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Structural VAR and Applications

Jean-Paul Renne

Banque de France

ENSTA, 22 January 2010

Jean-Paul Renne (Banque de France) Structural VAR and Applications ENSTA, 22 January 2010 1 / 55


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Overview of the presentation

1 Vector Auto-Regressions

DefinitionEstimationTests

2 Impulse responsesGeneral conceptApplication to Structural VAR

3 Applications

1 Blanchard and Quah (1989)2 Smets and Tsatsaronis (1997)3 Dedola and Lippi (2005)



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Structural VARVector Auto-Regressions: Short introduction

The VAR are widely used in economic analysis.

While simple and easy to estimate, they make it possible toconveniently capture the dynamics of multivariate systems.

VAR popularity is mainly due to Sims (1980) influential work.



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Structural VARVector Auto-Regressions: Notations

let yt denote an (n 1) vector of random variables. yt follows a pth

order Gaussian VAR if, for all t, we have

yt = c + 1yt1 + . . .pytp + t

where t N(0,).

Consequentlyyt | yt1,yt2, . . . ,yp+1 N(c +1yt1 + . . .pytp,).

Denoting with the matrix

c 1 2 . . . p

and with xt the

vector 1 yt1 yt2 . . . ytp , the log-likelihood is given byL(YT; ) = (Tn/2) log(2) + (T/2) log

1

1

2

T

t=1 yt

xt

1

yt

xt

.



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Structural VARVector Auto-Regressions: Maximum Likelihood Estimation (MLE)

The MLE of, denoted with is given by

= Tt=1

ytx

t T

t=1xtx

t1

. (1)

Exercise 1

After having computed the jth rows of , find an easy way to

estimate .



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Proof of equation (1)Lets rewrite the last term of the log-likelihood

T

t=1yt

xt

1

yt xt =

Tt=1

yt

xt + xt

xt

1yt

xt + xt

xt

=

Tt=1

t + (

)

xt1

t + (

)

xt

where the jth element of the (n 1) vector t is the sample residual forobservation t from an OLS regression of yjt on xt.



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T

t=1yt

xt

1

yt xt =

Tt=1

t1t + 2

Tt=1

t1( )xt

+

Tt=1

x

t( )1

( )

xt



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Lets apply the trace operator on the second term (that is a scalar):

T

t=1

t1( )xt = trace

T

t=1

t1( )xt

= trace

Tt=1

1( )xtt

= trace1

( )

Tt=1

xt

t



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Given that, by construction, the sample residuals are orthogonal to theexplanatory variables, this term is equal to zero.If xt = ( )

xt, we have

Tt=1

yt

xt

1yt

xt

=

T

t=1

t1t +

T

t=1

xt1xt

Since is a positive definite matrix, 1 is as well. Consequently, thesmallest value that the last term can take is obtained when xt = 0,ie when

= .



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Assume that we have computed , the MLE of is the matrix that

maximizes L(YT; ,).

Denoting with t the estimated residual yt xt, we have

L(YT; ,) = (Tn/2) log(2) + (T/2) log 1

12

Tt=1

t

1t

.

is a symmetric positive definite matrix. Fortunately, it turns out

that that the unrestricted matrix that maximizes the latter expressionis a symmetric postive definite matrix. Indeed,

()

=

T

2

1

2

T

t=1t

t = =

1

T

T

t=1t

t.



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Structural VARVector Auto-Regressions: Likelihood-Ratio test

The simplicity of the VAR framework and the tractability of its MLEcontribute to convenience of various econometric tests. We illustratethis here with the likelihhod ratio test.

The maximum value achieved by the MLE is

L(YT; ,) = (Tn/2) log(2) + (T/2) log1

1

2

Tt=1

t1

t

.



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The last term isTt=1

t1t = trace

Tt=1

t1t

= trace Tt=1

1tt

= trace1

Tt=1

tt

= trace1

T

= Tn.

Therefore

L(YT; ,) = (Tn/2) log(2) + (T/2) log1 Tn/2.

which is easy to calculate.



