Impulse Responses by Local Projections:

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Impulse Responses by Local Projections:

Practical Issues

by

Òscar Jordà

Economics Department, U.C. Davis e-mail: [email protected]

URL: www.econ.ucdavis.edu/faculty/jorda/

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Estimation and Inference of Impulse Responses by Local Projections

Motivation

Impulse Responses are important statistics that substantiate models of the economy

Is the DGP a VAR? Very likely it is not:

Zellner and Palm (1974) and Wallis (1977): a subset from a VAR follows a VARMA

Cooley and Dwyer (1998): many standard RBC models follow a VARMA

New solution techniques for nonlinear DSGE

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Disadvantages of VARs

 VARs approximate the data globally: best, linear, one-step ahead predictors.

Impulse responses are functions of multi-step forecasts

Standard errors for impulse responses from VARs are complicated: they are highly nonlinear functions of estimated parameters

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What this paper does

Because any model is a global approximation to the DGP in the sample, consider instead local

approximations for each forecast horizon of interest

Use local projections!

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Intuition

DGP

Projection

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Advantages of Local Projections

Can be estimated by single-equation OLS with standard regression packages

Provide simple, analytic, joint inference for impulse response coefficients

They are more robust to misspecification

Experimentation with very nonlinear and flexible

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Estimation

A definition of impulse response (Hamilton, 1994, Koop et al. 1996):

)

; 0

| (

)

;

| (

) ,

,

(t s _i E _t _s _t _i X_t E _t _s _t X_t IR d = y ₊ v = d − y ₊ v =

];

...

[

) ' (

,...)' ,

( 1 is

1

2 1

n i

t t

D E

X

n

d d

d v v

y y

y

=

Ω

=

×

−

(8)

Hence, consider

s s t p

s t p s t

s s

t₊ = + B ⁺ y ₋ + + B ⁺ y ₋ + u ₊

y α ₁ ¹ ₁ ... ¹

so that

h s

B s

t

IR( , ,d_i ) = ˆ₁^sd_i = 0,1,2,...

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Example: AR(1) vs. Local Projection

4 . 0 5

.

0 +

− +

= y

y ρ ε ε ε ; T = 180; p = 1, 100 reps

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Practical Comments

 The maximum lag p need not be common to all s projections (e.g. VMA(q))

The lag length and the IR horizon impose degree- of-freedom constraints for very small samples

 Consistency does not require that all n×^h

regressions be estimated jointly. It can be done by univariate regression for each n, and h.

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Variance Decompositions

1 '

1 ( ')

))

| (

(

) ' (

))

| (

(

,..., 1

, 0

)

| (

+ −

− + +

+ +

+

+ +

+

=

−

D E

D X

E MSE

E X

E MSE

h s

X E

s s s t

s t t

s t

s s s t

s t t

s t u

s s t t

s t s

t

u u

y

u u

y

u y

y

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Structural Identification for Linear Projections

Example: Cholesky Decomposition

Estimate a VAR (also, the first projection):

0 1

1 1

1

0 _t ... _p _t _p _t

t B y B y u

y = α + ₋ + + ₋ +

A A

Eu' u = Ω = ' Λ

−1

= A

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Alternatively:

When available instruments can be used to resolve endogeneity, structural impulse responses can be calculated directly:

s s t p

t s

p t

s t

S s

s

t₊ = α + A ⁺ ^y + A ⁺ ^y ₋ + + A ⁺ ^y ₋ + ε ₊

y ₀ ¹ ₁ ¹ ₁ ... ¹

so that the response to the i^th variable is simply h

s i

A i

s t

IR( , , ) = ˆ(., )₁^s⁺¹ = 0,1,2,...

