Documentos de Trabajo N.º 1944 - Bank of Spain...Documentos de Trabajo. N.º 1944 2019 (*) We thank...
Transcript of Documentos de Trabajo N.º 1944 - Bank of Spain...Documentos de Trabajo. N.º 1944 2019 (*) We thank...
DYNAMIC EFFECTS OF PERSISTENT SHOCKS
Mario Alloza, Jesús Gonzalo and Carlos Sanz
Documentos de Trabajo N.º 1944
2019
DYNAMIC EFFECTS OF PERSISTENT SHOCKS
Documentos de Trabajo. N.º 1944
2019
(*) We thank Fabio Canova, Jesús Fernández-Villaverde, Alessandro Galesi, Gergely Gánics, Juan F. Jimeno, Mikkel Plagborg-Møller, Juan Rubio-Ramírez, Enrique Sentana, and seminar participants at the I Workshop of the Spanish Macroeconomics Network (Universidad Pública de Navarra), Banco de España, CFE 2018 (University of Pisa), II Workshop in Structural VAR models (Queen Mary University of London), VII Workshop on Empirical Macroeconomics (Ghent University), 2019 American Meeting of the Econometric Society (University of Washington), and 2019 edition of the Padova Macro Talks for insightful comments. Alloza: [email protected]. Gonzalo: [email protected]. Sanz: [email protected].
Mario Alloza and Carlos Sanz
BANCO DE ESPAÑA
Jesús Gonzalo
UNIVERSIDAD CARLOS III DE MADRID
DYNAMIC EFFECTS OF PERSISTENT SHOCKS (*)
The Working Paper Series seeks to disseminate original research in economics and fi nance. All papers have been anonymously refereed. By publishing these papers, the Banco de España aims to contribute to economic analysis and, in particular, to knowledge of the Spanish economy and its international environment.
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Reproduction for educational and non-commercial purposes is permitted provided that the source is acknowledged.
© BANCO DE ESPAÑA, Madrid, 2019
ISSN: 1579-8666 (on line)
Abstract
We show that several shocks identifi ed without restrictions from a model, and frequently
used in the empirical literature, display some persistence. We demonstrate that the two
leading methods to recover impulse responses to shocks (moving average representations
and local projections) treat persistence differently, hence identifying different objects.
In particular, standard local projections identify responses that include an effect due to
the persistence of the shock, while moving average representations implicitly account
for it. We propose methods to re-establish the equivalence between local projections and
moving average representations. In particular, the inclusion of leads of the shock in local
projections allows to control for its persistence and renders the resulting responses
equivalent to those associated to counterfactual non-serially correlated shocks. We apply
this method to well-known empirical work on fi scal and monetary policy and fi nd that
accounting for persistence has a sizable impact on the estimates of dynamic effects.
Keywords: impulse response function, local projection, shock, fi scal policy, monetary policy.
JEL classifi cation: C32, E32, E52, E62.
Resumen
En este documento mostramos que varios shocks identifi cados sin restricciones de un
modelo, y usados frecuentemente en la literatura empírica, son persistentes. Demostramos
que los dos principales métodos para recuperar respuestas a impulsos de shocks
(representaciones de media móvil y proyecciones locales) tratan la persistencia de distinta
manera y, por tanto, identifi can objetos diferentes. En particular, las proyecciones locales
estándar identifi can respuestas que incluyen un efecto debido a la persistencia del
shock, mientras que las representaciones de media móvil implícitamente controlan por
la persistencia. Proponemos métodos para restablecer la equivalencia entre proyecciones
locales y representaciones de media móvil. En particular, la inclusión de adelantos del
shock en las proyecciones locales permite controlar por su persistencia y hace que las
respuestas resultantes sean equivalentes a las asociadas a shocks contrafactuales no
autocorrelacionados. Aplicamos este método a trabajos empíricos sobre política fi scal y
monetaria y encontramos que controlar por la persistencia tiene un impacto considerable
sobre las estimaciones de efectos dinámicos.