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For instance, assume that we want to test the null hypothesis that aset of variable follows a VAR(p0) against the alternative specificationof p1 lags (with p1 > p0).

Let us respectively denote with L0 and L1 the maximum log-likelihoodsobtained withp0 and p1 lags. Under the null hypothesis, we have

2

L1 L0

= T

log11

log

10

which asymptotically has a 2

distribution with degrees of freedomequal to the number of restrictions imposed under H0 (compared withH1), ie n

2(p1 p0).



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Structural VARVector Auto-Regressions: Unconditional variance

The unconditional matrix of variance-covariance of yt is

Var(y) = limt

E0((yt yt)(yt yt))

where yt denotes the unconditional mean of y.

Let denote with yt the vector

yt y

t1 . . . y

tp

, we have

y

t =

c0

...0

+

1 2 p1 0 0

0 . . . 0 00 0 1 0

yt1 +

t0

...0

yt = c + yt1 +

t



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It is then easy to get the Wolds decomposition of yt :

yt = c + c + yt2 + t1+ t

= c + t +(c + t1) + . . . +

k(c + tk) + . . .

The ts being iid, we have

Var(y) = +

+ . . . + k

k + . . .



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Structural VARVector Auto-Regressions: Criteria

In a VAR, adding lags quickly consume degrees of freedom. If laglength is p, each of the n equations contains n p coefficients plusthe intercept term.

Adding lengths improve in-sample fit, but is likely to result inover-parameterization and affect the out-of-sample prediction

performance.To select appropriate lag length, some criteria can be used (they haveto be minimized)

AIC = log

+

2

TN

SBIC = log+ log T

TN

where N = n p2 + p.


S l V


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Structural VARVector Auto-Regressions: Granger Causality

Granger (1969) developed a method to analyze the causal relationshipamong variables systematically.

The approach consists in determining whether the past values of y1,tcan help to explain the current y2,t.

Let us denote three information sets

I1,t = {y1,t,y1,t1, . . .}

I2,t = {y2,t,y2,t1, . . .}

It = {y1,t,y1,t1, . . .y2,t,y2,t1, . . .} .

We say that y1,t Granger-causes y2,t if

E [y2,t | I2,t1] = E [y2,t | It1] .


S l VAR


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To get the intuition behind the testing procedure, consider thefollowing bivariate VAR(p) process:

y1,t = 10 + pi=111(i)y1,ti +

pi=112(i)y2,ti + u1,t

y2,t = 20 + pi=121(i)y1,ti +

pi=122(i)y2,ti + u2,t.

Then,y1,t does not Granger-cause y2,t if

21(1) = 21(2) = . . . = 21(p) = 0.

Therefore the hypothesis testing isH0 : 21(1) = 21(2) = . . . = 21(p) = 0

HA : 21(1) = 0 or 21(1) = 0 or . . .21(p) = 0.


S l VAR


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Rejection of H0 implies that some of the coefficients on the laggedy1,ts are statistically significant.

This can be tested using the F-test or asymptotic chi-square test.The F-statistic is F = (RSSUSS)/p

USS/(T2p1) (where RSS: restricted residual

sum of squares, USS: unrestriced residual sum of squares)Under H0, the F-statistic is distributed as F(p, T 2p 1)In addition, pF 2(p).


St t l VAR


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Structural VARVector Auto-Regressions: Impulse responses

Objective: analyzing the effect of a given shock on the endogenousvariables.

Let us consider a random variable yt that presents the followingWolds decomposition:

yt =k=0

ktk.

The impulse response function of the shock t on yt,yt+1, . . . is givenby the matrices

k.

Formally, the impulse response of the shock t on the variable y isdefined as

yt+kt

= k.


St t l VAR


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Dynamics of yt,yt+1,yt+2, . . . when t = 1, t+1 = 0, t+2 = 0, . . .

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St ct al VAR


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Exercise 2

Consider the process

yt = 1 + 0.9yt1 + t.

Compute the unconditional mean and variance of yt.

Exercise 2

Consider the process

yt = 1 + 0.5yt1 + 0.4yt2 + t.