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Inference and Relation to VARs

VAR(1)

hence

;

0 0

0

; 0

VAR(p)

'

1

1 1

1

t t

p p

p t

t t

t

FW W

I F I

W

X

ν ν

µ µ µ

+

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡Π Π Π

≡

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

≡

+ Π

+

=

−

+

−

M

K

M M

K M

L L M

v y

y

v y

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From the VAR(1) representation of the VAR(p),

(

^µ

) (

^µ

)

µ

− +

+

−

+ +

=

−

+ − + −

− + +

+

p s t

s t s

t s

t

F F

y y

v v

v y

1 1 1 1

1 1 1

1

...

hence, as s → ∞

...

... ₁

1 1

1 + + +

+ +

= _t _t₋ ^s _t₋_s

t v F v F v

y γ

and therefore

s i

i F

s t

IR( , ,d ) = ₁ d

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IR coefficients from a VAR(p)

F₁¹  ¹

F

₁²

 

¹

F

₁¹

 

²



F

₁^s

 

¹

F

₁^s⁻¹

 

²

F

₁^s⁻²

. . . 

^p

F

₁^s^−p

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Compare this with the linear projection

s s t p

s t p s t

s s

t₊ = + B ⁺ y ₋ + + B ⁺ y ₋ + u ₊

y α ₁ ¹ ₁ ... ¹

then

(

^s

)

s

s s

ps s

s

F F

F B

F F

I

v v

v

u ¹

1 1 1 1

1

...

) ...

(

+ +

+

=

−

=

− + +

+

+ +

α

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Define ^Y_t ≡

(

y_t₊₁^,..., y_t₊_h

)

^'^;^V_t ≡

(

v_t₊₁^,..., v_t₊_h

)

^'^; ^X _t

Under the assumption that the DGP is a VAR(p), consider the system:

Φ +

Ψ

= _t _t

t X V

Y with

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

= Φ

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

= Ψ

n h n

h p p

h

I F I

F F

L

M M

0

...

; ...

... ₁

1

1 1

1

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Further define:

(

⊗ Ω

)

Φ ≡ Σ Φ

= Ω

= , then ( ' ) '

) '

( _t _t _v E V_tV_t I_h _v E v v

then

( ) ( )

[

^'

] ⁽ ⁾

^' ⁽ ⁾

ˆ )

( I X ¹ I X ¹ I X ¹vec Y

vec Ψ = ⊗ Σ⁻ ⊗ ⁻ ⊗ Σ⁻ The usual impulse responses and their correct

standard errors are rows 1-n and columns 1-nh of Ψˆ

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What does all this math mean?

It establishes the equivalence between the IR

coefficients from a VAR and from local projections

It shows how to impose the VAR constraints to

jointly estimate the local projections by block-GLS

The GLS estimates deliver efficient analytic inference for IR coefficients through time and

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Other remarks

Monte Carlos show little loss of efficiency in

estimating univariate local projections and using HAC robust standard errors (such as Newey-West)

Denote Σˆ the HAC, VCV matrix of _L Bˆ in the ₁^s

linear projection, then a 95% CI is 1.96 ±

(

di 'Σˆ Ldi

)

 Also, could use the s-1 stage residuals as regressors in the s stage projection

Linear projections are a type of general misspecification test!

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The Constrains of Linearity

 Symmetry

 Shape invariance

 History independence

 Multidimensionality

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Flexible Local Projections

Generally,

,...) ,

( ₋₁ Φ

= _t _t

t v v

y

A Taylor series approximation to Φ is the Volterra series expansion (the non-linear Wold):

y_t 

∑

_i^_0 ^ⁱ^v ^t−i 

∑

_i^_0

∑

_j^_0 ^^ij^v ^t−iv _t_−j 

∑

_i^_0

∑

_j^_0

∑

_k^_0 ^^ijk^v^t−iv_t_−jv_t_−k . . .

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It is natural to use polynomials for the local projections as well, for example:

s s t p

s t p s t

s t s t

s t s s

t

B B

C Q

B

+ + −

+ −

+ − + −

+ − +

+ +

+

+ +

=

u y

y

y y

2 1 2 1

3 1 1 1

2 1 1 1

1 1 1

...

