an empirical test for the identifying assumptions of the
TRANSCRIPT
An empirical test for the identifying assumptions
of the Blanchard and Quah (1989) model
Hyeon-seung Huh
School of Economics Yonsei University
262 Seongsanno, Seodaemun-Gu Seoul 120-749
Korea Tel: +82-2-2123-5499
Email: [email protected]
Abstract
In their vector autoregression model, Blanchard and Quah (1989) employed as identifying assumptions uncorrelatedness between aggregate supply and aggregate demand shocks and the long-run output neutrality condition. This paper examines the empirical consistency of these assumptions with actual data. To derive a testable form, the Blanchard and Quah model is transformed into a cointegration representation that produces identical results. This alternative setup is extended to allow for a joint test of the uncorrelatedness and long-run output neutrality assumptions. Empirical results indicate that the joint assumptions when taken together are accepted by the data for Japan and the U.S. but rejected for Germany. Further analysis reveals that the imposition of the long-run output neutrality assumption is responsible for such rejection in Germany.
Key words: Blanchard and Quah; Cointegration; Shocks; Identifying assumptions
JEL classification: C32; C52; E32
Acknowledgement: The author thanks the Department of Economics at the University of Washington for its hospitality. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-013-B00007).
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1. Introduction
In an influential study, Blanchard and Quah (BQ, 1989) developed a vector autoregression
(VAR) model that identifies the effects of aggregate supply (AS) and aggregate demand (AD)
shocks on real output and the unemployment rate. Two assumptions are employed for
identification of structural shocks. The first assumption is that AS and AD shocks are
mutually uncorrelated. The rationale for this is that structural shocks are primitive, exogenous
forces and hence have no common causes. The second assumption is that an AD shock has no
long-run effects on real output. This is based on the long-run output neutrality condition,
according to which the AS curve is vertical and independent of AD factors in the long run, so
that only AS shocks can have long-run effects on real output.1
Both assumptions are not without criticism, however. Cover et al. (2006) suggest
reasons why AS and AD shocks may be correlated with each other. Shiller (1986) and
Pesaran and Smith (1998) argue that assuming uncorrelatedness between shocks may not be
reasonable in many econometric applications of the structural VAR methodology.2 Cooley
and Dwyer (1998) and Giordani (2004) provide examples in which this assumption produces
specification errors and biased impulse response estimators. Campbell and Mankiw (1987)
and Keating (2005) summarize a variety of possible mechanisms through which aggregate
1 Uncorrelatedness between structural shocks is a common assumption that most structural VAR studies impose in the identification process. The absence of this assumption entails n(n–1)/2 additional restrictions for exact identification in an n-variable model. The long-run output neutrality assumption has also been extensively adopted for identifying AS and AD shocks individually. 2 The structural shocks are generally permitted to correlate with each other in the simultaneous equation macroeconometric model (e.g. the Cowles Commission approach). The dynamic structure is restricted, instead, for identification, typically by excluding contemporaneous and lagged regressors. See Pagan and Robertson
3
demand shocks may have long-run effects on the level of real output. In fact, BQ
acknowledged some of these effects, such as hysteresis (à la Blanchard and Summers, 1986)
and endogenous models of growth. The hysteresis hypothesis suggests that the natural
unemployment rate depends on the past levels of the unemployment rate, allowing aggregate
demand shocks to influence the natural rate and hence, the level of real output in the long
run.3 Within the context of an endogenous growth model, any aggregate demand shock can
have long-run effects on the level of real output as long as it produces temporary changes in
the amount of resources allocated to growth (Stadler, 1986; King et al., 1988).
The present paper examines the extent to which the uncorrelatedness and long-run
output neutrality assumptions of the BQ model are consistent with actual data. This issue is
important because the model implications are dependent on the adequacy of the assumptions
in use. Where the assumptions are inconsistent with data, their imposition may result in
misrepresentation of the true dynamic structure of the model. Nevertheless, the
uncorrelatedness and long-run output neutrality assumptions in the BQ model have rarely
been evaluated for their empirical relevance. The main reason is that the two assumptions are
exact-identifying restrictions, and hence they cannot be tested.
To derive a testable form, this paper presents the BQ model in a cointegration
framework. The starting point is that real output is difference-stationary and the
(1995) and Jacobs and Wallis (2005) for a comparison between this traditional approach and the VAR modeling. 3 For empirical applications, see Dolado and Jimeno (1997) and Astley and Yates (1999).
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unemployment rate is stationary in levels. This implies that the two variables are cointegrated
where the unemployment rate itself is the cointegration relation. The corresponding vector
error correction model is structurally identified by using a procedure that decomposes the
shocks into those with permanent effects and those with transitory effects. The results are
exactly the same as those from the BQ model.4
The key feature is that this cointegration representation can be extended to allow for
relaxation of the two identifying assumptions underlying the BQ model. The correlation
between AS and AD shocks and the long-run response of real output to an AD shock are not
restricted but determined freely by the data. Thus, comparison of the results with those from
the BQ model can be used as a joint test for evaluating the empirical relevance of the two
identifying assumptions with respect to the data. This paper also introduces two companion
representations. One allows for correlation between AS and AD shocks under the long-run
output neutrality assumption. The other admits that an AD shock can have potentially long-
run effects on real output under the uncorrelatedness assumption between AS and AD shocks.