Draw the impulse response of yt to t (up to yt+3). What is thecumulated impact of a shock (t = 1, t+1 = 0, t+2 = 0, . . .) onyt?


Structural VAR


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Structural VARVector Auto-Regressions: Causal mechanisms

Assume that the GDP growth gt is affected by some real shocks ur,tfollowing

gt = 0.3(it1 t) + ur,t

where it denotes the nominal interest rate and t denotes the inflationrate.

Besides, assume that we have

it = 0.9it1 + 1.5t + ump,t

t = 0.9t1 + 0.2gt1 + un,t

where ump,t and un,t are respectively some monetary-policy andcost-push shocks.

The strucural shocks ut are uncorrelated (i.e., the covariance matrix ofut, denoted with u is diagonal) and ut is serially uncorrelated (i.e.Cov(utk, ut) = 0 for any t and k > 0).


Structural VAR


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The structural model reads

gt = 0.3(it1 t) + ur,t

it = 0.9it1 + 1.5t + ump,t

t = 0.9t1 + 0.2gt1 + un,

t.

To get it in a reduced-form, let us substitute t in the right-handsides of the first two equations:

gt = 0.06gt1 0.3it1 + 0.27t1 + 0.3un,

t + ur,

t

it = 0.9it1 + 1.35t1 + 0.3gt1 + ump,t + 1.5un,t

t = 0.9t1 + 0.2gt1 + un,t.


Structural VAR


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Exercise 3

Write the model in matrix form.

Is this economy stationary?

Propose a way of estimating the model.

How to recover the structural shocks ut?


Structural VAR


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Structural VARVector Auto-Regressions:Impulse responses

In matrix form

gtitt

=

0.06 0.3 0.270.3 0.9 1.350.2 0 0.9

gt1it1t1

+

g,ti,t,t

with

g,ti,t

,t

=

1 0 0.30 1 1.5

0 0 1

ur,tump,t

un,t

= B

ur,tump,t

un,t

.

With the procedure described above, one only gets an estimate of

where

= Var

g,ti,t

,t

.


Structural VAR


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Note however that we must have

= BuB

where u is diagonal positive.

In addition, given the structural framework, one knows that B is anupper-triangular matrix.

The Choleshy decomposition can therefore be used to get the Bmatrix.


Structural VAR


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Whereas the VAR model is able to capture efficiently the interactionsbetween the different variables, it does not allow to reveal theunderlying causal mecanisms since two different causal schemes cancorrespond to the same reduced forms.

By taking into account certain economic relationships, a StructuralVAR model (SVAR) makes it possible to identify structural shockswhile letting play the interactions between the different variables (seeGali, 1992 or Gerlach and Smets 1995).

Formally, let assume that the residuals t are some linear combinations

of the structural shocks ut, that is:

t = But.


Structural VAR


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How it has been shown previously, a SVAR is based on a structuralmodel that draws from a theoretical framework.

As a starting point, we always have = BuB that provides us with

n(n + 1)/2 restrictions to recover the B matrix.

Consequently, to get the B matrix, one have to impose n(n 1)/2additional restrictions.

There exist two kinds of restrictions that can be easily implemented ina SVAR: short-run and long-run restrictions:

a short-run restriction prevents a structural shock from affecting anendogenous variable contemporaneously;a long-run restriction prevents a structural shock from affecting anendogenous variable in a cumulative way.


Structural VAR


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Concretely, the short-run restrictions consists in setting to zero some

entries of B.

The long-run restrictions require additional computations to beapplied. More precisely, one needs to implement the computation ofthe cumulative effect of one of the structural shocks ut on one of the

endogenous variable.Assume that we have the (reduced-form) VAR

yt = c +1yt1 + . . .pytp + t.

As was shown previously, one can always write a VAR(p) as a VAR(1),by stacking yt,yt1, . . .ytp+1 in a vector y

t . Consequently, let usconsider only the VAR(1) case:

yt = c + yt1 + t.


Structural VAR


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Once more, let us consider the Wolds form of yt:

yt = c + t + (c + t1) + . . . + k (c + tk) + . . .

