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Universit`a di Pavia

Impulse Response Functions

Eduardo Rossi


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VAR

VAR(p):

y t = ( y1t , . . . , y Nt )

(L )y t = t

(L ) = IN 1L . . . p L p t = (1t , . . . , Nt )

t independent V W N (0 , )The process is stable if

det (I N

1z

. . .

p z p ) = 0 for

|z

| 1

On the assumption that the process has been initiated in the innitepast ( t = 0 , 1, 2, . . . ) it generates stationary time series that havetime-invariant means, variances, and covariances.

Eduardo Rossi c - Econometrics 10 2


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VAR

Because VAR models represent the correlations among a set of

variables, they are often used to analyze certain aspects of therelationships between the variables of interest.



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Granger-Causality

Granger (1969) has dened a concept of causality which,undersuitable conditions, is fairly easy to deal with in the context of VARmodels. Therefore it has become quite popular in recent years. The

idea is that a cause cannot come after the effect .

If a variable x affects a variable z, the former should help improvingthe predictions of the latter variable.

t is the information set containing all the relevant information inthe universe available up to and including period t.zt (h

|t ) be the optimal (minimum MSE) h-step predictor of the

process zt at origin t, based on the information in t .

The corresponding forecast MSE: t (h |t ).



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Granger-Causality

The process x t is said to cause zt in Grangers sense if

t (h |t ) < t (h |t {x s : s t}) for at least one h = 1 , 2, . . .t {x s : s t} is the set containing all the relevant information inthe universe except for the information in the past and present of thex t process.If zt can be predicted more efficiently if the information in the x tprocess is taken into account in addition to all other information in

the universe, then x t is Granger-causal for zt .



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Instantaneous Causality

If x t : (N 1) causes z t : (M 1) and z t also causes x tthe process ( z t , x

t ) is called a feedback system .

We say that there is instantaneous causality between zt and x t if

z (1|t x t +1 ) = z (1|t )In other words, in period t, adding x t +1 to the information set helpsto improve the forecast of z t +1 .

This concept of causality is really symmetric, that is, if there isinstantaneous causality between z t and x t , then there is alsoinstantaneous causality between x t and z t .



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A possible criticism of the foregoing denitions could relate to the

choice of the MSE as a measure of the forecast precision. Of course,the choice of another measure could lead to a different denition of causality.Equality of the MSEs will imply equality of the correspondingpredictors.In that case a process z t is not Granger-caused by x t if the optimalpredictor of z t does not use information from the x t process. This

result is intuitively appealing.



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A more serious practical problem is the choice of the information sett . Usually all the relevant information in the universe is notavailable to a forecaster and, thus, the optimal predictor given

tcannot be determined. Therefore a less demanding denition of causality is often used in practice.



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Instead of all the information in the universe, only the information inthe past and present of the process under study is considered relevant

and t is replaced by {z s , x s |s t}. Furthermore, instead of optimalpredictors, optimal linear predictors are compared.



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Impulse responses functions

Granger-causality may not tell us the complete story about theinteractions between the variables of a system.In applied work, it is often of interest to know the responseof one variable to an impulse in another variable in a system

that involves a number of further variables as well.One would like to investigate the impulse response relationshipbetween two variables in a higher dimensional system. Of course, if there is a reaction of one variable to an impulse in another variablewe may call the latter causal for the former.

We will study this type of causality by tracing out the effect of an

exogenous shock or innovation in one of the variables on some or allof the other variables.

This kind of impulse response analysis is called multiplier analysis .



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For instance, in a system consisting of an ination rate and aninterest rate, the effect of an increase in the ination rate may be of

interest.

In the real world, such an increase may be induced exogenously fromoutside the system by events like the increase of the oil price in1973/74 when the OPEC agreed on a joint action to raise prices.

Alternatively, an increase or reduction in the interest rate may beadministered by the central bank for reasons outside the simple twovariable system under study.