These models are utilized for examining the question of which assumption, i.e.
uncorrelatedness of the structural shocks or long-run output neutrality, has greater impact
when the joint assumptions taken together are rejected by the data.
The remainder of this paper is organized as follows. Section 2 presents a cointegration
4 Englund et al. (1994) and Fry and Pagan (2005) gave a general indication of this possibility, but they did not prove equivalence between the BQ model and its cointegration representation, due to a different focus.
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representation of the BQ model and an extension is made to allow for relaxation of the
uncorrelatedness and long-run output neutrality assumptions. Empirical applications to the
U.S., Germany, and Japan are provided in Section 3. Section 4 presents the two companion
models together with their empirical results. Section 5 concludes the paper. The appendix
formally proves equivalence between the BQ model and its cointegration representation.
2. A joint testing procedure
2-1. The Blanchard and Quah model
The Blanchard and Quah (BQ) model begins with estimation of a reduced-form VAR model
given as:
p pt yy,i t i yu,i t i 1t
i 1 i 1y f y f u e− −
= =Δ = Δ + +∑ ∑ (1)
p pt uy,i t i uu,i t i 2t
i 1 i 1u f y f u e− −
= == Δ + +∑ ∑ (2)
where ty is real output, tu is the unemployment rate, Δ is the first difference operator,
t 1t 2te (e ,e ) '= is a vector of reduced-form shocks and is iid with a mean of zero and a
covariance matrix of 't tE(e e ) = Ω .5 Let tz be the vector t t( y ,u ) 'Δ . The vector moving
average (VMA) representation for (1) and (2) can be expressed in compact form as:
t tz C(L)e= (3)
where 20 1 2C(L) C C L C L = + + + , 0C I= , i
ii 0C(1) C L∞== ∑ , and L is the lag operator.
5 The constant term is suppressed for the sake of illustration.
6
Subject to identification, a structural VMA representation corresponding to (3) is
given as:
t tz (L)= Γ ε (4)
where 20 1 2(L) L L Γ = Γ + Γ + Γ + , i
ii 0(1) L∞=Γ = Γ∑ , and t 1t 2t( , ) 'ε = ε ε is a vector of
structural shocks. Following BQ, 1tε and 2tε are denoted as an aggregate supply (AS)
shock and an aggregate demand (AD) shock, respectively. These are assumed to have a mean
of zero and a covariance matrix of the form:
12't t
12
1E( )
1σ⎛ ⎞
ε ε = Σ = ⎜ ⎟σ⎝ ⎠ (5)
where each structural shock is normalized to have unit variance without loss of generality.
From (3) and (4), the relationships between the reduced-form and structural parameters are:
0(L) C(L)Γ = Γ (6)
and
1t 0 te−ε = Γ . (7)
For exact identification of the structural shocks, the four parameters in 0Γ or 10−Γ
must be uniquely determined. The two identifying assumptions now come into action.
Uncorrelatedness between AS and AD shocks leads to 12 0σ = in (5), and then '0 0Γ Γ = Ω
in (7) yields three restrictions on 0Γ . The long-run output neutrality assumption implies that
the element (1,2) of the long-run impact matrix (1)Γ in (4) is zero. (1)Γ then becomes
lower triangular. This provides one remaining restriction on 0Γ by setting the (1,2) element
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of 0C(1)Γ in (6) to zero. Accordingly, solving '0 0Γ Γ = Ω under long-run output neutrality
gives the four parameters in 0Γ .6 Once 0Γ is estimated, the impulse responses and the
forecast-error variances of the series can be computed from (6).
2-2. A cointegration representation of the Blanchard and Quah model
The BQ model can be cast in a cointegated VAR framework. This is possible because the
model assumes that real output is difference-stationary and the unemployment rate is
stationary in levels. One implication is that the two variables are cointegrated where the
unemployment rate itself is the cointegration relation, i.e. [0, 1]'β = . The error correction
term is given as:
t t'z uβ = (8)
and a vector error correction (VEC) representation for (1) and (2) may be written as:
p p 1t yy,i t i yu,i t i 1 t p 1t
i 1 i 1y g y g u u e
−− − −
= =Δ = Δ + Δ + α +∑ ∑ (9)
p p 1t uy,i t i uu,i t i 2 t p 2t
i 1 i 1u g y g u u e
−− − −
= =Δ = Δ + Δ + α +∑ ∑ (10)
where 1 2[ , ]'α = α α is a vector of error correction coefficients.