= c + But + (c + But1) + . . . + k (c + Butk) + . . .

Consequently, the cumulated effect of the first structural shock u1,t onthe endogenous variables is obtained by computing

(B +

B + . . . +k

B + . . .)

0

...0

if the initial shock of u1,t is of magnitude .


Structural VAR


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Vector Auto-Regressions:Impulse responses

In this context, consider the following long-run restriction: the jth

structural shock does not affect,in a cumulative way, the ith

endogenous variable.

Then, denoting with the matrix (I ++ . . . +k + . . .)B, it comesthat the entry (i,j) of must be equal to zero.


Structural VAR


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A simple SVAR

It can be noted that short-run restrictions are simpler to implementthan long-run one.

There are particular cases in which some well-known matrixdecompostion can be used to easily estimate some specific SVAR.

Imagine a context in which you can argue that there exists an

ordering of the shocks:A first shock (say, 1,t) can affect instantaneously (i.e., in t) only oneof the endogenous variable (say, y1,t);A second shock (say, 2,t) can affect instantaneously (i.e., in t) thefirst two endogenous variables (say, y1,t and y2,t);

...

Exercise 4

In such a context, what is the form of the matrix B?

Suggest a methodology to estimate such a SVAR.


Structural VAR


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Cholesky decomposition: Illustration

Dedola and Lippi (The Monetary Transmission Mechanism: Evidencefrom the Industries of Five OECD Countries, 2005) estimate 5structural VAR for the US, the UK, Germany, France and Italy toanalyse the monetary-policy transmission mechanisms.

They estimate an SVAR over the period 1975-1997, using 5 lags in

VAR.

The shock-identification scheme is based on Cholesky decompositions,the ordering of the endogenous variables being: the industrialproduction, the consumer price index, a commodity price index, the

short-term rate, a monetary aggregate and (except for the US).This ordering implies that monetary policy reacts to the shocksaffecting the first three variables but that the latter react to monetarypolicy with a one-period lag.


Structural VAR


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Responses of Macro-variables to a monetary policy shock


Structural VAR


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Responses of Macro-variables to a monetary policy shock


Structural VAR


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Illustration: Blanchard and Quah (1989)

The article The Dynamics Effects of Aggregate Demand and SupplyDisturbances (AER, 1989) implements long-run restrictions in asmall-sized VAR.

Two variables are considered GDP and unemployment.

Consequently, the VAR is affected by two types of shocks.

The authors want to identify supply shocks (that xan have apermanent effect on output) and demand shocks (that can not have a

permanent effect on output).


Structural VAR


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The motivation of the authors regarding their long-run restrictions can

be obtained from a traditional Keynesian view of fluctuations.The authors propose a variant of a model from Stanly and Fisher(1977)

Yt = Mt Pt + a.t (2)

Yt = Nt + t (3)

Pt = Wt t (4)

Wt = W |

Et1Nt = N

(5)

To close the model, the aithors assume the following dynamics for themoney supply and the productivity

Mt = Mt1 + dt

t = t1 + st


Structural VAR


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In this context, it can be shown that

gnpt = dt

dt1 + a.(

st

st1) +

st

ut = dt a

st

Then, it appears that the demand shocks have no long-run impact onoutput. Besides, neither shocks have a long-run impact onunemployment.

The endogenous variable is ( gnpt ut) where gdpt denotes thelogarithm of GNP.

It is assumed to be stationary. Therefore, neither disturbances has along-run effect on unemployment or the rate of change in output.

The long-run restriction implies that the demand shocks also have nolong-run effect on the output level gnp itself.


Structural VAR


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Estimation data: quarterly, from 1950:2 to 1987:4.8 lags.