Palabras clave: función de respuesta a un impulso, proyecciones locales, shock, política
fi scal, política monetaria.
Códigos JEL: C32, E32, E52, E62.
BANCO DE ESPAÑA 7 DOCUMENTO DE TRABAJO N.º 1944
h
t + h t
BANCO DE ESPAÑA 8 DOCUMENTO DE TRABAJO N.º 1944
t t + h
t
t + h
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BANCO DE ESPAÑA 10 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 11 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 12 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 13 DOCUMENTO DE TRABAJO N.º 1944
yt = δxt + ut
xt = γxt−1 + εt,
BANCO DE ESPAÑA 14 DOCUMENTO DE TRABAJO N.º 1944
xt
yt xt
E(xtut) = 0 ut εt
ut ∼ (μu, σ2u) εt ∼ (με, σ
2ε)
γ δ
xt yt
yt xt
t
R(h) h
R(h) = E [yt+h|xt = 1,Ωt−1]− E [yt+h|xt = 0,Ωt−1] ,
Ωt−1 εt xt
t − 1
xt γ �= 0 xt xt+j = 0
xt
R(0) =∂yt∂xt
= δ
R(1) =∂yt+1
∂xt
= δγ
R(2) =∂yt+2
∂xt
= δγ2
. . .
R(h)∗
R(h)∗ = E [yt+h|xt = 1, xt+1, ..., xt+h,Ωt−1]− E [yt+h|xt = 0, xt+1, ..., xt+h,Ωt−1] .
xt
γ
xt
R(h) = E [yt+h|εt = 1, εt+1 = 0, ..., εt+h = 0,Ωt−1] −E [yt+h|εt = 0, εt+1 = 0, ..., εt+h = 0,Ωt−1]
BANCO DE ESPAÑA 15 DOCUMENTO DE TRABAJO N.º 1944
h = 0, 1, . . . , H
ξt+h
γ = 0 R(h) = R(h)∗ ∀ h
γ �= 0 R(h) �= R(h)∗ ∀ h > 0
R(h)MA
yt = θ0xt + θ1xt−1 + θ2xt−2 + θ3xt−3 + θ4xt−4 + . . .+ et,
R(h)MA = ∂yt+h
∂xt= θh ∀ h
h = 0, 1, . . . , H
yt+h = δhxt + ξt+h,
{δh}Hh=0
R(h)LP = δh h
xt
γ = 0 R(h)MA = R(h)LP = R(h)∗ = R(h) ∀h
γ = 0
R(0)∗ =∂yt∂xt
= δ
R(1)∗ =∂yt+1
∂xt
∣∣∣xt+1
= 0
R(2)∗ =∂yt+2
∂xt
∣∣∣xt+1,xt+2
= 0
. . .
BANCO DE ESPAÑA 16 DOCUMENTO DE TRABAJO N.º 1944
yt yt = ρyt−1+ δxt+ut
δ1 = δρ+ δγ
h > 0
γ �= 0 h = 0 R(h)MA = R(h)LP = R(h)∗ = R(h)
γ �= 0 h ≥ 1 R(h)MA = R(h)∗ �= R(h)LP = R(h)
γ �= 0
xt yt+h
xt yt
xt yt+h xt+h cov(xt, xt+h) �= 0 γ �= 0
γ
xt R(h)MA R(h)LP
R(h)LP = R(h) = δγh R(h)MA = R(h)∗ = 0 h ≥ 1
xt h = 1
yt+1 = δ1xt + ξt+1,
δ1 = R(1)LP xt yt+1 xt δ1
γ �= 0 yt+1 xt
yt+1 = δxt+1 + ut+1
= δ (γxt + εt+1) + ut+1
= δγxt + u∗t+1,
u∗t+1 = δεt+1+ut+1 δ1
xt δ1 = δγ t
t+1 xt xt+1 γ �= 0 xt+1
δ1
xt
t+ 1
yt+1 = θ0xt+1 + θ1xt + θ2xt−1 + θ3xt−2 + θ4xt−3 + . . .+ et+1.