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Three-variables system:

y t =y1,t

y2,t

y3,t

investmentincome

consumption

the effect of an innovation in investment. To isolate such an effect,suppose that all three variables assume their mean value prior totime t = 0, y t = , t < 0, and investment increases by one unit in

period t = 0, i.e. 10 = 1.



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Now we can trace out what happens to the system during periodst = 1 , 2, . . . if no further shocks occur, that is,

20 = 30 = 0

and

2 = 0, 3 = 0, . . .

We assume that all three variables have mean zero and set c = 0,then

y t = 1y t 1 + t



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y t =

y1,t

y2,t

y3,t

=

.5 0 0

.1 .1 .3

0 .2 .3

y1 ,t 1

y2 ,t 1

y3 ,t 1

+

1 ,t

2 ,t

3 ,t

Tracing a unit shock in the rst variable in period t = 0 in thissystem we get

y 0 =y1 ,0y2 ,0

y3 ,0

=1 ,02 ,0

3 ,0

=10

0

y 1 =

y1 ,1

y2 ,1

y3 ,1

= 1y 0 =0.5

0.1

0



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y 2 =

y1 ,2

y2 ,2

y3 ,2

= 1y 1 = 21y 0 =

0.5

0.06

0.02it turns out that y i = ( y1 ,i , y2,i , y3 ,i ) is just the rst column of i1

y i = i1u i

where

u i = (0 , 0, . . . , 1, . . . , 0)

the elements of i1 represent the effects of unit shocks in thevariables of the system after i periods. They are called impulse responses or dynamic multipliers .



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Recall that i1 = i just the i-th coefficient matrix of the MA

representation of a VAR(1) process.The MA coefficient matrices contain the impulse responses of thesystem. This result holds more generally for higher order VAR(p)processes as well.V M A () representation:

y t =

i=0

i t i 0 = In



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Impulse-response function

y t + n =

i =0

i t + n i

{ n }i,j = yit + n

jt

the response of yi,t + n to a one-time impulse in yj,t with all othervariables dated t or earlier held constant.

The response of variable i to a unit shock (forecast error) in variable j is sometimes depicted graphically to get a visual impression of thedynamic interrelationships within the system.



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If the variables have different scales, it is sometimes useful to considerinnovations of one standard deviation rather than unit shocks.

For instance, instead of tracing an unexpected unit increase ininvestment in the investment/income/consumption system withV ar (1 ,t ) = 2 .25, one may follow up on a shock of 2.25 = 1 .5 unitsbecause the standard deviation of 1,t is 1.5.Of course, this is just a matter of rescaling the impulse responses.

The impulse responses are zero if one of the variables does notGranger-cause the other variables taken as a group.

An innovation in variable k has no effect on the other variables if theformer variable does not Granger-cause the set of the remaining

variables.Eduardo Rossi c - Econometrics 10 18


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The Orthogonalized Impulse-Response Function

A problematic assumption in this type of impulse response analysis isthat a shock occurs only in one variable at a time. Such anassumption may be reasonable if the shocks in different variables areindependent.

If they are not independent one may argue that the error termsconsist of all the inuences and variables that are not directly

included in the set of y variables.

Thus, in addition to forces that affect all the variables, there may be

forces that affect variable 1, say, only. If a shock in the rst variableis due to such forces it may again be reasonable to interpret the icoefficients as dynamic responses.



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On the other hand, correlation of the error terms may indicate that ashock in one variable is likely to be accompanied by a shock inanother variable.

In that case, setting all other errors to zero may provide a misleadingpicture of the actual dynamic relationships between the variables.



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This is the reason why impulse response analysis is often performedin terms of the MA representation:

= PP Cholesky decomposition

P is a lower triangular matrix. Fromy t = t + 1 t 1 + 2 t 2 + . . . 0 = In

to

y t = 0w t + 1w t 1 + 2w t 2 + . . .

with

i = i Pw t = P

1 t

E [w t w

t ] = IN



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It is reasonable to assume that a change in one component of w t hasno effect on the other components because the components areorthogonal (uncorrelated).