Structural VAR


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Dynamic effects of demand disturbancesHE AMERICAN ECONOMIC REVIEW SEPTEMBER 1989

positions onlymic effects ofces.d and Supply

and and sup-in Figures 1gures 1 and 2of output andthe horizontal. Figures 3-6but now withs around thehump-shaped

oyment. Their

1.401.201.00-0.80 --0.60-0.400.200.00 ,,,

-0.20 0 10 20 30 40-0.40-0.60 -

FIGURE 1. RESPONSE TO DEMAND,-= OUTPUT,- = UNEMPLOYMENT


Structural VAR

but now withs around the

- . -FIGURE 1. RESPONSE TO DEMAND,-= OUTPUT,


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Dynamic effects of supply disturbanceshump-shapedoyment. Theirquarters. Theine to vanishThe responsesare mirror im-to this aspectussing the ef-llest when thellowing for anemploymentcays the mostchange in theis allowed, thehanges in un-ively unimpor-mand distur-onsistent withmic effects of

- = UNEMPLOYMENT

1.000.800.60 -/0.40-0.20 -0.00 l+q ii i0 10 20 3 0 40-0.20 -

-0.40-0.60-

FIGURE 2. RESPONSE TO SUPPLY, OUTPUT,- = UNEMPLOYMENT


Structural VARV A R i I l

In the interveningnd output devia- 7.607.40


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Uutput-gap deifnition

The output gap is the component of GNP that is explained only by demandshocks.

. At the peak re-s an implied coef-four, higher in ab-coefficient. Thatoefficient is higherthan for demandat we expect. Sup-to affect the rela-employment, andtle or no change in

sof Demand andbances.mic effects of eachext step is to assessto fluctuations innt. We do this inrmal, and entails arical time-series ofof output to thesiness cycles. Thedecompositions ofnt in demand and components of unemployment are station-

7.207.001950 1960 1970 1980

FIGURE 7. OUTPUT FLUCTUATIONSABSENTDEMAND

0.100.080.060.040.02 v0.007

-0.04-0.06-0.08-0.101950 1960 1970 1980

FIGURE 8. OUTPUT FLUCTUATIONSDUE TODEMAND


Structural VARV i d i i


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Variance decomposition

The k quarter-ahead forecast error in output is defined as thedifference between the actual value of output and its VAR-basedforecast.

Variance decompositionConsider the VAR(1): yt = c + yt1 + t where the residuals aresome linear combinations of structural shocks ut (t = But).

Compute k = yt+k Et (yt+k) with respect to ut+1, ut+2, . . . ut+k.

How to compute the contribution of the ith

structural shock on the jth

endogenous variable?


Structural VARV t A t R i I l


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66 THE AMERICAN ECONOMIC RE VIEW SEPTEMBER 1989TABLE 2-VARIANCE DECOMPOSITION OF OUTPUT ND UNEMPLOYMENT

(CHANGE IN OUTPUT GROWTH AT 1973/1974; UNEMPLOYMENT DETRENDED)Percentage of Variance Due to Demand:Horizon(Quarters) Output Unemployment

1 99.0 51.9(76.9,99.7) (35.8,77.6)2 99.6 63.9(78.4,99.9) (41.8,80.3)3 99.0 73.8(76.0,99.6) (46.2,85.6)4 97.9 80.2(71.0,98.9) (49.7,89.5)8 81.7 87.3(46.3,87.0) (53.6,92.9)12 67.6 86.2(30.9,73.9) (52.9,92.1)40 39.3 85.6(7.5,39.3) (52.6,91.6)

TABLE 2A-VARIANCE DECOMPOSITION OF OUTPUT AND UNEMPLOYMENT(No DUMMY BREAK, TIME TREND IN UNEMPLOYMENT)Jean-Paul Renne (Banque de France) Structural VAR and Applications ENSTA, 22 January 2010 45 / 55

Structural VARSmets and Tsatsaronis (1997)


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Smets and Tsatsaronis (1997)

Why does the yield curve predict economic activity? BIS WorkingPaper No.49.

Objective: Investigating why the slope of the yield curve predictsfuture economic activity in Germany and the United States.

Methodology: A structural VAR is used to identify aggregate supply,aggregate demand, monetary policy and inflation scare shocks and toanalyse their effects on the real, nominal and term premiumcomponents of the term spread and on output.

Findings: In both countries demand and monetary-policy shockscontribute to the covariance between output growth and the laggedterm spread, while inflation scares do not.


