Impulse Response Functions

Wouter J. Den Haan

University of Amsterdam

April 28, 2011

General de…nition IRFs

The IRF gives the j

-period response when the system is shocked by a one-standard-deviation shock.

Suppose

y

= _ρy

+ _ε

_{and ε}

has a variance equal to σ

Consider a sequence of shocks f ^¯ε

g

. Let the generated series for y

be given by f ^¯y

g

.

Consider an alternative series of shocks such that

˜ε

= ^¯ε

+ σ if t = τ

¯ε

o.w.

The IRF is then de…ned as

IRF ( j ) = ˜y

¯y

IRFs for linear processes

Linear processes: The IRF is independent of the particular draws for ¯ε

Thus we can simply start at the steady state (that is when ¯ε

has been zero for a very long time) IRF ( j ) = σρ

Often you cannot get an analytical formula for the impulse response function, but simple iteration on the law of motion (driving process) gives you the exact same answer

Note that the IRF is not stochastic

IRFs in theoretical models

When you have solved for the policy functions then it is trivial to get the IRFs by simply giving the system a one standard deviation shock and iterating on the policy functions.

Shocks in the model are structural shocks, such as productivity shock

preference shock

monetary policy shock

IRFs in empirical models

What we are going to do?

Describe an empirical model that has turned out to be very useful (for example for forecasting)

Reduced form VAR

Make clear we do not have to worry about variables being I(1) Describe a way to back out structural shocks (this is the hard part)

Structural VAR

Reduced Form Vector AutoRegressive models (VARs)

Let y

be an n 1 vector of n variables (typically in logs)

y

=

∑

A

y

+ u

where A

is an n n matrix.

This system can be estimated by OLS (equation by equation) even if y

contains I ( ₁ ) variables

constants and trend terms are left out to simplify the notation

Spurious regression

Let z

and x

be I ( 1 ) variables that have nothing to do with each other

Consider the regression equation z

= _ax

+ _u

The least-squares estimator is given by

ˆa

= ^∑

Tt=1

x

z

∑

x

Problem:

T

lim

ˆa

6= ⁰

Source of spurious regressions

The problem is not that z

and x

are I(1)

The problem is that there is not a single value for a such that u

is stationary

If z

and x

are cointegrated then there is a value of a such that z

ax

is stationary

Then least-squares estimates of a are consistent

but you have to change formula for standard errors

How to avoid spurious regressions?

Answer: Add enough lags.

Consider the following regression equation z

= _ax

+ _bz

+ _u

Now there are values of the regression coe¢ cients so that u

is stationary, namely

a = _{0 and b} = ₁

So as long as you have enough lags in the VAR you are …ne

(but be careful with inferences)

Structural VARs

Consider the reduced-form VAR y

=

∑

A

y

+ u

For example suppose that y

contains the interest rate set by the central bank real GDP

residential investment What a¤ects

the error term in the interest rate equation?

the error term in the output equation?

the error term in the housing equation?

Structural shocks

Suppose that the economy is being hit by "structural shocks", that is shocks that are not responses to economic events Suppose that there are 10 structural shocks. Thus

u

= Be

where B is a 3 10 matrix.

Without loss of generality we can assume that

E [ e

e

] = I

Structural shocks

Can we identify B from the data?

E [ u

u

] = BE [ e

e

] B

= BB

We can get an estimate for E [ _u

_u

] using

Σ ˆ =

∑

ˆu

ˆu

/ ( _{T J} )

But B has 30 and ˆ Σ only 9 elements.

Identi…cation of B

Can we identify B if there are only three structural shocks?

B has 9 distinct elements Σ is symmetric ˆ

Not all equations are independent. Σ

= _Σ

. For example Σ

= b

b

+ b

b

+ b

b

but also

Σ

= b

b

+ b

b

+ b

b

In other words, di¤erent B matrices lead to the same Σ matrix

Identi…cation of B

We need additional identi…cation assumptions 2

4 u

u

u

3 5 = B

2 4 e

e

e

3 5

And suppose we impose

B = 2

4 0 0

0 3 5

Then I can solve for the remaining elements of B from ˆB

ˆB = _Σ ^ˆ

In Matlab use B=chol(S)

