slides svar - washington

7/30/2019 Slides SVAR - Washington

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Structural VARs

Structural Representation

Consider the structural VAR (SVAR) model

y1t = 10 b12y2t + 11y1t1 + 12y2t1 + 1t

y2t = 20 b21y1t + 21y1t1 + 22y2t1 + 2t

where 1t2t

! iid

00

!,

21 00 22

!!.

Remarks:

1t and 2t are called structural errors

In general, cov(y2t, 1t) 6= 0 and cov(y1t, 2t) 6= 0

All variables are endogenous - OLS is not appropri-

ate!


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In matrix form, the model becomes

" 1 b12b21 1

# " y1ty2t

#=

"1020

#+

"11 1221 22

# "y1t1y2t1

#+

"1t2t

#or

Byt = 0 + 1yt1 + t

E[t0t] = D =

21 00 22

!In lag operator notation, the SVAR is

B(L)yt = 0 + t,B(L) = B 1L.


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Reduced Form Representation

Solve for yt in terms ofyt1 and t :

yt = B1

0 + B11yt1 + B

1t

= a0 + A1yt1 + uta0 = B

10,A1 = B

11,ut = B

1t

orA(L)yt = a0 + utA(L) = I2 A1L

Note that

B1 =1

"1 b12

b21 1 # , = det(B) = 1b12b21The reduced form errors ut are linear combinations of thestructural errors t and have covariance matrix

E[utu0t] = B

1E[t0t]B10

= B1DB10

= .

Remark: Parameters of RF may be estimated by OLSequation by equation


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Identification Issues

Without some restrictions, the parameters in the SVARare not identified. That is, given values of the reducedform parameters a0,A1 and , it is not possible touniquely solve for the structural parameters B,0,1and D.

10 structural parameters and 9 reduced form para-meters

Order condition requires at least 1 restriction on theSVAR parameters

Typical identifying restrictions include

Zero (exclusion) restrictions on the elements ofB;e.g., b12 = 0.

Linear restrictions on the elements ofB; e.g., b12 +b21 = 1.


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MA Representations

Wold representation

Multiplying both sides of reduced form by A(L)1 =

(I2 A1L)1 to give

yt = +(L)ut(L) = (I2 A1L)

1

=X

k=0

kLk, 0 = I2,k = A

k1

= A(1)1a0

E[utu0t] =

Remark: Wold representation may be estimated using

RF VAR estimates


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Structural moving average (SMA) representation

SMA ofyt is based on an infinite moving average of the

structural innovations t. Using ut = B1

t in the Wold

form gives

yt = +(L)B1

t

= +(L)t

(L) =X

k=0

kLk

= (L)B1

= B1 +1B1L +

That is,

k = kB1 = Ak1B

1, k = 0, 1, . . . .

0 = B1 6= I2


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Example: SMA for bivariate system"y1ty2t

#=

"12

#+

(0)11

(0)12

(0)21

(0)22

"

1t2t

#

+

(1)11

(1)12

(1)21

(1)22

"

1t12t1

#+

Notes

0 = B1 6= I2. 0 captures initial impacts of

structural shocks, and determines the contempora-

neous correlation between y1t and y2t.

Elements of the k matrices, (k)ij , give the dynamic

multipliers or impulse responses of y1t and y2t to

changes in the structural errors 1t and 2t.


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Impulse Response Functions

Consider the SMA representation at time t + s"y1t+sy2t+s

#=

"12

#+

(0)11

(0)12

(0)21

(0)22

"

1t+s2t+s

#+

+

(s)11

(s)12

(s)21 (s)22

" 1t2t # + .The structural dynamic multipliers are

y1t+s1t

= (s)11 ,

y1t+s2t

= (s)12

y2t+s

1t

= (s)21 ,

y2t+s

2t

= (s)22

The structural impulse response functions (IRFs) are the

plots of (s)ij vs. s for i, j = 1, 2. These plots summa-

rize how unit impulses of the structural shocks at time timpact the level of y at time t + s for different values ofs.

Stationarity of yt implies

lims

(s)ij = 0, i , j = 1, 2


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The long-run cumulative impact of the structural shocks

is captured by

(1) =

"11(1) 12(1)21(1) 22(1)

#=

P

s=0 (s)11

Ps=0

(s)12P

s=0 (s)21

Ps=0

(s)22

(L) =

"11(L) 12(L)21(L) 22(L)

#

=

P

s=0 (s)11 L

s Ps=0

(s)12 L

sPs=0

(s)21 L

s Ps=0

(s)22 L

s


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Digression: Dynamic Regression Models

In the SVAR every variable is engodenous. Suppose, for

example, y2t is strictly exogenous which implies b21 = 0

and 21 = 0. Then, the first equation is an ADL(1,1)

y1t = + y1t1 + 0y2t + 1y2t1 + 1t

cov(y2t, 1t) = 0

In lag operator notation the equation becomes

(L)y1t = + (L)y2t + 1t

(L) = 1 L, (L) = 0 + 1L

The second equation is an AR(1) model for y2t

y2t = c + y2t1 + 2t

Stationarity now only requires || < 1 and || < 1.