θh
h = 1 R(1)MA = θ1
BANCO DE ESPAÑA 17 DOCUMENTO DE TRABAJO N.º 1944
yt xt = 1t xt = 0
γ
( )∂yt+1
∂xt= δγ θ1
∂yt+1
∂xt
∣∣∣xt+1
= 0
xt+1 xt
R(h)MA = E [yt+h|xt = 1,Ωt−1, xt+h−1, ..., xt+1]− E [yt+h|xt = 0,Ωt−1, xt+h−1, ..., xt+1] ,
R(h)LP = E [yt+h|εt = 1,Ωt−1]− E [yt+h|εt = 0,Ωt−1] .
RLP RMA γ > 0 γ < 0
γ
RLP (h) �= RMA(h) γ �= 0 h ≥ 1
xt
xt
εt = xt − γxt−1.
R∗(h) xt εt
t t + h
xt+h
BANCO DE ESPAÑA 18 DOCUMENTO DE TRABAJO N.º 1944
yt+h = δh,0xt + δh,1xt+1 + ξt+h,
δh,0 h xt
RF (h)
xt
R(h)F = R(h)∗ = R(h)MA ∀ γ h
xt
δh,0∂xt+1
∂xt�= 0 γ �= 0
h
h
R(h)MA = R(h)∗ γ
R(h)
εt xt
xt
yt = δγtx0 + δt∑
i=0
γiεt−i + ut.
yt εt R(h)MA−per
θ̃h
yt = θ̃0εt + θ̃1εt−1 + θ̃2εt−2 + θ̃3εt−3 + θ̃4εt−4 + . . .+ et.
yt εt
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E(εt−sut−r) = 0 ∀s, r ≥ 0 ut εt N (0, 1)
B0 = 1.5 B1 = 1 ρ = 0.9 σ2ε = σ2
u
ρ xt yt
B1
yt xt
yt+h = ρyt−1 + βh,0xt + βh,1xt−1 + βh,fxt+1 + ξt+h.
θ̃h = δh ∀ h
xt
θ0 = θ̃0 = δ θ1 = θ̃1 − γθ̃0, . . . , θh = θ̃h − γθ̃h−1 yt+1
xt δ1 = θ̃1 xt yt+1
yt+1 xt γδ
xt
xt
R(h)MA−per = R(h) = R(h)LP ∀ γ h
yt = ρyt−1 +B0xt +B1xt−1 + ut
xt = γxt−1 + εt,
γ = 0yt
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γ = 0.2ρ
γ = 0
βh,f = 0 γ = 0.2 βh,f = 0
γ = 0.2 βh,f �= 0
xt
B1 xt
β̂1,0 = 1.5
ρ B1
R̂(h)LP → R(h)∗
xt
yt
βh,f = 0
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
= 0 0 0 & leads
γ = 0γ = 0.2
γ = 0.2
R̂(h)LP > R(h)∗
h > 0 xt
γ ρ
BANCO DE ESPAÑA 21 DOCUMENTO DE TRABAJO N.º 1944
xt
R̂(h)F → R(h)∗
xt
xt
R̂MA(h) → R∗(h) γ
ρ < 1
yt = (1− ρL)−1 (B0 +B1L) xt + (1− ρL)−1 ut,
L
ut xt
yt xt ϑh
yt = ϑ0xt + ϑ1xt−1 + ϑ2xt−2 + ϑ3xt−3 + . . .+ ϑHxt−H + ξt,
H
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
γ = 0γ = 0.2
xt ε̂t εt
BANCO DE ESPAÑA 22 DOCUMENTO DE TRABAJO N.º 1944
γ = 0
γ = 0.2
xt ε̂t
xt ϑh
xt γ
R̂(h)MA → R(h)∗ γ
R̂(h)LP γ = 0 R̂(h)F
εt
xt
xt εt
xt
γ �= 0 R̂(h)MA−per = R̂(h)LP → R(h)
zt
zt
yt = βgt + ut
ut = mt + at
gt = λxt + (1− λ)mt
zt = xt + νt
xt = γxt−1 + εt,
at νt εt N (0, 1)
gt yt