Let ts be the vector t t( y , u ) 'Δ Δ . The Granger Representation Theorem shows that
the VEC model in (9) and (10) can be inverted to obtain the VMA representation (for details,
see Engle and Granger, 1987):
6 An equivalent but simpler solution is to combine (6) and (7) as ' 'C(1) C(1) (1) (1)Ω = Γ Γ . Taking the Choleski
8
t ts D(L)e= (11)
where 20 1 2D(L) D D L D L = + + + , 0D I= , and
i 'ii 0D(1) D L ∞
⊥ ⊥== = β ψα∑ (12)
where 1 2[ , ]'⊥ ⊥ ⊥α = α α and 1 2[ , ]'⊥ ⊥ ⊥β = β β are vectors orthogonal to α and β ,
respectively (i.e. ' 0⊥α α = and ' 0⊥β β = ), ' 1( G(1) )−⊥ ⊥ψ = α β , and G(1) is the short-run
impact matrix of reduced-form shocks from (9) and (10), given by:
p p 1yy,i yu,ii 1 i 1
p p 1uy,i uu,ii 1 i 1
1 g gG(1)
g 1 g
−= =
−= =
⎡ ⎤− −⎢ ⎥= ⎢ ⎥− −⎢ ⎥⎣ ⎦
∑ ∑∑ ∑
.
A detailed derivation is found in Johansen (1991).
The presence of one cointegrating relationship in the model implies that one shock
exhibits permanent effects, whereas the other shock demonstrates only transitory effects.7
The first ( 1tε ) is designated as an AS shock and the second ( 2tε ) as an AD shock. This
interpretation is plausible in that the AS shock has permanent effects on real output but the
AD shock has only transitory effects. To decompose the shocks into those with permanent
effects and those with transitory effects, we employ the Beveridge and Nelson (1981)
procedure of a type proposed for VEC models by Mellander et al. (1992), Englund et al.
(1994), Levtchenkova et al. (1996), and Fisher et al. (2000). The model posits for the
relationship between the reduced-form and structural shocks of (7) that:
decomposition of C(1) C(1) 'Ω provides all estimates in (1)Γ , and hence 0Γ , from (6). 7 See King et al. (1991), Mellander et al. (1992), and Levtchenkova et al. (1996) for the implications of cointegration in the structural identification of VEC models.
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'1
0 1'⊥−−
⎡ ⎤α⎢ ⎥Γ =⎢ ⎥α Ω⎣ ⎦
. (13)
Equation (13) is modified for each structural shock to have unit variance, which is given as:
'11
0 1 ' 11
⊥−− −
⎡ ⎤Λ α⎢ ⎥Γ =⎢ ⎥Φ αΩ⎣ ⎦
(14)
where 1 T '1 1− −
⊥ ⊥Λ Λ = α Ωα and ' ' 11 1
−Φ Φ = αΩ α .
Combining (7) and (14) reveals that '1 te⊥Λ α and 1 ' 1
1 te− −Φ αΩ are permanent AS
( 1tε ) and transitory AD ( 2tε ) shocks, respectively. It is also straightforward to show that
'' '1 1' 1 T
t t 0 0 1 ' 1 1 ' 11 1
E( ) I⊥ ⊥− −− − − −
⎡ ⎤ ⎡ ⎤Λ α Λ α⎢ ⎥ ⎢ ⎥ε ε = Γ ΩΓ = Ω =⎢ ⎥ ⎢ ⎥Φ αΩ Φ αΩ⎣ ⎦ ⎣ ⎦
. (15)
The correlation between AS and AD shocks is zero. The inverse matrix of 10−Γ is calculated
as:
' T0 1 1 −
⊥⎡ ⎤Γ = Ωα Λ αΦ⎣ ⎦ . (16)
The application of (6), (12), and (16) demonstrates that
' ' T 10 1 1 1(1) D(1) [ ] [ 0]− −
⊥ ⊥ ⊥ ⊥Γ = Γ = β ψα Ωα Λ αΦ = β ψΛ . (17)
As the implied cointegrating vector of [0, 1]'β = gives [1, 0]'⊥β = , (1)Γ of
dimension (2x2) becomes:
11 0(1)
0 0
−⎡ ⎤ψΛΓ = ⎢ ⎥⎢ ⎥⎣ ⎦
. (18)
Equation (18) confirms that the AS shock has permanent effects on real output, while the AD
shock has only transitory effects. Both shocks have transitory effects on the unemployment
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rate, which is consistent with the assumption that the rate is stationary in levels. This
cointegration representation matches the BQ model exactly, and they generate identical
results. A formal proof is provided in the Appendix.
