(3)
20
58
(5)
13 (4)
8
(1)
20
(3)
1
6
(6)
1
(3)
81
(6)
11
(2)
4
40
(8)
7
(3)
40
(6)
13
(2)
10
50
(7)
23 (5)
23
(4)
3
(1)
20
48
(8)
32 (6)
19
(3)
1
(0)
1
16 (10)
1
(3)
34
(7)
49
(7)
4
35
(7)
14 (4)
14
(3)
38
(5)
10
23
(6)
40 (5)
9
(2)
28
(4)
20
12
(3)
69 (4)
4
(1)
15
(3)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 7
Variance Decomposition: Model 3
Type of Shock
Variable Horizon
y
i
∆m
∆p
m−p
i − ∆p
AS
IS
MD
MS
1
34
(9)
33 (6)
20
(6)
12
(2)
4
54
(8)
14 (4)
16
(5)
15
(3)
10
78
(5)
6
(2)
9
(3)
7
(2)
20
88
(3)
3
(1)
5
(2)
4
(1)
1
56
(8)
42 (8)
0
(0)
2
(0)
4
19
(7)
33 (6)
11
(4)
36
(5)
10
7
(4)
35 (6)
19
(6)
40
(6)
20
4
(3)
35 (7)
22
(6)
39
(6)
1
7
(5)
13 (6)
0
(1)
80
(6)
4
21
(7)
19 (5)
3
(1)
58
(6)
10
19
(7)
25 (5)
3
(1)
53
(6)
20
16
(6)
23 (5)
7
(2)
54
(6)
1
19
(6)
8
(4)
73
(6)
0
(0)
4
15
(5)
5
(3)
51
(6)
29
(5)
10
9
(3)
3
(2)
48
(6)
40
(6)
20
7
(2)
4
(2)
45
(7)
44
(6)
1
19 (19)
3
(3)
17
(7)
61
(9)
4
48
(9)
12 (4)
25
(6)
15
(4)
10
34
(8)
19 (5)
43
(8)
4
(1)
20
22
(8)
21 (6)
52
(8)
4
(1)
1
8
(6)
33 (7)
5
(3)
55
(7)
4
10
(3)
44 (6)
20
(4)
26
(4)
10
8
(4)
64 (5)
14
(3)
14
(2)
20
6
(4)
80 (4)
7
(2)
7
(1)
Standard errors are given in parentheses. They were calculated using Runkle’s (1987)
bootstrapping method based on 200 replications.
Table 8
Variance Decomposition: Model 4
Type of Shock
Variable
Horizon
y
1
5
(4)
69 (5)
26
(4)
4
18 (7)
54 (7)
32
(5)
10
47 (7)
31 (6)
22
(3)
20
75 (4)
14 (3)
10
(2)
1
32 (9)
35 (7)
33
(6)
4
23 (8)
66 (8)
11
(2)
10
17 (8)
77 (7)
5
(1)
20
14 (7)
83 (7)
3
(1)
1
83 (3)
0
(0)
17
(3)
4
77 (4)
9
(2)
14
(2)
10
64 (6)
18 (4)
18
(3)
20
62 (7)
19 (5)
19
(3)
1
11 (7)
0
(2)
89
(7)
4
44 (6)
13 (3)
43
(6)
10
29 (7)
37 (6)
34
(5)
20
15 (4)
67 (5)
18
(3)
i
∆p
i − ∆p
AS
IS
MM
Standard errors are given in parentheses. They were calculated
using Runkle’s (1987) bootstrapping method based on 200 replications.
Table 9
Variance Decomposition: Model 5
Type of Shock
Variable
Horizon
y
1
10
(6)
90
(6)
4
24
(7)
76
(7)
10
50
(5)
50
(5)
20
75
(3)
25
(3)
1
92
(2)
8
(2)
4
84
(3)
16
(3)
10
67
(5)
33
(5)
20
64
(6)
36
(6)
∆p
AS
AD
Standard errors are given in parentheses. They
were calculated using Runkle’s (1987) bootstrapping method based on 200 replications.
Figure 1
Model 1
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 2
Model 2
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 3
Model 3
Responses to AS-Shock
Responses to IS-Shock
Responses to MD-Shock
Responses to MS-Shock
Figure 4
Model 4
Responses to AS-Shock
Responses to IS-Shock
Responses to MM-Shock
Figur