BANCO DE ESPAÑA 23 DOCUMENTO DE TRABAJO N.º 1944
gt
mt zt
λ �= 0
xt γ �= 0
β = 2
λ γ gt yt
γ = 0 γ = 0.2
1 2 3 4 50
0.5
1
1.5
2
2.5
3
No endogeneityOLSIV
1 2 3 4 50
0.5
1
1.5
2
2.5
3
No endogeneity, =0OLS, =0.2IV, =0.2IV with leads, =0.2
γ = 0 γ = 0.2
γ = 0 λ = 1
gt yt
λ = 0.5
γ = 0
gt
zt
gt
xt γ = 0
γ = 0.2
BANCO DE ESPAÑA 24 DOCUMENTO DE TRABAJO N.º 1944
γ = λ = 1
gt
zt
β = 2
BANCO DE ESPAÑA 25 DOCUMENTO DE TRABAJO N.º 1944
R(h)∗
R(h)∗
R(h) xt εt
γ �= 0
R(h)∗
R(h)
R(h)∗
R(h)∗
R(h)
BANCO DE ESPAÑA 26 DOCUMENTO DE TRABAJO N.º 1944
18.4%
yt gt
yt,h = βyhshockt +
P∑j=1
ρzj,hzt−j +h∑
f=1
γf,hshockt+f + ξt
gt,h = βghshockt +
P∑j=1
ρzj,hzt−j +h∑
f=1
γf,hshockt+f + εt,
BANCO DE ESPAÑA 27 DOCUMENTO DE TRABAJO N.º 1944
h
zt P yt gt shockt
h shockt
h h
γf,h = 0 ∀ f, h
γf,h �= 0
Mt,h =
∑hi=1 β
yh∑h
i=1 βgh
.
BANCO DE ESPAÑA 28 DOCUMENTO DE TRABAJO N.º 1944
0 5 10 15 20
quarter
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35GDP
0 5 10 15 20
quarter
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45GOV
h
xt+h = St−1
[αA,h +
P∑j=1
ρA,j,hzt−j + βA,hshockt +h∑
f=1
δA,f,hshockt+f
]+
(1− St−1)
[αB,h +
P∑j=1
ρA,j,hzt−j + βB,hshockt +h∑
f=1
δB,f,hshockt+f
]+ ξt+h,
xt St
St = 1 St = 0
δA,f,h = δB,f,h = 0 ∀f, h
BANCO DE ESPAÑA 29 DOCUMENTO DE TRABAJO N.º 1944
0 5 10 15 20
quarter
-0.1
0
0.1
0.2
0.3GDP - EXPANSION
0 5 10 15 20
quarter
-0.1
0
0.1
0.2
0.3
0.4GOV - EXPANSION
0 5 10 15 20
quarter
-0.4
-0.2
0
0.2
0.4
0.6
0.8GDP - RECESSION
0 5 10 15 20
quarter
-0.5
0
0.5
1GOV - RECESSION
h
BANCO DE ESPAÑA 30 DOCUMENTO DE TRABAJO N.º 1944
St
St
St
0 2 4 6 8 10 12 14 16 18 20
quarter
-2
-1.5
-1
-0.5
0
0.5
1
cum
ulat
ive
mul
tiplie
r
EXP, no leadsEXP, with leadsREC, no leadsREC, with leads
BANCO DE ESPAÑA 31 DOCUMENTO DE TRABAJO N.º 1944
yi,t μh,i
λh,t Xt
βh,0
h
βh,f = 0 ∀h, f
βh,f �= 0
yi,t+h = μh,i + λh,t + βh,0shocki,t +h∑
f=1
βh,fshocki,t+f + βh,sXi,t + ξi,t+h,
BANCO DE ESPAÑA 32 DOCUMENTO DE TRABAJO N.º 1944
βh,f = 0 ∀h, f
βh,f �= 0 h
yt+h − yt−1yt−1
= βh,0shockt +h∑
f=1
βh,fshockt+f + ξt+h.