2-3. Relaxation of the uncorrelatedness and long-run output neutrality assumptions
Now we present an extended model that can be used to jointly test the empirical relevance of
the uncorrelatedness and long-run output neutrality assumptions. This model, denoted as BQ-
E, shares the VEC representation in (9) and (10) but takes a different decomposition for the
identification of structural shocks. Specifically, it is assumed that
'21
0 1 ' 12
⊥−− −
⎡ ⎤Λ β⎢ ⎥Γ =⎢ ⎥Φ αΩ⎣ ⎦
, (19)
and its inverse is calculated as:
' 1 1 ' 1 10 2 2[ ( ) ( ) ]− − − −
⊥ ⊥ ⊥Γ = Ωα Λ β Ωα β Φ αΩ β (20)
where 1 T '2 2− −
⊥ ⊥Λ Λ = β Ωβ and ' ' 12 2
−Φ Φ = αΩ α . From (7), '2 te⊥Λ β and 1 ' 1
2 te− −Φ αΩ
correspond to AS and AD shocks, respectively.
The covariance matrix of the structural shocks is:
' T2 2' 1 T
t t 0 0 ' T2 2
1E( )
1
−⊥− −
−⊥
⎡ ⎤Λ β αΦ⎢ ⎥ε ε = Γ ΩΓ =⎢ ⎥Λ β αΦ⎣ ⎦
. (21)
The long-run impact matrix of the structural shocks is obtained from (12) and (20) as:
' ' 1 ' 1 ' 1 10 2 2(1) D(1) [ ( ) ( ) ]− − − −
⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥Γ = Γ = β ψα Ωα Λ β Ωα β ψα β Φ αΩ β . (22)
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On using [1, 0]'⊥β = , (22) is summarized as:
' ' 1 ' 1 ' 1 12 2
0( ) ( )(1) D(1)
0 0
− − − −⊥ ⊥ ⊥ ⊥ ⊥⎡ ⎤ψα Ωα Λ β Ωα ψα β Φ αΩ βΓ = Γ = ⎢ ⎥
⎢ ⎥⎣ ⎦. (23)
Equations (21) and (23) show key differences from the BQ model. The correlation between
two structural shocks is not restricted to being zero, nor is the long-run effect of the AD shock
on real output. Instead, they are allowed to be determined freely by the data. This provides
one way of jointly testing their empirical relevance. If results from the BQ model are
different from those of the BQ-E model, this can be taken as evidence that the joint
assumptions of uncorrelatedness and long-run output neutrality, taken together, are not
consistent with the actual data. If the results do not differ, the two assumptions are
empirically supported, justifying their use in the BQ model. A more detailed discussion will
be provided in the section to follow.
If '⊥β α in (21) and '
⊥α β in (23) are zero, both the uncorrelatedness and the long-
run output neutrality become valid for the BQ-E model.8 This occurs when real output is
weakly exogenous to the cointegrating vector in the VEC model of (9) and (10), i.e.
2[0, ]'α = α and hence 1[ , 0]'⊥ ⊥α = α .9 In fact, the BQ and BQ-E models coincide exactly.
This can be easily demonstrated by noting that when 2[0, ]'α = α , ⊥α can be
reparameterized as [1, 0]'⊥α = .10 Because ⊥ ⊥α = β , all equations in the BQ-E model
8 Note that 2 2, Λ Φ , and ψ cannot be zero by definition. 9 See Johansen and Juselius (1990) for the implications of weak exogeneity in the cointegrated system. 10 Consider the error correction coefficient matrix α of dimension (nxr) in an n-variable model with r
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become identical to their counterparts in the BQ model. This equivalence between the BQ-E
and BQ models offers stronger evidence in support of the joint assumptions of
uncorrelatedness and long-run output neutrality. Note that the weak exogeneity of real output
is a sufficient condition, but not a necessary one, for accepting the two assumptions jointly.
Even if real output is not weakly exogenous, it is possible for the uncorrelatedness and long-
run output neutrality assumptions to be empirically supported, as determined by the data.
3. Testing results of the identifying assumptions in Blanchard and Quah
The analysis outlined above is applied to quarterly observations of real GDP ( ty ) and the
unemployment rate ( tu ) in three major economies: Germany, Japan, and the U.S. Data were
taken from the IMF International Financial Statistics. The sample period is 1960:Q1 to
2006:Q4. The real GDP series was transformed using natural logarithms and differenced once
to achieve stationarity. For estimation of the VEC models in (9) and (10), the lag length p was
chosen on the basis of the Sims likelihood ratio test. The chosen lengths are p=8 and p=11 for
Germany and Japan, respectively. In the case of the U.S., p=8 is used to match the lag
structure in the original BQ model. All lag lengths are also consistent with the results of
Breusch-Godfrey Lagrange Multiplier tests, which indicate the absence of serial correlation
cointegration relationships present. Partition it as ' ' '
m r[ , ]α = α α , where mα and rα are, respectively, the first m (=n-r) and last r rows of α . Using Gaussian elimination with complete pivoting gives
1 'm m r[I , ]−
⊥α = − α α . Because m 0α = in our case, this becomes [1, 0]'⊥α = . For a full derivation, see Fisher and Huh (2007, p. 186).
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in both real output growth and unemployment rate equations at the 5% significance level.