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The first equation may then be solved for y1t as a function

of y2t and 1t

y1t =

(1)+ (L)1(L)y2t + (L)

11t

= + (L)y2t + (L)t

=

(1)

(L) = (L)1(L), (L) = (L)1

Since y2t is exogenous, we have two sources of shocks.

Note: there can be four types of dynamic multipliers :

y1t+sy

2t

,y1t+s

2t

,y1t+s

1t

,y2t+s

2t


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The short-run dyamic multipliers with respect to y2t and

1t are

y1t+sy2t

=y1t

y2ts= ,s

y1t+s1t

=y1t

1ts= s

In the steady state or long-run equilibrium all variablesare constant

y1 = + (L)y2 = + (1)y

2

y2 =c

1

(1) = (1)1(1) =

0+

11

The long-run impact of a change in y2 on y1 is then

y1y2

= (1) =0 + 1

1 =X

s=0

y1t+sy2t


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Identification issues

In some applications, identification of the parameters of

the SVAR is achieved through restrictions on the para-

meters of the SMA representation.

Identification through contemporaneous restrictions

Suppose that 2t has no contemporaneous impact on y1t.

Then (0)12 = 0 and

0 =

(0)11 0

(0)21

(0)22

.

Since 0 = B1 then

(0)11 0

(0)21

(0)22

= 1

"1 b12b21 1

# b12 = 0

Hence, assuming (0)12 = 0 in the SMA representation is

equivalent to assuming b12 = 0 in the SVAR representa-

tion.


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Identification through long-run restrictions

Suppose 2t has no long-run cumulative impact on y1t.

Then

12(1) =X

s=0

(s)12 = 0

(1) = " 11(1) 021(1) 22(1)

# .This type of long-run restriction places nonlinear restric-

tions on the coefficients of the SVAR since

(1) = (1)B1 = A(1)1B1

= (I2 B11)

1B1


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Estimation Issues

In order to compute the structural IRFs, the parameters

of the SMA representation need to be estimated. Since

(L) = (L)B1

(L) = A(L)1 = (I2 A1L)1

the estimation of the elements in (L) can often be

broken down into steps:

A1 is estimated from the reduced form VAR.

Given cA1, the matrices in (L) can be estimatedusing ck = cAk1.

B is estimated from the identified SVAR.

Given B and k, the estimates ofk, k = 0, 1, . . . ,

are given by k =ckB1.


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Forecast Error Variance Decompositions

Idea: determine the proportion of the variability of the

errors in forecasting y1 and y2 at time t + s based on

information available at time t that is due to variability

in the structural shocks 1 and 2 between times t and

t + s.

To derive the FEVD, start with the Wold representation

for yt+s

yt+s = + ut+s +1ut+s1 +

+s1ut+1 +sut +s+1ut1 + .

The best linear forecast of yt+s based on information

available at time t is

yt+s|t = +sut +s+1ut1 +

and the forecast error is

yt+syt+s|t = ut+s +1ut+s1 + +s1ut+1.


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Using

t = B1ut, k = kB

1

The forecast error in terms of the structural shocks is

yt+s yt+s|t = B1

t+s +1B1

t+s1 +

+s1B1

t+1

= 0t+s +1t+s1 + +s1t+

The forecast errors equation by equation are"y1t+s y1t+s|ty2t+s y2t+s|t

#=

(0)11

(0)12

(0)21

(0)22

"

1t+s2t+s

#+

+ (s

1)11 (s

1)12

(s1)21

(s1)22

" 1t+12t+1

#


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For the first equation

y1t+s y1t+s|t = (0)11 1t+s + +

(s1)11 1t+1

+(0)12 2t+s + +

(s1)12 2t+1

Since it is assumed that t i.i.d. (0,D) where D is

diagonal, the variance of the forecast error in may be

decomposed as

var(y1t+s y1t+s|t) = 21(s)

= 21

(0)11

2+ +

(s1)11

2!

+2

2(0)12 2

+ + (s1)12 2

! .The proportion of 21(s) due to shocks in 1 is then

1,1(s) =

21

(0)11

2+ +

(s1)11

2!2

1

(s)


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the proportion of 21(s) due to shocks in 2 is

1,2(s) =

22

(0)12

2+ +

(s1)12

2!21(s)

.


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The forecast error variance decompositions (FEVDs) for

y2t+s are

2,1(s) =

21

(0)21

2+ +

(s1)21

2!22(s)

,

2,2(s) =

2

2(0)22 2

+ + (s1)22 2

!22(s)

,

where

var(y2t+s y2t+s|t) = 22(s)

= 21(0)21 2 + + (s1)21 2!+22

(0)22

2+ +

(s1)22

2!.