α

Hence

( )

^⎪⎭^⎪^⎬^⎫

⎪⎩

⎪⎨

⎧

+ +

= +

−

3 1 2

2 1 1

1 2 1

1

3 ˆ 3

ˆ 2 ˆ

) ,

, (

i i

t i

s t

i i

s t s i

i C

Q s B

t IR

d d

y d

y

d d

y d d

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Remarks

IRs no longer symmetric nor shape invariant

IRs no longer history independent – they depend on

−1

yt . Evaluate at y_t₋₁ = y to evaluate at linearity.

Define

)' 3

3 2

( _i _t ₁ _i _i² ²_t ₁ _i _t ₁ _i² _i³

i ≡ d y ₋ d + d y ₋ d + y ₋ d + d

λ ^then

a 95% CI is approximately

(

^λ ^Σ ^λ

)

± ˆ

96 .

1 ^'

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Flexible projections in general

Since what matters are the terms associated with y_t₋₁ (but not the remaining lags), then

s s t t

s t s

t₊ = m y ₋ X ₋ + u ₊

y ( ₁; ₁)

Thus, any parametric, semi-parametric and non- parametric conditional mean estimator will do.

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Monte Carlo Simulations

Two experiments:

1. Robustness to lag-length misspecification, consistency and efficiency: Christiano,

Eichenbaum and Evans (1996)

2. Robustness to nonlinearities: Jordà and Salyer (2003)

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Robustness, Consistency and Efficiency

Estimate the CEE VAR with 12 lags. Save the coefficients to produce the Monte Carlos

Three experiments:

1. Fit a VAR(2) and local-linear and –cubic projections: Robustness

2. Fit local-linear and –cubic projections with 12 lags: Consistency

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Experiment 1

-.4 -.3 -.2 -.1 .0 .1

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of EM

-.3 -.2 -.1 .0 .1

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of P

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of PCOM

-.4 -.2 .0 .2 .4 .6 .8

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of FF

-.008 -.006 -.004 -.002 .000 .002 .004

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of NBRX

-.3 -.2 -.1 .0 .1 .2 .3 .4

2 4 6 8 10 12 14 16 18 20 22 24

VAR (2) Linear (2) Cubic (2) Response of M2

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Experiment 2

-.4 -.3 -.2 -.1 .0 .1

2 4 6 8 10 12 14 16 18 20 22 24

Linear Cubic Response of EM

-.30 -.25 -.20 -.15 -.10 -.05 .00 .05 .10

2 4 6 8 10 12 14 16 18 20 22 24

Linear Cubic Response of P

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5

2 4 6 8 10 12 14 16 18 20 22 24

Linear Cubic Response of PCOM

-.4 -.2 .0 .2 .4 .6 .8

Response of FF

-.008 -.006 -.004 -.002 .000 .002 .004

Response of NBRX

-.3 -.2 -.1 .0 .1 .2 .3 .4

Response of M2

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Experiment 3

EM P PCOM

s

True- MC

Newey- West (Linear)

Newey- West (Cubic)

True- MC

Newey- West (Cubic)

True- MC

Newey- West (Cubic) 1 0.000 0.007 0.008 0.0000 0.007 0.007 0.000 0.089 0.096

… … … … … … … … … …

12 0.046 0.044 0.048 0.042 0.042 0.045 0.390 0.380 0.416

… … … … … … … … … …

24 0.064 0.063 0.068 0.086 0.081 0.086 0.371 0.431 0.484

FF NBRX ∆M2

1 0.000 0.022 0.024 0.0005 0.0005 0.0005 0.014 0.012 0.014

… … … … … … … … … …

12 0.077 0.075 0.083 0.0009 0.0009 0.0010 0.082 0.077 0.085

… … … … … … … … … …

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Nonlinearities

Simulate:

t t t

t t

t

t t

t t t t

t t t

h u

h

I N

h Bh

y y y A y

y y

1 1 1

, 2 1

1 1

3 2

1 1 1

1 3

1 2

1 1

3 2 1

; 5

. 0 3

. 0 5

. 0

) , 0 (

~

;

ε ε ε

ε ε

= +

+

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡ + +

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

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Compare:

A VAR(1)

Local-linear projections with one lag

Local-cubic projections with one lag

A Bayesian, time-varying parameter/volatility VAR à la Cogley and Sargent (2001,2003) - TVPVAR

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The Monte-Carlo design for the TVPVAR:

• 100 obs. used to calibrate the prior

• Gibbs-sampler initialized with 2,000 draws

• Additional 5,000 draws to ensure convergence

• Select the quintiles of the distribution of the residuals of the first equation to identify 5 dates.