The estimated VEC models are expanded to models in the levels of the series. They
are then inverted numerically to generate estimates of the reduced-form shocks. After these
estimates have been obtained, the structural shocks are identified and their impacts on the
series are calculated utilizing the procedures outlined in Section 2. Figure 1 depicts the
responses of the levels of the series to a one-standard-deviation shock in each structural
disturbance. The BQ and BQ-E models produce responses that are consistent with standard
economic theory. A favorable AS shock causes real output to increase across the horizons.
This shock lowers the unemployment rate at short horizons. In response to a favorable AD
shock, real output increases and the unemployment rate declines. Real output in the BQ
model eventually returns to its pre-shock levels as a consequence of the long-run output
neutrality assumption. The BQ-E model exhibits a similar pattern in Japan and the U.S.,
although it allows the AD shock to have long-run effects on real output. Germany is an
exception in that the responses remain positive at long horizons. For the unemployment rate,
the long-run responses to AS and AD shocks are all zero in the three countries, reflecting the
assumption that it is a stationary process.
The issue is how significantly the responses in the BQ model differ from those of the
BQ-E model. To evaluate this, depicted together in Figure 1 are 95% confidence bands
generated by 500 bootstrap replications of the BQ model. The Japan and U.S. cases show that
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all responses from the BQ-E model reside well within the 95% confidence bands of the
responses in the BQ model. In fact, the BQ and BQ-E models produce impulses of a quite
similar shape and magnitude. As they are statistically indistinguishable, this can be taken as
evidence that the uncorrelatedness and long-run output neutrality assumptions underlying the
BQ model are jointly accepted by the data.
The results for Germany differ considerably. The BQ-E model shows that AD shocks
have positive long-run effects on real output. The responses are large and above the 95%
upper bound of the responses from the BQ model at virtually all horizons. The AD shock also
leads to a larger fall in the unemployment rate than the BQ model implies. The effect is more
pronounced for short horizons at which the responses are significantly below the 95% lower
bound of the responses from the BQ model. The BQ model is statistically different from the
BQ-E model. This suggests that the joint assumptions of uncorrelatedness and long-run
output neutrality, when taken together, are not consistent with the data. It is thus likely that
the application of the BQ model to Germany produces misleading inferences on the dynamic
responses of the variables.
As a further check, Table 1 reports estimates for the error correction coefficients 1α
and 2α in (9) and (10). Based on the t-test, the coefficient 1α is not statistically different
from zero in Japan and the U.S., implying that real output is weakly exogenous to the
cointegration relationship. The null hypothesis of 2 0α = is rejected, as expected from the
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Granger Representation Theorem. A consequence is that the BQ and BQ-E models produce
identical results for these countries (see Section 2).11 This offers strong evidence supporting
the use of the uncorrelatedness and long-run output neutrality assumptions in the BQ model.
For the case of Germany, the coefficient 1α is statistically different from zero, while the
coefficient 2α is not. The unemployment rate, not real output, is the variable that is weakly
exogenous to the cointegrating relationship. As a result, the BQ and BQ-E models do not
coincide. In fact, Figure 1 revealed that the two models were statistically different and the
joint assumptions of uncorrelatedness and long-run neutrality, when taken together, were
rejected by the data.
4. Uncorrelatedness of the shocks versus long-run output neutrality
This section examines the question of which assumption, uncorrelatedness of the structural
shocks or long-run output neutrality, has greater impact leading to rejection of the BQ model
in Germany. Two companion models are developed along the lines of the BQ-E model of
Section 2. One allows for correlation between AD and AS shocks under the long-run output
neutrality assumption. The other admits that an AD shock can have potentially long-run
effects on real output under the uncorrelatedness assumption between AD and AS shocks.
The two models, denoted as BQ-C and BQ-L, respectively, may be regarded as restricted
11 To save space, we have not reported the impulse responses of the model imposing weakly exogeneity of real output. They are available upon request.
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cases of the BQ-E model.
The BQ-C model assumes, for the relationship between reduced-form and structural
shocks of (7), that
'31
0 1 ' 13
⊥−− −
⎡ ⎤Λ α⎢ ⎥Γ =⎢ ⎥Φ βΩ⎣ ⎦
(24)
and its inverse is calculated as:
' 1 1 ' 1 10 3 3[ ( ) ( ) ]− − − −
⊥ ⊥ ⊥Γ = Ωβ Λ α Ωβ α Φ βΩ α (25)
where 1 T '3 3− −
⊥ ⊥Λ Λ = α Ωα and ' ' 13 3
−Φ Φ = βΩ β . From (7) and (24), '3 te⊥Λ α and
1 ' 13 te− −Φ βΩ correspond to permanent AS ( 1tε ) and transitory AD ( 2tε ) shocks, respectively.