0 1 2 3 4 5 6 7 8
year
-2
-1.5
-1
-0.5
0
0.5
perc
ent
GDP
BANCO DE ESPAÑA 33 DOCUMENTO DE TRABAJO N.º 1944
βh,f = 0 ∀h, f
βh,f �= 0
0 2 4 6 8 10 12
quarter
-5
-4
-3
-2
-1
0
1
2
3
perc
ent
GDP
βh,f = 0
βh,f �= 0h
BANCO DE ESPAÑA 34 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 35 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 36 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 37 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 38 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 39 DOCUMENTO DE TRABAJO N.º 1944
yt+h = δhxt + ξt+h,
δh = R(h)LP xt yt+h
δh
δh =cov(yt+h, xt)
var(xt).
xt yt+h
yt = θ0xt + θ1xt−1 + θ2xt−2 + θ3xt−3 + θ4xt−4 . . .+ ut.
∀t
yt+h = θ0xt+h + θ1xt+h−1 + θ2xt+h−2 + θ3xt+h−3 + . . .+ θhxt + ut,
θh = R(h)MA h
θh =cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
.
γ = 0 xt = εt ∼ (με, σ2ε)
θh =cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
=cov(yt+h, xt)
var(xt).
δh = θh R(h)LP = R(h)MA ∀hR(h)MA = R(h)∗ ∀h, γ
R(h)∗ =∂yt+h
∂xt
∣∣∣xt+1,...,xt+h
=cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
.
γ �= 0
δh =cov(yt+h, xt)
var(xt)=
cov(δxt+h + ut+h, xt)
var(xt)= δ
cov(xt+h, xt)
var(xt)= δγh,
xt+h = γhxt +h−1∑j=0
γjεt+h−j.
BANCO DE ESPAÑA 40 DOCUMENTO DE TRABAJO N.º 1944
xt
θh =cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
= δcov(xt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
.
h = 0 θ0 = δ cov(xt,xt)var(xt)
= δ h > 0
θh = δ cov(xt+h,xt)
var(xt)
∣∣∣xt+1,...,xt+h
= 0 γ �= 0 R(h)LP =
R(h)MA h = 0 �
yt+h = δh,0xt + δh,1xt+1 + ξt+h,
δh,0 = R(h)F xt yt+h
δh,0
δh,0 =cov(yt+h, xt)
var(xt)
∣∣∣xt+1
= δcov(xt+h, xt)
var(xt)
∣∣∣xt+1
.
xt
xt+1
θh =cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
= δcov(xt+h, xt)
var(xt)
∣∣∣xt+1
= δh,0.
h = 0
θ0 =cov(yt, xt)
var(xt)
∣∣∣xt+1,...,xt+h
=cov(yt, xt)
var(xt)= δ = δ0,0.
h > 0
δh,0 = δcov(xt+h, xt)
var(xt)
∣∣∣xt+1
= δγh−1 cov(xt+1, xt)
var(xt)
∣∣∣xt+1
= 0.
θh =cov(yt+h, xt)
var(xt)
∣∣∣xt+1,...,xt+h
= δγh−1 cov(xt+1, xt)
var(xt)
∣∣∣xt+1
= 0.
R(h)F = R(h)MA ∀ h γ R(h)MA = R(h)∗
�
BANCO DE ESPAÑA 41 DOCUMENTO DE TRABAJO N.º 1944
yt = θ̃0εt + θ̃1εt−1 + θ̃2εt−2 + θ̃3εt−3 + θ̃4εt−4 . . .+ ut,
θ̃0 = R(h)MA−per εt yt+h εt
yt εt
εt ∼ (με, σ2ε)
θ̃h =cov(yt+h, εt)
var(εt)
∣∣∣εt+1,...,εt+h
=cov(yt+h, εt)
var(εt)= δ
cov(xt+h, εt)
var(εt).
θ̃h = δh R(h)MA−per =
R(h)LP xt+h
θ̃h = δγhcov(xt +
∑h−1j=0 γ
jεt+h−j, εt)
var(εt)= δγh cov(xt, εt)
var(εt)= δγh.
