impulse responses by local projections: practical issues · impulse responses by local projections:...

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/

Estimation and Inference of Impulse Responses by Local Projections

Òscar Jordà U.C. Davis

2

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 models produce polynomial difference equations



3

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



4

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!



5

Intuition

VAR

DGP

Projection



6

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 models is straight-forward



7

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

);0|();|(),,( ttsttitsti XEXEstIR =−== ++ vydvyd

];......[)'(

,...)',(1 is

1

21

ni

tt

ttt

t

DEX

n

dddvv

yyy

=Ω=

=×

−−



8

Hence, consider

sstpt

spt

ssst BB +−

+−

++ ++++= uyyy 1

11

1 ...α

so that

hsBstIR is

i ,...2,1,0 ˆ),,( 1 == dd



9

Example: AR(1) vs. Local Projection

411 4.05.0 −−− +−+= ttttt yy εεερ ; T = 180; p = 1, 100 reps



10

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.



11

Variance Decompositions

1'1 )'())|((

)'())|((

,...,1,0 )|(

−++

−+

+++

+++

=

=

==−

DEDXEMSE

EXEMSE

hsXE

sst

ssttst

sst

ssttstu

ssttstst

uuy

uuy

uyy



12

Structural Identification for Linear Projections Example: Cholesky Decomposition Estimate a VAR (also, the first projection):

011

11

0 ... tptptt BB uyyy ++++= −−α

AAE Λ=Ω= ''uu

Then 1−= AD . This is the D for all subsequent projections.



13

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

sstpt

spt

st

Ssst AAA +−

+−

+++ +++++= εα yyyy 1

11

11

0 ...

so that the response to the ith variable is simply

hsiAistIR s ,...2,1,0 )(.,ˆ),,( 11 == +



14

Inference and Relation to VARs

VAR(1) hence ;0

00

00;

VAR(p) '

1

11

1

ttt

t

pp

pt

t

t

ttt

FWW

I

IFW

X

νν

µ

µ

µ

+=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ΠΠΠ

≡⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−≡

+Π+=

−

−

+−

M

K

MMKM

L

L

M

v

y

y

vy



15

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

( ) ( )µµ

µ

−++−

++++=−

−+

−+

−+++

ptspt

st

sststst

FF

FF

yy

vvvy1

11

1

1111

...

...

hence, as ∞→s

...... 1111 +++++= −− st

sttt FF vvvy γ

and therefore

is

i FstIR dd 1),,( =



16

IR coefficients from a VAR(p)

F11 1

F12 1F1

1 2

F1s 1F1

s−1 2F1s−2 . . . pF1

s−p



17

Compare this with the linear projection

sstpt

spt

ssst BB +−

+−

++ ++++= uyyy 1

11

1 ...α

then

( )tsstst

sst

ss

sp

ss

FF

FB

FFI

vvvu 1111

11

11

1

...

)...(

+++=

=

−−−=

−+++

++

α



18

Define ( ) ( ) thttthttt XVY ;',...,;',..., 11 ++++ ≡≡ vvyy Under the assumption that the DGP is a VAR(p), consider the system:

Φ+Ψ= ttt VXY with

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=Ψ

n

hn

hpp

h

I

FI

FF

FF

L

MMMMMM

0

...;

...

... 1

1

11

1



19

Further define:

( ) Σ≡ΦΩ⊗Φ=Ω= ')'(then,)'( vhttvtt IVVEE vv then

( ) ( )[ ] ( ) )('')ˆ( 111 YvecXIXIXIvec −−− Σ⊗⊗Σ⊗=Ψ

The usual impulse responses and their correct standard errors are rows 1-n and columns 1-nh of Ψ



20

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 across responses.



21

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 LΣ the HAC, VCV matrix of sB1ˆ in the linear projection, then a 95% CI is ( )iLi dd Σ± ˆ'96.1

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

Linear projections are a type of general misspecification test!



22

The Constrains of Linearity Symmetry

Shape invariance

History independence

Multidimensionality



23

Flexible Local Projections Generally,

,...),( 1−Φ= ttt vvy A Taylor series approximation to Φ is the Volterra series expansion (the non-linear Wold):

y t ∑ i0 iv t−i ∑ i0

∑ j0 ijv t−iv t−j

∑ i0 ∑ j0

∑ k0 ijk v t−iv t−jv t−k . . .



24

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

sstpt

spt

st

st

st

ssst

BB

CQB

+−+

−+

−+

−+

−+

+

+++

++++=

uyy

yyyy1

21

2

31

11

21

111

11

...