Moreover, the variances of the components are one. Thus, a unitinnovation is just an innovation of size one standard deviation.

The elements of the i are interpreted as responses of the system tosuch innovations.

{ i }jkis assumed to represent the effect on variable j of a unit innovation inthe k-th variable that has occurred i periods ago.



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To relate these impulse responses to a VAR model, we consider thezero mean VAR(p)process:

(L )y t = t

This process can be rewritten in such a way that the disturbances of different equations are uncorrelated. For this purpose, we choose adecomposition of the white noise covariance matrix

= WW

is a diagonal matrix with positive diagonal elements and W is alower triangular matrix with unit diagonal.



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This decomposition is obtained from the Choleski decomposition = PP by dening a diagonal matrix D which has the same maindiagonal as P and by specifying

W = PD 1

= DD

then from

= PP

and

P = WD

= PP = WDD W = WW



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Premultiplying the VAR(p) by A W 1

Ay t = A

1 y t 1 + . . . + A

p y t p + e t

where

A i = A i

e t = A t



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e t has a diagonal var-cov matrix:

E [e t e

t ] = A E [ t

t ]A = AA

Adding ( I N A )y t to both sides gives

(I N A )y t + Ay t = ( I N A )y t + A

1 y t 1 + . . . + A

p y t p + e t

y t = A

0 y t + A

1 y t 1 + . . . + A

p y t p + e t

Because W is lower triangular with unit diagonal, the same is truefor A .



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Hence,

A 0 = ( I N A )

=

0 0 . . . 0 0

12 0 . . . 0 0...

. . . . . . ...

......

N, 1 N, 2 . . . N,N 1 0

is a lower triangular matrix with zero diagonal and, thus, in therepresentation of the VAR(p) process, the rst equation contains noinstantaneous ys on the right-hand side. The second equation maycontain y1,t and otherwise lagged ys on the right-hand side. Moregenerally,the k-th equation may contain y1,t , . . . , y k 1 ,t and noty

k,t, . . . , y

N,t on the right-hand side.



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Thus, if the VAR(p) with instantaneous effects reects the actual

ongoings in the system, yst cannot have an instantaneous impact onykt for k < s .

In the econometrics literature such a system is called a recursive model (Theil (1971)). Wold has advocated these models where theresearcher has to specify the instantaneous causal ordering of thevariables. This type of causality is therefore sometimes referred to as

Wold-causality .



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If we trace eit innovations of size one standard error through thesystem, we just get the impulse responses. This can be seen bysolving the system for y t :

(I N A

0 )y t = A

1 y t 1 + . . . + A

p y t p + e t

y t = ( I N A

0 ) 1A 1y t 1 + . . . +( I N A

0 ) 1A p y t p +( I N A

0 ) 1e t

Noting that ( IN

A 0) 1 = W = PD 1 shows that the instantaneous

effects of one-standard deviation shocks ( eit s of size one standarddeviation) to the system are represented by the elements of

WD = P = 0

because the diagonal elements of D are just standard deviations of the components of e t .



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The i may provide response functions that are quite different fromthe i responses.

Note that 0 = P is lower triangular and some elements below the

diagonal will be nonzero if has nonzero off-diagonal elements.

For the investment/income/consumption example

0 = P =1.5 0 0

0 1 0

0 0.5 0.7

indicates that an income ( y2) innovation has an immediate impact onconsumption ( y3).



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If the white noise covariance matrix contains zeros, somecomponents of t are contemporaneously uncorrelated. Suppose,forinstance, that 1t is uncorrelated with it , i = 2 , . . . , N .In this case, A = W 1 and thus A 0 has a block of zeros so that y1has no instantaneous effect on yi , i = 2 , . . . , N . In theexample,investment has no instantaneous impact on income andconsumption because

=2.25 0 0

0 1 0.5

0 0.5 0.74

1t is uncorrelated with 2t and 3t . This, of course, is reected in thematrix of instantaneous effects 0 (impact multipliers ).



