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
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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
Estimation and Inference of Impulse Responses by Local Projections
Òscar Jordà U.C. Davis
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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
Estimation and Inference of Impulse Responses by Local Projections
Òscar Jordà U.C. Davis
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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!
Estimation and Inference of Impulse Responses by Local Projections
Òscar Jordà U.C. Davis
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Intuition
VAR
DGP
Projection
Estimation and Inference of Impulse Responses by Local Projections
Òscar Jordà U.C. Davis
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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 models is straight-forward
Estimation and Inference of Impulse Responses by Local Projections
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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=
=×
−−
Estimation and Inference of Impulse Responses by Local Projections
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Hence, consider
sstpt
spt
ssst BB +−
+−
++ ++++= uyyy 1
11
1 ...α
so that
hsBstIR is
i ,...2,1,0 ˆ),,( 1 == dd
Estimation and Inference of Impulse Responses by Local Projections
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Example: AR(1) vs. Local Projection
411 4.05.0 −−− +−+= ttttt yy εεερ ; T = 180; p = 1, 100 reps
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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Variance Decompositions
1'1 )'())|((
)'())|((
,...,1,0 )|(
−++
−+
+++
+++
=
=
==−
DEDXEMSE
EXEMSE
hsXE
sst
ssttst
sst
ssttstu
ssttstst
uuy
uuy
uyy
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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 == +
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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),,( =
Estimation and Inference of Impulse Responses by Local Projections
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IR coefficients from a VAR(p)
F11 1
F12 1F1
1 2
F1s 1F1
s−1 2F1s−2 . . . pF1
s−p
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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 across responses.
Estimation and Inference of Impulse Responses by Local Projections
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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 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!
Estimation and Inference of Impulse Responses by Local Projections
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The Constrains of Linearity Symmetry
Shape invariance
History independence
Multidimensionality
Estimation and Inference of Impulse Responses by Local Projections
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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 . . .
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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 '
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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)
Estimation and Inference of Impulse Responses by Local Projections
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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 3. Check standard errors from 2: Efficiency
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
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
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−
−
−
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
with 8, 900 Mhz processors and 16GB RAM
Estimation and Inference of Impulse Responses by Local Projections
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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
TRUENo GARCHVAR
Linear ProjectionCubic ProjectionBayesian VAR
Response of Y1 Response of Y2
Response of Y3
Estimation and Inference of Impulse Responses by Local Projections
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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
percent at annual rate
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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%
Estimation and Inference of Impulse Responses by Local Projections
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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 Inflation Regime
High Inflation Regime
High Inf lation Regime
Estimation and Inference of Impulse Responses by Local Projections
Òscar Jordà U.C. Davis
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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
Applications to non-Gaussian data
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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The usual solution approach Assume: ttt uxx += −1α Conjecture: 111 −−− ++= tttt dcxbzz ε Undetermined Coeffs.: 1=d
( )( )αµβγα
+−−=
bc
1
( ) 02 =−+−− µµβ bb
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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.”
Estimation and Inference of Impulse Responses by Local Projections
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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.”
Estimation and Inference of Impulse Responses by Local Projections
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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.
Estimation and Inference of Impulse Responses by Local Projections
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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 Regional Development Fund?
Estimation and Inference of Impulse Responses by Local Projections
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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
Estimation and Inference of Impulse Responses by Local Projections
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Estimation and Inference of Impulse Responses by Local Projections
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Estimation and Inference of Impulse Responses by Local Projections
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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
These are the same coefficients as in the Tables!
Estimation and Inference of Impulse Responses by Local Projections
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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)