α

Hence

( )( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

++

+++=

−−

−32

12

11

2111

33ˆ

2ˆˆ),,(

iitits

iits

is

iC

QBstIR

ddydy

ddydd



25

Remarks

IRs no longer symmetric nor shape invariant IRs no longer history independent – they depend on

1−ty . Evaluate at yy =−1t to evaluate at linearity. Define

)'33 2 ( 321

21

21 iititiitii ddydyddyd +++≡ −−−λ then

a 95% CI is approximately

( )iCi λλ Σ± ˆ96.1 '



26

Flexible projections in general Since what matters are the terms associated with 1−ty (but not the remaining lags), then

ssttt

sst Xm +−−+ += uyy );( 11

Thus, any parametric, semi-parametric and non-parametric conditional mean estimator will do. Notice this can be done univariately.



27

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)



28

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 3. Check standard errors from 2: Efficiency



29

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


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


Response of PCOM

-.4

-.2

.0

.2

.4

.6

.8

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


Response of FF

-.008

-.006

-.004

-.002

.000

.002

.004

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


Response of NBRX

-.3

-.2

-.1

.0

.1

.2

.3

.4

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


Response of M2



30

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

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

Linear Cubic

Response of FF

-.008

-.006

-.004

-.002

.000

.002

.004

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

Linear Cubic

Response of NBRX

-.3

-.2

-.1

.0

.1

.2

.3

.4

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

Linear Cubic

Response of M2



31

Experiment 3 EM P PCOM s

True-MC

Newey-West

(Linear)

Newey-West

(Cubic)

True-MC

Newey-West

(Linear)

Newey-West

(Cubic)

True-MC

Newey-West

(Linear)

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… … … … … … … … … …24 0.077 0.087 0.095 0.0006 0.0009 0.0010 0.078 0.088 0.096



32

Nonlinearities Simulate:

tttttt

t

t

t

tt

t

t

t

t

t

t

t

huhuh

INh

Bhyyy

Ayyy

111,12

11

3

2

11

1

13

12

11

3

2

1

;5.03.05.0

),0(~ ;

ε

εεεε

=++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

−

−



33

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



34

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

with 8, 900 Mhz processors and 16GB RAM



35

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

TRUENo GARCHVAR

Linear ProjectionCubic ProjectionBayesian VAR

Response of Y1 Response of Y2

Response of Y3



36

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

percent at annual rate



37

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.



38

The Data

-12

-8

-4

0

4

8

55 60 65 70 75 80 85 90 95 00

Output Gap

0

4

8

12

16

20

55 60 65 70 75 80 85 90 95 00

Federal Funds Rate

0

2

4

6

8

10

12

14

55 60 65 70 75 80 85 90 95 00

Inflation

4.75%

6%



39

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 Inf lation Regime



40

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

Applications to non-Gaussian data



41

An Efficient Moment-Matching Estimator for RE models Example (Fuhrer and Olivei, 2004):

( ) ttttttt xEzEzz εγµβµ ++−+= +− 11

• 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



42

The usual solution approach Assume: ttt uxx += −1α Conjecture: 111 −−− ++= tttt dcxbzz ε Undetermined Coeffs.: 1=d

( )( )αµβγα

+−−=

bc

1

( ) 02 =−+−− µµβ bb



43

An alternative Assume: ∑ ∑∞

=

∞

= −− +=0 0i i itiitit ux θερ

Conjecture: ∑ ∑∞

=

∞

= −− +=0 0i i itiitit ubaz ε

U.C.: ( ) ( )

( )( ) ssss

ssss

bbbaaa

bbaa

γθµβµγρµβµ

µβγβ

+−+=+−+=

−=+−=

+−

+−

11

11

1010

... ;1



44

Define: ( )( )( )( )'...,,0,,...,,0

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

',...,,,...,,1

1,1113

11112

10101

010

−−

++

−−

===

−=

hh

hh

hh

hh

XbbaaX

bbaaXbbaaaY

θθρρ

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

( ) ωγµβµ =+−+− 321 )( XXXY



45

Let the covariance matrix of the impulse responses be:

YYVAR Ω=)( Then, an efficient estimator for β,µ, and γ is:

ωω YΩ'min

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



46

Identification with Long-Run Restrictions Blanchard and Quah Let ∑∞

= −==0

)(i ititt ALA εεy with IE =εε ' .

B-Q impose zero coefficient restrictions on A(1). Using the reduced form ∑∞

= −==1

)(i ititt CLC uuy

and Σ=uu'E we can recover the matrix D.



47

How to estimate an approximation to C(1) with linear projections? Option 1: Add the estimated linear projection coefficients,

∑ =≅ h

ssBC

1 1)1(ˆ

for h “large.”



48

Option 2: Define: ∑ = ++ =

h

s stht 0yY .

Then

hhtpt

hpt

hht GG +−−+ +++= uyyY ...11

from which hG1ˆ is an estimate of ∑ =

h

i iC1

. Choose h “large.”



49

Example:

ttt yy ε+= −175.0

-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.



50

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 Regional Development Fund?



51

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



54

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

These are the same coefficients as in the Tables!



55

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: practical issues · impulse responses by local projections:...

Documents