θ̃h = δh ∀ h �
BANCO DE ESPAÑA 42 DOCUMENTO DE TRABAJO N.º 1944
xt xt
εt = xt −γxt−1 εt xt
B1 = 0
yt+h = ρhyt−1 + λhεt + ξt+h.
yt εt
εt
h = 0 h = 1
λ0 =∂yt+1
∂εt= B0
λ1 =∂yt+1
∂εt= ρ
∂yt∂εt
+B0∂xt+1
∂εt= ρB0 +B0γ = B0(γ + ρ).
xt
xt yt+1 yt+2
yt xt
B1 �= 0 ε̂t
εt
yt+h = ρyt−1 + λh,0ε̂t + λh,1ε̂t−1 + ξt+h.
ε̂t
γ = 0
BANCO DE ESPAÑA 43 DOCUMENTO DE TRABAJO N.º 1944
ε̂t
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
γ = 0γ = 0.2
yt = ρyt−1 + δ0xt + δ1xt−1 + εyt
xt = γxt−1 + εxt .
A0Yt = B∗Yt−1 + εt
[1 0
−δ0 1
][xt
yt
]=
[γ 0
δ1 ρ
][xt−1yt−1
]+
[εxt
εyt
].
BANCO DE ESPAÑA 44 DOCUMENTO DE TRABAJO N.º 1944
Yt = BYt−1 + ut,
B = A−10 B∗ ut = A−1
0 εt
xt
ut
γ �= 0 yt εxt
xt ρ = δ1 = 0
xt
yt R(h)V AR = δ0∑∞
k=0 γkεxt−k
xt γ �= 0
ρ = 0.9 δ0 = 1.5 δ1 = 1 γ = 0 γ = 0.2
yt xt εxt
yt
γ xt
xt xt
xt
xt
xt
yt xt
yt xt
xt xt
γ
BANCO DE ESPAÑA 45 DOCUMENTO DE TRABAJO N.º 1944
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
VAR = 0VAR = 0.2VAR-X = 0VAR-X = 0.2
yt εxtγ
εxt
yt εxt xt εxt
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
LP = 0LP = 0.2VAR = 0VAR = 0.2
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
VAR = 0VAR = 0.2
yt xt εxtγ
γ = 0 γ = 0.2γ = 0
γ = 0.2
BANCO DE ESPAÑA 46 DOCUMENTO DE TRABAJO N.º 1944
yt xt
yt = ρyt−1 +B0xt +B1xt−1 + ut,
xt
ut ut ∼ N (0, 1) ρ = 0.9
B0 = 1.5 B0 = 1
497
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
3
3.5
True IRFLP - 0 forwardsLP - 1 forwardLP - 20 forwards
BANCO DE ESPAÑA 47 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 48 DOCUMENTO DE TRABAJO N.º 1944
yt
shockt
xt
yt+h = βh,0shockt + θh(L)xt + βh,f
h∑f=1
shockt+f + ξt+h,
BANCO DE ESPAÑA 49 DOCUMENTO DE TRABAJO N.º 1944
0 10 20 30 40 50
month
-3
-2
-1
0
1
2
3
perc
ent
FEDERAL FUNDS
0 10 20 30 40 50
month
-4
-3
-2
-1
0
1
2
perc
ent
INDUSTRIAL PRODUCTION
0 10 20 30 40 50
month
-0.5
0
0.5
1
perc
ent
UNEMPLOYMENT
0 10 20 30 40 50
month
-4
-3
-2
-1
0
1
perc
ent
CPI
h
βh,f = 0 ∀h, f
βh,f �= 0
BANCO DE ESPAÑA 50 DOCUMENTO DE TRABAJO N.º 1944
xt βh,f = 0
βh,f �= 0
0 10 20 30 40 50
month
-3
-2
-1
0
1
2
3
perc
ent
FEDERAL FUNDS
0 10 20 30 40 50
month
-4
-3
-2
-1
0
1
2
perc
ent
INDUSTRIAL PRODUCTION
0 10 20 30 40 50
month
-0.5
0
0.5
1
1.5
perc
ent
UNEMPLOYMENT
0 10 20 30 40 50
month
-4
-3
-2
-1
0
1
2
perc
ent
CPI
h
BANCO DE ESPAÑA 51 DOCUMENTO DE TRABAJO N.º 1944
yt+h = βh,0shockt + θh(L)xt + βh,f
min{h,12}∑f=1
shockt+f + ξt+h,