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The fact that 0 is lower triangular shows that the ordering of thevariables is of importance, that is, it is important which of the

variables is called y1 and which one is called y2 and so on.One problem with this type of impulse response analysis is that theordering of the variables cannot be determined with statisticalmethods but has to be specied by the analyst. The ordering has tobe such that the rst variable is the only one with a potentialimmediate impact on all other variables.

The second variable may have an immediate impact on the last

N 2 components of y t but not on y1t and so on. To establish suchan ordering may be a quite difficult exercise in practice.


Th O h li d I l R F i


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The choice of the ordering, the Wold causal ordering, may, to a largeextent, determine the impulse responses and is therefore critical forthe interpretation of the system.

For the investment/income/consumption example it may bereasonable to assume that an increase in income has an immediateeffect on consumption while increased consumption stimulates theeconomy and, hence, income with some time lag.


Th O th g li d I l R F ti


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Besides specifying the relevant impulses to a system, there are anumber of further problems that render the interpretation of impulse

responses difficult.A major limitation of our systems is their potentialincompleteness.Although in real economic systems almost everythingdepends on everything else, we will usually work withlow-dimensional VAR systems.All effects of omitted variables are assumed to be in the innovations.If important variables are omitted from the system,this may lead to

major distortions in the impulse responses and makes them worthlessfor structural interpretations.


The Orthogonalized Impulse Response Function


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Consider a system y t which is partitioned in vectors z t and x t , withVMA representation

y t =z t

xt

= 1

2

+ 11 (L ) 12 (L )

21

(L ) 22

(L )

1t

2t

If the z t variables are considered only and the x t variables areomitted from the analysis, we get a prediction error MArepresentation:

z t = 1 +

i =0

11 ,i 1t i +

i =1

12 ,i 2t i

z t = 1 +

i =0F i v t i

The actual reactions of the z t components to innovations 1t may begiven by the 11 ,i matrices.


The Orthogonalized Impulse Response Function


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On the other hand,the F i or corresponding orthogonalized impulseresponse are likely to be interpreted as impulse responses if theanalyst does not realize that important variables have been omitted.The F i will be equal to the 11 ,i , if and only if x t does notGranger-cause z t .


MA Representation


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MA Representation

The stable VAR(p) has a Wold MA representation:

y t = t + 1 t 1 + 2 t 2 + . . .

where s = sj =1 s j j s = 1 , 2, . . .

with 0 = IK . The elements of the j are the forecast error impulse

responses .Different ways to orthogonalize the impulses. Choleskydecomposition of . Such an approach is arbitrary, unless there are

special reasons for a recursive structure.Different ways to use nonsample information in specifying uniqueinnovations and hence unique i-r.


SVAR


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SVAR

Structural form model

Ay t = 1y t 1 + 2y t 2 + . . . + p y t p + Be t

The matrix A contains the instantaneous relations between theleft-hand-side variables. It has to be invertible.



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SVAR


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To identify the structural form parameters, we must placerestrictions on the parameter matrices.

Even if A = IN the assumption of orthogonal shocks

E [e t e t ] = IN

is not sufficient to achieve identication.

For a N -dimensional system, N (N 1)2

restrictions are necessary fororthogonalizing the shocks because there are N (N 1)2 potentiallydifferent instantaneous covariances.

An example of such an identication scheme is the triangular (orrecursive) identication suggested by Sims (1980). Such a scheme isalso called Wold causal chain system and is often associated with acausal chain from the rst to the last variable in the system.


Identification


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Because the i-r functions computed from these models depend on the

ordering of the variables, nonrecursive identication schemes thatalso allow for instantaneous effects of the variables

A = IN

have been suggested in the literature (Sims (1986), Bernanke (1986)).Restrictions on the long-run effects of some shocks are also used toidentify SVAR models (Blanchard and Quah (1989), Gal (1999),

King, Plosser, Stock, and Watson (1991)).



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