Identi…cation of B

Suppose instead we use 2 4 u

u

u

3 5 = _D

2 4 e

e

e

3 5

And that we impose

D = 2

4 0 0

0 3 5

This corresponds with imposing B =

2

4 0

0 0 3 5

This does not a¤ect the IRF of e

. All that matters for the

IRF is whether a variable is ordered before or after r

First-order VAR

y

= A

y

+ Be

Now we can calculate IRFs, variances, and autocovariances analytically

Mainly because you can easily calculate the MA representation

y

= Be

+ A

Be

+ A

Be

+

State-space notation

Every VAR can be presented as a …rst-order VAR. For example let y

y

= A

y

y

+ A

y

y

+ B e

e

2

6 6 4

y

y

y

y

3 7 7

5 = ^A

^A

I

0

2 6 6 4

y

y

y

y

3 7 7

5 + ^{B 0}

0

0

2 6 6 4

e

e

0 0

3

7 7

5

VAR used by Gali

z

=

∑

A

z

+ Bε

with

z

= ^{∆ ln} ( _y

_/h

)

∆ ln ( h

) ε

= ^ε

ε

Identifying assumption (Blanchard-Quah)

Non-technology shock does not have a long-run impact on productivity

Long-run impact is zero if

Response of the level goes to zero

Responses of the di¤erences sum to zero

Get MA representation

z

= _A ( _L ) _z

+ _Bε

= ( _{I A} ( _L ))

_Bε

= D ( L ) ε

= D

ε

+ D

ε

+

Note that D

= _B

Sum of responses

∑

D

= D ( 1 ) = ( I A ( 1 ))

B Blanchard-Quah assumption:

∑

D

= ⁰

De…nition Reduced form VAR Reduced form VAR Trick Blanchard-Quah Critique

If you ever feel bad about getting too much

criticism ....

If you ever feel bad about getting too much criticism ....

be glad you are not a structural VAR

Structural VARs & critiques

From MA to AR Lippi & Reichlin

From prediction errors to structural shocks Fernández-Villaverde, Rubio-Ramirez, Sargent Problems in …nite samples

Chari, Kehoe, McGratten

From MA to AR

Consider the two following di¤erent MA(1) processes

y

= ε

+ ¹

2 ε

, E

[ ε

] = 0, E

h ε

i

= σ

x

= e

+ 2e

, E

[ e

] = 0, E

h e

i

= σ

/4

Di¤erent IRFs

Same variance and covariance

E y

y

= E x

x

From MA to AR

AR representation:

y

= ( 1 + θL ) ε

1

( 1 + _θL ) ^y

= _ε

1

( 1 + _θL ) =

∑

a

L

Solve for a

s from

1 = a

+ ( a

+ a

θ ) L + ( a

+ a

θ ) L

+

From MA to AR

Solution:

a

= ₁ a

= _a

_θ

a

= _a

_θ = _a

_θ

You need

j ^θ j < ¹

Prediction errors and structural shocks

Solution to economic model

x

= _Ax

+ _Bε

y

= _Cx

+ _Dε

x

: state variables

y

: observables (used in VAR)

ε

: structural shocks

Prediction errors and structural shocks

From the VAR you get

e

= y

E

[ y

]

= Cx

+ Dε

E

[ Cx

]

= C ( x

E

[ x

]) + Dε

Problem: Not guaranteed that

x

= E

[ x

]

Prediction errors and structural shocks

Suppose: y

= x

that is, all state variables are observed Then

x

= E

[ x

]

Prediction errors and structural shocks

Suppose: y

6= ^x

F-V,R-R,S show that x

= E

[ x

] if

the eigenvalues of A BD

C

must be strictly less than 1 in modulus

Finite sample problems

Summary of discussion above

Life is excellent if you observe all state variables But,

we don’t observe capital (well)

even harder to observe news about future changes

If ABCD condition is satis…ed, you are still ok in theory Problem: you may need ∞-order VAR for observables

recall that k

has complex dynamics

Finite sample problems

1

Bias of estimated VAR

apparently bigger for VAR estimated in …rst di¤erences

2

Good VAR may need many lags

Alleviating …nite sample problems

Do with model exactly what you do with data:

NOT: compare data results with model IRF YES:

generate N samples of length T calculate IRFs as in data

compare average across N samples with data analogue

This is how Kydland & Prescott calculated business cycle stats