yt
shockt
xt
h
h < 12 12
βh,f = 0
∀h, f βh,f �= 0
BANCO DE ESPAÑA 52 DOCUMENTO DE TRABAJO N.º 1944
0 10 20 30 40 50
month
-2
0
2
4
6
8
perc
ent
ONE-YEAR RATE
0 10 20 30 40 50
month
-0.05
0
0.05
0.1
0.15
0.2
0.25
perc
ent
INDUSTRIAL PRODUCTION
0 10 20 30 40 50
month
-4
-2
0
2
4
perc
ent
EXCESS BOND PREMIUM
0 10 20 30 40 50
month
-0.06
-0.04
-0.02
0
0.02
0.04
perc
ent
CPI
hh = 12 12
BANCO DE ESPAÑA 53 DOCUMENTO DE TRABAJO N.º 1944
BANCO DE ESPAÑA 54 DOCUMENTO DE TRABAJO N.º 1944
−0.
30−
0.20
−0.
100.
000.
100.
20A
utoc
orre
latio
ns o
f sho
ck
0 10 20 30 40Lag
−0.
20−
0.10
0.00
0.10
0.20
Aut
ocor
rela
tions
of s
hock
0 10 20 30 40Lag
−0.
20−
0.10
0.00
0.10
0.20
0.30
Aut
ocor
rela
tions
of s
hock
0 10 20 30 40Lag
−0.
100.
000.
100.
200.
30A
utoc
orre
latio
ns o
f sho
ck
0 10 20 30 40Lag
−0.
20−
0.10
0.00
0.10
0.20
Aut
ocor
rela
tions
of s
hock
0 10 20 30 40Lag
−0.
20−
0.10
0.00
0.10
0.20
Aut
ocor
rela
tions
of s
hock
0 10 20 30 40Lag
BANCO DE ESPAÑA 55 DOCUMENTO DE TRABAJO N.º 1944
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
date
-1.5
-1
-0.5
0
0.5
1
1.5
2
perc
ent
1975 1980 1985 1990 1995 2000 2005
date
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
perc
ent
1990 1995 2000 2005 2010
date
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
rate
1900 1920 1940 1960 1980 2000
date
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
perc
ent
1970 1975 1980 1985 1990 1995
date
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
perc
enta
ge p
oint
s
1950 1960 1970 1980 1990 2000
date
-2
-1.5
-1
-0.5
0
0.5
1
perc
ent
BANCO DE ESPAÑA 56 DOCUMENTO DE TRABAJO N.º 1944
0 5 10 15 20
quarter
-0.1
0
0.1
0.2
0.3
0.4
0.5GDP
0 5 10 15 20
quarter
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7GOV
h
0 2 4 6 8 10 12 14 16 18 20
quarter
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
cum
ulat
ive
mul
tiplie
r
BANCO DE ESPAÑA 57 DOCUMENTO DE TRABAJO N.º 1944
0 2 4 6 8 10 12 14 16 18 20
quarter
-2
-1.5
-1
-0.5
0
0.5
1
cum
ulat
ive
mul
tiplie
r
EXP, no leadsEXP, with leadsEXP, with leads (fixed state)REC, no leadsREC, with leadsREC, with leads (fixed state)
0 1 2 3 4 5 6 7 8
year
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
perc
ent
CONSUMPTION
βh,f = 0
βh,f �= 0 h
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Unidad de Servicios AuxiliaresAlcalá, 48 - 28014 Madrid
E-mail: [email protected]