Structural VARYield curve slope (10yr 3mth) vs Output gap Euro area data Source: OECD


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Yield-curve slope (10yr-3mth) vs. Output gap, Euro area data, Source: OECD

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Underlying theoretical model

The authors argue that their model and in particular the restrictionsthey use is consistent with a semi-structural model that reads:

yt = yt + st (Aggregate supply)

yt = 1yt1 + 2yt2 + (1 1 2)yt3 4t1 + st(IS)

it t = 1 (it1 t1) + 2t + 3 (yt yt) + it(Taylor rule)

Rt = t +1

N

Ni=1

Ett+i + t(Fisher equation)

t = 1t + (1 1)Ett+

1 + 2(yt1 yt1)(Phillips curve)

Rt = Et

Ni=1

it+i

(expectation hypothesis)




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The identification suggested by the model calls for a mixture of shortand long-run zero restrictions.

The assumption of a vertical long-run Phillips curve implies thatdemand shocks and nominal shocks (which include monetary-policyand inflation-scare shocks) have no long-run impact on the level of

real output.Supply innovations are thus the source of all permanent shocks tooutput.

Demand shocks are distinguished from nominal shocks by the

assumption that the latter do not contemporaneously aff

ect realoutput.

Finally, the authors assume that monetary authorities do not respondcontemporaneously (i.e. within the quarter) to inflation scares.


Structural VARThe effect of a supply shock


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The effect of a supply shock

1.6

0.8

0

0.8

1.6

1.0

0.5

0

0.5

1.0

0.8

0.4

0

0.4

0.8

0.8

0.4

0

0.4

0.8

Germany GermanyUnited States United States

Output

Inflation

Term spread

Real component


Structural VARThe effect of a supply shock


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The effect of a supply shock

1.0

0.5

0

0.5

1.0

1.0

0.5

0

0.5

1.0

0.8

0.4

0

0.4

0.8

0.8

0.4

0

0.4

0.8

Short rate

Long rate

Inflation component

Term premium


Structural VARThe effect of a demand shock


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1.6

0.8

0

0.8

1.6

1.0

0.5

0

0.5

1.0

0.8

0.4

0

0.4

0.8

0.8

0.4

0

0.4

0.8

Germany GermanyUnited States United States

Output

Inflation

Term spread

Real component


Structural VARThe effect of a demand shock


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1.0

0.5

0

0.5

1.0

1.0

0.5

0

0.5

1.0

0.8

0.4

0

0.4

0.8

0.8

0.4

0

0.4

0.8

Short rate

Long rate

Inflation component

Term premium


Structural VARDecomposition of term-spread/output-growth covariance


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p p / p g

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9

Covarianceof which:

SupplyDemand

PolicyInflation scare

Lags (in quarters)

Note: The unconditional covariance between the current output growth and lagged term spread (solid line) is decomposed tothe contributions of the four structural shocks identified in the VAR.

Germany United States


References


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Bhar, R. and Hamori, S. (2005). Empirical Techniques in Finance. Ed. SpringerFinance.

Blanchard, O. and Quah, D. (1989). The Dynamic Effects of Aggregate Demandand Supply Disturbances. American Economic Review, vol. 79.

Dedola, L. and Lippi, F. (2005). The monetary transmission mechanism: Evidencefrom the industries of five OECD countries. European Economic Review, vol. 49(6).

Gal, J. (1992). How well does the IS-LM model fit postwar US data? QuarterlyJournal of Economics.

Gerlach, S. and F. Smets (1995). The monetary transmission mechanism: evidencefrom the G7 countries. CEPR Discussion Paper, no. 1219.

Granger, C. (1969). Prediction with a Generalized Cost of Error Function.Operational Research Quarterly, vol. 20.

Hamilton, J. (1994). Time Series Analysis. Princeton University Press.

Sims, C. (1980). Macroeconomics and Reality. Econometrica, vol. 48.

Smets, F. and Tsatsaronis, K. (1997). Why does the yield curve predict economicactivity? BIS Working Paper No. 49.


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