• Given the local histories of the 5 dates, calculate 100 Monte Carlo forecasts 1-8 steps ahead

• Obtain 5 impulse responses as the average of each 100 replications.

• Time of run: 9 days, 2 hours, 17 min. on a Sun Sunfire

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Nonlinearities

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2

1 2 3 4 5 6 7 8

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

1 2 3 4 5 6 7 8

-0.4 0.0 0.4 0.8 1.2

1 2 3 4 5 6 7 8

TRUE No GARCH

Linear Projection Cubic Projection

Response of Y1 Response of Y2

Response of Y3

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A New-Keynesian-Type Model of the Economy

Rudebusch and Svensson (1999) model:

• percentage gap between real GDP and potential GDP (from CBO)

• quarterly inflation in the GDP, chain-weighted price index in percent, annual rate

• quarterly average of the federal funds rate in

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Asymmetries and Thresholds

Does the effectiveness of monetary policy depend on:

The stage of the business cycle

Whether inflation is high or low

Whether interest rates are close to the zero bound or not.

I test for threshold effects with Hansen’s (2000) test.

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The Data

-12 -8 -4 0 4 8

55 60 65 70 75 80 85 90 95 00 Output Gap

4 8 12 16 20

0 2 4 6 8 10 12 14

55 60 65 70 75 80 85 90 95 00 Inflation

4.75%

6%

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Thresholds in the New-Keynesian Model

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

1 2 3 4 5 6 7 8 9 10 11 12

Response of Y-Gap to Fed Funds Shock

-1.2 -0.8 -0.4 0.0 0.4 0.8

1 2 3 4 5 6 7 8 9 10 11 12

Response of Inflation to Fed Funds Shock

-0.8 -0.4 0.0 0.4 0.8 1.2

1 2 3 4 5 6 7 8 9 10 11 12

Response of Fed Funds to Fed Funds Shock

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4

1 2 3 4 5 6 7 8 9 10 11 12

Response of Y-Gap to Fed Funds Shock

-.8 -.6 -.4 -.2 .0 .2 .4 .6 .8

1 2 3 4 5 6 7 8 9 10 11 12

Response of Inflation to Fed Funds Shock

-0.8 -0.4 0.0 0.4 0.8 1.2

1 2 3 4 5 6 7 8 9 10 11 12

Response of Fed Funds to Fed Funds Shock Inflation Threshold: 4.75%

Fed Funds Threshold: 6%

Low Inflation Regime

High Inflation Regime

Low Inflation Regime

Low Inflation Regime Low Inflation Regime

Low Inf lation Regime

Low Inf lation Regime High Inflation Regime

High Inflation Regime

High Inf lation Regime

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Future Research

Estimation of deep parameters in rational

expectations models by efficient matching of MA coefficients.

Applications to Panel Data and treatment effects.

 Efficiency improvements: using stage s-1 residuals as regressors in stage s projections

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An Efficient Moment-Matching Estimator for RE models

Example (Fuhrer and Olivei, 2004):

( )

_t _t _t _t _t

t

t z E z E x

z = µ ₋₁ + β − µ ₊₁ + γ + ε

• z and x are the structural and driving processes

• ε is the stochastic shock

• For IS curve set z the output gap x the interest rate

• For AS curve set z inflation, x the output gap

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The usual solution approach

Assume: x_t = αx_t₋₁ + u_t

Conjecture: z_t ₌ bz_t₋₁ ₊ cx_t₋₁ ₊ dε_t₋₁ Undetermined Coeffs.: d = 1

(

β γαµ

)(

α

)