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Identification Using Recursive Causal Orderings

Consider the bivariate SVAR. We need at least one re-

striction on the parameters for identification. Suppose

b12 = 0 so that B is lower triangular. That is,

B = " 1 0b21 1 #B1 = 0 =

"1 0b21 1

#The SVAR model becomes the recursive model

y1t = 10 + 11y1t1 + 12y2t1 + 1t

y2t = 20 b21y1t + 21y1t1 + 22y2t1 + 2t

The recursive model imposes the restriction that the value

y2t does not have a contemporaneous effect on y1t. Since

b21 6= 0 a priori we allow for the possibility that y1t has

a contemporaneous effect on y2t.


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The reduced form VAR errors ut = B1t become

ut =" u1t

u2t

#=" 1 0b21 1

# " 1t2t

#=

"1t

2t b211t

#.

Claim: The restriction b12 = 0 is sufficient to just identify

b21 and, hence, just identify B.


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To establish this result, we show how b21 can be uniquely

identified from the elements of the reduced form covari-ance matrix . Note"

21 1212

22

#=

"1 0b21 1

# "21 00 22

# "1 b210 1

#

= "21 b21

21

b2121

22 + b

221

21 # .

Then, we can solve for b21 via

b21 = 1221

= 21

,

where = 12/12 is the correlation between u1 and

u2. Notice that b21 6= 0 provided 6= 0.


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Estimation Procedure

1. Estimate the reduced form VAR by OLS equation by

equation:

yt =

ba0 +

cA1yt1 +

but

b = 1TT

Xt=1butbu0t

2. Estimate b21 and B fromb :

bb21 = b12b

21,

bB = " 1 0bb21 1 # .3. Estimate SMA from estimates ofa0,A1 and B:

yt = b +c(L)btb =

ba0(I2c

A1)1

ck = cAk1bB1, k = 0, 1, . . .cD = bBbbB0.


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Remark:

Above procedure is numerically equivalent to estimating

the triangular system by OLS equation by equation:

y1t = 10 + 11y1t1 + 12y2t1 + 1t

y2t = 20 b21y1t + 21y1t1 + 22y2t1 + 2t

Why? Since cov(1t, 2) = 0 by assumption, cov(y1t, 2t) =

0


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Recovering the SMA representation using the Choleski

Factorization of.

Claim: The SVAR representation based on a recursive

causal ordering may be computed using the Choleski fac-

torization of the reduced form covariance matrix .

Recall, the Choleski factorization of the positive semi-

definite matrix is given by

= PP0

P = "p11 0p

21p

22 #A closely related factorization obtained from the Choleski

factorization is the triangular factorization

= TT0

T = "1 0

t21 1 # , = "1 0

0 2 # ,i 0, i = 1, 2.


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Consider the reduced form VAR

yt = a0 + A1yt1 + ut,

= E[utu0t]

= TT0

Construct a pseudo SVAR model by premultiplying by

T1

:

T1yt = T1a0 +T

1A1yt1 +T1ut

or

Byt = 0 + 1yt1 + t

where

B = T1, 0 = T1a0,

1 = T1A1, t = T

1ut.


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The pseudo structural errors t have a diagonal covari-

ance matrix

E[t0t] = T

1E[utu0t]T10

= T1T10

= T1TT0T10

= .

In the pseudo SVAR,

B =

"1 0

b21 1

#= T1 =

"1 0t21 1

#b12 = 0, b21 = t21


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Ordering of Variables

The identification of the SVAR using the triangular fac-

torization depends on the ordering of the variables in yt.

In the above analysis, it is assumed that yt = (y1t, y2t)0

so that y1t comes first in the ordering of the variables.

When the triangular factorization is conducted and thepseudo SVAR is computed the structural B matrix is

B = T1 =

"1 0

b21 1

# b12 = 0

If the ordering of the variables is reversed, yt = (y2t, y1t)0,

then the recursive causal ordering of the SVAR is reversed

and the structural B matrix becomes

B = T1 =

"1 0

b12 1

# b

21= 0


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Sensitivity Analysis

Ordering of the variables in yt determines the recur-

sive causal structure of the SVAR,

This identification assumption is not testable

Sensitivity analysis is often performed to determine

how the structural analysis based on the IRFs and

FEVDs are influenced by the assumed causal order-

ing.

This sensitivity analysis is based on estimating the

SVAR for different orderings of the variables.

If the IRFs and FEVDs change considerably for dif-

ferent orderings of the variables in yt then it is clearthat the assumed recursive causal structure heavily

influences the structural inference.


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Residual Analysis

One way to determine if the assumed causal ordering in-

fluences the structural inferences is to look at the resid-

ual covariance matrix from the estimated reduced form

VAR. If this covariance matrix is close to being diagonal

then the estimated value ofB will be close to diagonaland so the ordering of the variables will not influence the

structural inference.

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