Gonzalo and Ng (2001) introduced a similar expression to (24) but adopted a different
normalization to render structural shocks uncorrelated with each other. In the BQ-C model,
the covariance matrix of structural shocks is given as:
' T3 3' 1 T
t t 0 0 ' T3 3
1E( )
1
−⊥− −
−⊥
⎡ ⎤Λ α βΦ⎢ ⎥ε ε = Γ ΩΓ =⎢ ⎥Λ α βΦ⎣ ⎦
. (26)
The correlation between AS and AD shocks is not restricted to being zero but is determined
freely by the data. The long-run impact matrix of structural shocks is obtained from (12) and
(25) as:
10 3(1) D(1) [ 0]−
⊥Γ = Γ = β ψΛ . (27)
Imposition of [1, 0]'⊥β = gives:
13
00(1) D(1)
0 0
−⎡ ⎤ψΛΓ = Γ = ⎢ ⎥⎢ ⎥⎣ ⎦
. (28)
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The AD shock has no long-run effect on real output, just as for the BQ model. In fact, the
BQ-C and BQ models share the same long-run impact matrix (1)Γ as 1 3Λ = Λ from
1 T 1 T '1 1 3 3− − − −
⊥ ⊥Λ Λ = Λ Λ = α Ωα .
Proceeding to the BQ-L model, it is assumed that
'41
0 1 ' 14
⊥−− −
⎡ ⎤Λ β⎢ ⎥Γ =⎢ ⎥Φ βΩ⎣ ⎦
(29)
where 1 T '4 4− −
⊥ ⊥Λ Λ = β Ωβ and ' ' 14 4
−Φ Φ = βΩ β . Escribano and Pena (1994) proposed a
similar decomposition procedure but did not pursue the identification of structural shocks.
Here, '4 te⊥Λ β and 1 ' 1
4 te− −Φ βΩ correspond to AS and AD shocks, respectively. The
remaining formulae can be calculated similarly, as before, and are therefore listed without
detailed derivations. They are:
' T0 4 4[ ]−
⊥Γ = Ωβ Λ βΦ , (30)
' 1 Tt t 0 0E( ) I− −ε ε = Γ ΩΓ = , (31)
' ' ' T0 4 4(1) D(1) [ ]−
⊥ ⊥ ⊥ ⊥ ⊥Γ = Γ = β ψα Ωβ Λ β ψα βΦ , (32)
and, with imposition of [1, 0]'⊥β = ,
' ' ' T4 4
0 (1) D(1)
0 0
−⊥ ⊥ ⊥⎡ ⎤ψα Ωβ Λ ψα βΦΓ = Γ = ⎢ ⎥
⎢ ⎥⎣ ⎦. (33)
The BQ-L model in (31) admits uncorrelatedness between AD and AS shocks, as for the BQ
model. However, (32) and (33) show that long-run output neutrality is not imposed a priori
and the AD shock is allowed to have long-run effects on real output.
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Figure 2 presents the responses of the levels of the series generated from the BQ-C
and BQ-L models for Germany. Those from the BQ model and their 95% confidence bands
are reproduced for comparison. The BQ-C model shows responses that are of a very similar
shape and magnitude to those in the BQ model. The only possible exceptions are the
responses of unemployment rate to the AS shock, but these are within the 95% confidence
bands of the responses in the BQ model across the horizons. By contrast, the BQ-L model
produces different implications. Again, AD shocks have positive long-run effects on real
output and the responses are larger than the 95% upper bound of those from the BQ model at
virtually all horizons. Both AS and AD shocks also show larger effects on the unemployment
rate than the BQ model suggests at short to medium horizons. On taking the results together,
the uncorrelatedness assumption is consistent with the data, but the long-run output neutrality
assumption is not. It appears that the imposition of long-run output neutrality was too
restrictive, and this seems to have contributed to the rejection of the BQ model in Germany.
To ascertain how the results change with the long-run output neutrality assumption,
Table 2 reports the forecast error variance of the series attributable to each structural shock in
the BQ and BQ-L models. Both models show that the AS shock is the major determinant of
the movement in real output across the horizons. One particular difference arises, however. In
the BQ model, the AD shock accounts for some fraction of the forecast error variance at short
horizons, but the contribution eventually converges to zero as a result of the long-run output
19
neutrality assumption. The BQ-L model indicates that this shock affects real output in the
long run. It accounts for around 36% of the forecast error variance at a horizon of 100
quarters. The results differ more considerably for the unemployment rate. The BQ model
demonstrates that the AS shock explains the vast majority of the variation in the rate at all
horizons. The contribution of the AD shock is limited to accounting for no more than 15% of
the forecast error variance. In the BQ-L model, the AD shock is more important for
explaining the variation in the unemployment rate. It accounts for more than 74% of the
forecast error variance across the horizons. In contrast, the AS shock plays only a small role
in influencing the rate.
The BQ-C and BQ-L models were not applied for Japan and the U.S. because the
uncorrelatedness and long-run neutrality assumptions were jointly accepted by the data in
Section 3. An obvious prediction is that both BQ-C and BQ-L would not be statistically
different from the BQ model. This can easily be checked by recalling that real output is
weakly exogenous to the cointegrating vector in Japan and the U.S. Since ' 0⊥α β = (see
Section 2), the BQ-C model in (26) has correlation between AD and AS shocks of zero. The
AD shock has no long-run effect on real output in the BQ-L model of (32) and (33). A
consequence is that the BQ-C and BQ-L models preserve both the uncorrelatedness and the
long-run output neutrality underlying the BQ model. In fact, the BQ, BQ-C, and BQ-L
models become identical. When ⊥ ⊥α = β and hence α = β , the BQ-C and BQ-L models
20
have exactly the same equations as the BQ model.12 The equivalence between these three
models confirms our earlier finding that the joint assumptions of uncorrelatedness and long-
run output neutrality are consistent with the actual data for Japan and the U.S.
5. Concluding remarks
One of the most well-known structural VAR models is Blanchard and Quah (BQ, 1989). They
developed a model that identifies the effects of aggregate supply and aggregate demand
shocks on real output and the unemployment rate. Since then, numerous applications and
extensions have followed. The BQ model employs as identifying assumptions
uncorrelatedness between aggregate supply and aggregate demand shocks and the long-run
output neutrality condition. Presumably, the model implications would be dependent on the
adequacy of these assumptions. Where the assumptions are inconsistent with data, their
imposition may result in misrepresentation of the true dynamic structure of the model.
Nevertheless, the uncorrelatedness and long-run output neutrality assumptions have rarely
been tested for their empirical relevance. Further, several studies have provided evidence
against these two assumptions.
The present paper examines the extent to which the uncorrelatedness and long-run
output neutrality assumptions of the BQ model are consistent with actual data in Germany,
12 See footnote 9 for the proof of ⊥ ⊥α = β .
21
Japan, and the U.S. To derive a testable form, the BQ model is transformed into a
cointegration representation that produces identical results. This alternative setup is extended
to allow for the possibility that the two types of structural shock are mutually correlated and
an aggregate demand shock has long-run effects on real output. Comparison of the results
with those from the BQ model has shown no significant difference in Japan and the U.S.
However, the two models produce different implications for Germany. In this case, the joint
assumptions of uncorrelatedness and long-run output neutrality are rejected by the data, thus
questioning the appropriateness of application of the BQ model to Germany. Further
examination reveals that the imposition of long-run output neutrality is responsible for such
rejection.
22
Appendix
The BQ VAR model of (1) and (2) can be expressed in matrix form as:
t tF(L)z e= . (A-1)
Similarly, the VEC model of (9) and (10) can be presented in matrix form as:
t t p t tG(L)s u (L)s e−− α = Π = (A-2)
where
11 12
21 22
G (L) G (L)G(L)
G (L) G (L)⎡ ⎤
= ⎢ ⎥⎣ ⎦
and p 1
11 12 1p 1
21 22 2
G (L) G (L) L (1 L)(L)
G (L) G (L) L (1 L)
−
−
⎡ ⎤− α −⎢ ⎥Π =⎢ ⎥− α −⎣ ⎦
.
To derive the relationship between the two models, consider the following
transformation on (A-2):
1
t t1 1
1 0 1 0(L) s e
0 (1 L) 0 (1 L)
−
− −
⎡ ⎤ ⎡ ⎤Π =⎢ ⎥ ⎢ ⎥
− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦.13 (A-3)
This yields
pt11 12 1
tp t21 22 2
yG (L) G (L)(1 L) Le
uG (L) G (L)(1 L) L
⎡ ⎤ Δ− − α ⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥ ⎣ ⎦− − α⎣ ⎦
. (A-4)
Comparison with (A-1) implies that
p11 12 11 12 1
p21 22 21 22 2
F (L) F (L) G (L) G (L)(1 L) LF(L)
F (L) F (L) G (L) G (L)(1 L) L
⎡ ⎤− − α⎡ ⎤⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦ − − α⎣ ⎦
. (A-5)
Recall the relationship between the long-run impact matrices of reduced-form and
structural shocks in (6):
13 Campbell and Shiller (1988) and Mellander et al. (1992) provide useful formulae for transforming VEC models to VAR counterparts in a general setup.
23
0(1) C(1)Γ = Γ , (6)
or, equivalently,
1 1 10(1) ( C(1) )− − −Γ = Γ . (A-6)
Equations (3), (A-1), and (A-5) imply that
11 11
21 2
G (1)C(1) F(1)
G (1)− −α⎡ ⎤
= = ⎢ ⎥−α⎣ ⎦. (A-7)
10−Γ is given in (14) as:
'11
0 1 ' 11
⊥−− −
⎡ ⎤Λ α⎢ ⎥Γ =⎢ ⎥Φ αΩ⎣ ⎦
. (14)
By using (A-7) and (14) with ' 0⊥α α = , it can be shown that
1 1 1 2 11 11 10
1 22 2 12 2 11 1 12 21 2
G (1) 0C(1)
( ) ( ) G (1)⊥ ⊥− − Λ α Λ α −α ∗⎡ ⎤ ⎡ ⎤ ⎡ ⎤
Γ = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥Ξ α Ω −α Ω Ξ α Ω −α Ω −α ∗ ∗⎣ ⎦⎣ ⎦ ⎣ ⎦ (A-8)
where 11(1/ | |) −Ξ = Ω Φ , 11Ω , 22Ω , and 12Ω are the variances of 1te and 2te and their
covariance in Ω , respectively, and ∗ denotes a non-zero element. Because 1 10 C(1)− −Γ is
lower triangular, so is (1)Γ from (A-6). The long-run output neutrality condition is evident.
The lower triangularity of (1)Γ together with uncorrelatedness between AS and AD shocks
in (15) satisfy the assumptions of the BQ model. The BQ model and its cointegrated
representation become equivalent and they generate identical empirical results.
24
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Table 1: Tests for error-correction coefficients Real output Unemployment
Country 1α t-test 2α t-test
Germany − 0.062 (0.03) 0.04 − 0.005 (0.01) 0.35
Japan − 0.050 (0.14) 0.72 − 0.021 (0.01) 0.03
U.S. 0.048 (0.05) 0.32 − 0.034 (0.01) 0.02 The second column reports the estimate for the error correction coefficient 1α in the output equation (9). Figures in parentheses are the standard errors. The third column reports the marginal significance level (p-value) of t-test statistic for the null hypothesis that 1α is equal to zero. The fourth and fifth columns do the same for the equation of the unemployment rate in (10).
28
Table 2: Forecast error variance decompositions for Germany BQ model BQ-L model
Real output Unemployment Real output Unemployment
Qtrs AS AD AS AD AS AD AS AD
1 54.2 45.8 87.0 13.0 100.0 0.0 19.6 80.4
2 62.0 38.0 87.5 12.5 98.4 1.6 20.2 79.8
4 70.4 29.6 89.8 10.2 95.8 4.2 23.5 76.5
8 78.4 21.6 91.4 8.6 91.5 8.5 25.8 74.2
10 82.2 17.8 90.5 9.5 87.7 12.3 24.5 75.5
20 90.5 9.5 86.5 13.5 76.8 23.2 19.7 80.3
30 93.3 6.7 86.0 14.0 73.4 26.6 18.9 81.1
40 94.8 5.2 85.6 14.4 71.1 28.9 18.5 81.5
50 95.8 4.2 85.5 14.5 69.2 30.8 18.3 81.7
60 96.5 3.5 85.4 14.6 67.7 32.3 18.2 81.8
70 97.1 2.9 85.3 14.7 66.5 33.5 18.1 81.9
80 97.4 2.6 85.3 14.7 65.4 34.6 18.0 82.0
90 97.7 2.3 85.2 14.8 64.5 35.5 18.0 82.0
100 98.0 2.0 85.2 14.8 63.7 36.3 18.0 82.0 The figures are the fractions of the forecast error variance of the series attributable to aggregate supply (AS) and aggregate demand (AD) shocks.
Figure 1: (a) Responses of the series from the BQ and BQ-E models for Germany
-0.5
0
0.5
1
1.5
2
2.5
3
1 10 19 28 37 46 55 64 73 82 91 100
-0.5
0
0.5
1
1.5
2
2.5
3
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Real output AD shock Real output
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Unemployment rate AD shock Unemployment rate
BQ model 95% Confidence bands BQ-E model
30
Figure 1: (b) Responses of the series from the BQ and BQ-E models for Japan
-2
-1
0
1
2
3
4
5
6
7
8
1 10 19 28 37 46 55 64 73 82 91 100
-2
-1
0
1
2
3
4
5
6
7
8
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Real output AD shock Real output
-0.20
-0.15
-0.10
-0.05
0.00
0.05
1 10 19 28 37 46 55 64 73 82 91 100
-0.20
-0.15
-0.10
-0.05
0.00
0.05
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Unemployment rate AD shock Unemployment rate
BQ model 95% Confidence bands BQ-E model
31
Figure 1: (c) Responses of the series from the BQ and BQ-E models for the U.S.
-0.3
0
0.3
0.6
0.9
1.2
1.5
1.8
1 10 19 28 37 46 55 64 73 82 91 100
-0.3
0
0.3
0.6
0.9
1.2
1.5
1.8
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Real output AD shock Real output
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Unemployment rate AD shock Unemployment rate
BQ model 95% Confidence bands BQ-E model
32
Figure 2: Responses of the series from the BQ, BQ-C, and BQ-L models for Germany
-0.5
0
0.5
1
1.5
2
2.5
3
1 10 19 28 37 46 55 64 73 82 91 100
-0.5
0
0.5
1
1.5
2
2.5
3
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Real output AD shock Real output
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
1 10 19 28 37 46 55 64 73 82 91 100
AS shock Unemployment rate AD shock Unemployment rate
BQ model 95% Confidence bands BQ-C model BQ-L model