+

−

= − c b

1

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An alternative

Assume: xt ⁼

∑

i^∞₌0 ρiεt⁻i ⁺

∑

i^∞₌0θiut⁻i

Conjecture: zt ⁼

∑

i^∞₌0aiεt⁻i ⁺

∑

i^∞₌0biut⁻i

U.C.:

( ) ( )

( )

^s ^s

s s

b b

b

a a

a

b b

a a

γθ µ

β µ

γρ µ

β µ

µ β

γ β

+

− +

=

+

− +

=

−

= +

−

=

+

−1 1

1 0

...

; 1

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Define:

( )

(

⁰^, ^,...,^,..., ^, ^,^,...,⁰^, ^..., ^'

)

^'

' ,...,

, 0 , ,...,

, 0

' ,...,

, ,...,

, 1

1 ,

1 1

1 3

1 1

2

1 0

1

0 1

0

−

+ +

−

=

−

=

h h

X

b b

a a

X

b b

a a

X

b b

a a

a Y

θ θ

ρ ρ

Notice that the reduced form coefficients and the structural coefficients are related by:

(

µ + β − µ + γ

)

= ω

− X₁ X ₂( ) X ₃ Y

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Let the covariance matrix of the impulse responses be:

Y Y

VAR( ) = Ω

Then, an efficient estimator for β^,µ^{, and}γ^is:

ω ω'Ω_Y min

This also gives a natural metric for model fit even for models that would be rejected by FIML.

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Identification with Long-Run Restrictions

Blanchard and Quah

Let ⁼ ( ) ⁼

∑

^∞₌0 ⁻

i i t i

t

t A L ε A ε

y with Eε^'ε = I ^.

B-Q impose zero coefficient restrictions on A(1).

Using the reduced form ⁼ ( ) ⁼

∑

^∞₌1 ⁻

i i t i

t

t C L u C u

y Σ

=

(47)

How to estimate an approximation to C(1) with linear projections?

Option 1: Add the estimated linear projection coefficients,

∑

₌

≅ ^h_s B^s Cˆ(1) 1 1

for h “large.”

(48)

Option 2:

Define: t⁺h ⁼

∑

^hs₌ t⁺s 0 y

Y .

Then

h h t p

t h p t

h h

t₊ = G y ₋ + + G y ₋ + u ₊

Y ₁ ₁ ...

from which Gˆ is an estimate of ₁^h

∑

i^h₌ Ci

1 . Choose h

“large.”

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Example:

t t

t y

y = 0.75 ₋₁ + ε

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

2 4 6 8 10 12 14 16 18 20 22 24

Impulse Response ± 2 S.E.

0 1 2 3 4 5

2 4 6 8 10 12 14 16 18 20 22 24

Accumulated Response ± 2 S.E.

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A Panel Data Application

Example: Measuring the dynamic effect of a treatment.

“European Union Regional Policy and its Effects on Regional Growth and Labor Markets” by Florence Bouvet (Econ Dept, Graduate Student)

How does a poor region’s growth and unemployment respond to fund allocation from the European

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Data: panel of 111 regions from 8 EU countries, 1975-1999.

Instruments: political alignment between the regions, the national government and the EU Comission.

Estimation: TSLS with fixed country effects and fixed time effects.

Two examples: Response of Growth and Unemployment

(52)

(53)

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However,

Growth (∆%) Unemployment (∆%)

-1 -0.5 0 0.5 1 1.5

t t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 t+10

-3 -2 -1 0 1 2 3

t t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 t+10

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Conclusions

If the IR is the object of interest, concentrate on

fitting the long-horizon forecasts rather than fitting the data one-period ahead.

Projections can be estimated univariately – simplifies IR estimation and inference for

panel/longitudinal data and non-Gaussian data.

Consider graph analysis to resolve

contemporaneous causality (Demiralp and Hoover)

Impulse Responses by Local Projections: