structural var modelsmatthieustigler.github.io/lectures/lect8svar.pdf · 2020. 7. 16. · 1...
TRANSCRIPT
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Structural VAR models
Matthieu [email protected]
December 9, 2008
Matthieu Stigler [email protected] () Structural VAR models December 9, 2008 1 / 33
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Lectures list
1 Stationarity
2 ARMA models for stationary variables
3 Some extensions of the ARMA model
4 Non-stationarity
5 Seasonality
6 Non-linearities
7 Multivariate models
8 Structural VAR models
9 Cointegration the Engle and Granger approach
10 Cointegration 2: The Johansen Methodology
11 Multivariate Nonlinearities in VAR models
12 Multivariate Nonlinearities in VECM models
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1 Lectures
2 Impulse response function
3 Structural VAR modelsStuctural vector autoregressive model (SVAR)
Choleski decompositionBlanchard-Quah decomposition
Impulse-response functionForecast variance decomposition
4 Impulse response functionStable VAR case
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VMA representation
VAR(1) to VMA(∞):
yt = A1yt−1 + εt (1)
= A0 +∞∑i=0
Ai1εt−i (2)
VAR(p) in VAR(1) to VMA(∞):
Yt = A1Yt−1 + Et (3)
=∞∑i=0
Ai1Et−i (4)
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Interpretation of the VMA(∞) coefficient matrices
From the VMA(∞) of a VAR(1):
yt = A0 +∞∑i=0
Ai1εt−i
Elements of Ai represents effects of unit shocks in the variables after iperiods.Interpretation: the εt can be seen as 1 step-ahead forecast error so arecalled forecasr error impulse responses.
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> library(vars)
> data(Canada)
> var<- VAR(Canada[,c("e","rw")], p = 2, type = "const")
> imp<-irf(var, boot=FALSE, ortho=FALSE,n.ahead=200)
> plot(imp)
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Scaling of the impulse:
rescale the axes with 1 =√σ2
y
Relation with Granger causality:IRF of y1 to yi i 6= 1 is zero if y1 does not Granger cause the othersvariables
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Cumulated IRF
From MA representation:
yt = A0 +∞∑i=0
Φiεt−i
We now want to know the cumulated impact:Ψn =
∑ni=0 Φi accumulated responses over n periods
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problem with the interpretation
Previous assumption: schocks are independent (we force other shocks tobe zero).
But shocks may be correlated! See the var-cov matrix of the residuals.
So impulse is composite effect and interpretation is not direct.
Solution: Create independant (orthogonal) residuals.
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Triangular and Choleski decomposition of matrices
Theorem (Triangular decomposition)
Any positive definite symmetric matrix A has a unique representation ofthe form:
A = BDB′
where:
B is lower triangle with 1 along the principal diagonal
D is a diagonal matrix
Theorem (Choleski decomposition)
Any positive definite symmetric matrix A has a unique representation:
A = PP′
P is lower triangle with squares roots of D along the diagonal
Choleski decomposition just let P = BD1/2
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Choleski decomposition with R
> m <- matrix(c(5,1,1,3),2,2);m
[,1] [,2][1,] 5 1[2,] 1 3
> tP <- chol(m);tP
[,1] [,2][1,] 2.236068 0.4472136[2,] 0.000000 1.6733201
> #R gives P'P (and we saw PP')
> t(tP) %*% tP
[,1] [,2][1,] 5 1[2,] 1 3
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Triangular decomposition with RIt is not available directly so we find it from Choleski by setting:
C has the diag of PB = PC−1
D = CC′
> C<-matrix(0,2,2)
> diag(C)<-diag(tP)
> A<-t(tP)%*%solve(C);A #lower triangle
[,1] [,2][1,] 1.0 0[2,] 0.2 1
> D<-C%*%t(C);D #diagonal
[,1] [,2][1,] 5 0.0[2,] 0 2.8
> A%*%D%*%t(A) #original matrix
[,1] [,2][1,] 5 1[2,] 1 3
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Orthogonal IRFFrom the VMA:
yt = A0 +∞∑i=0
Φεt−i
Decompose Σε = PP′
where P is lower triangular matrix.Insert PP−1 into the VMA:
yt = A0 +∞∑i=0
ΦiPP−1εt−i
And let:
Θi ≡ ΦiP
wt ≡ P−1εt
So we have:
yt = A0 +∞∑i=0
Θwt−i
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Orthogonal IRF 2We rewrote
yt = A0 +∞∑i=0
Θwt−i
Proposition
The residuals wt are independant to each other: Var(wt) = I
Proof.
Var(wt) ≡ Σw = Var(P−1εt) = P−1ΣεP−1′= P−1PP
′P−1′
= I
Hence:
Innovations are independant
Variance is one
So unit innovation is just an innovation of size one standard deviation.
The (j,k) element of Θj is assumed to represent the effect on variable j ofa unit innovation of variable k thats has occured i period ago.
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Orthogonal IRF 3
This orthogonalization is called sometimes Wold
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Outline
1 Lectures
2 Impulse response function
3 Structural VAR modelsStuctural vector autoregressive model (SVAR)
Choleski decompositionBlanchard-Quah decomposition
Impulse-response functionForecast variance decomposition
4 Impulse response functionStable VAR case
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SVAR model
We now study a full system:
yt = δ1 + b12zt + γ11yt−1 + γ12zt−1 + ε1t
zt = δ2 + b21yt + γ21yt−1 + γ22zt−1 + ε2t
In matrix notation:[1 −b12
−b21 1
] [yt
zt
]=
[δ1δ2
]+
[γ11 γ12
γ21 γ22
] [yt−1
zt−1
]+
[ε1tε2t
]Compactly:
Bxt = ∆ + Γxt−1 + εt
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From SVAR to VARThis SVAR can be rewritten in a VAR by premultiplicating by B−1:
xt = A0 + A1xt−1 + ut
ut = B−1εt
A0 = B−1∆
A1 = B−1Γ
Under the assumption that the εt from SVAR are white noise, the ut fromVAR have:
Zero mean
fixed variance
individually not autocorelated
Correlated to each other if b12, b21 6= 0
Examine: ut = B−1εt =
u1t = (ε1t − b12ε2t)/(1− b12b21)
u2t = (ε2t − b21ε1t)/(1− b12b21)
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Problems with the estimation
SVAR: endogeneity problem
VAR to SVAR: unidentification of the structural parameters. Needrestrictions
Example
VAR: 9=2+4+3 parameters (intercept+slope+var/cov)
SVAR: 10=2+2+4+2 parameters (contemp+intercept+slope+var)
We have to make 1 restriction on the structural parameters
Proposition
We need k2−k2 restrictions to identify the SVAR from the VAR.
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Types of restrictions
Choleski
Triangular
Economic a priori
Blanchard-Quah
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Choleski restrictions
Remember, by setting:
wt ≡ P−1εt where Σε = PP′
we could go from a VMA with correlated residuals to a VMA with Σw = I :
yt = A0 +∞∑i=0
Θwt−i
This is implicitly a SVAR where B = P:
SVAR: Pxt = ∆ + Γxt−1 + εtVAR: xt = P−1∆ + P−1Γxt−1 + P−1εt = A∗ + A∗xt−1 + ut
The residuals are Σu = Var(P−1ε) = P−1ΣεP−1′= I
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Triangular restrictions
We can also use the triangular decomposition:
Σε = ADA′
and set B = A so:
SVAR: Axt = ∆ + Γxt−1 + εtVAR: xt = A−1∆ + A−1Γxt−1 + A−1εt = A∗ + A∗xt−1 + ut
The residuals are Σu = Var(Aε) = AΣεA′
= D
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IRF
Both triangular or Choleski imply that B is lower triangular:
B =
(1 0
b21 1
)That means that in the SVAR we have set b12 = 0:
yt = δ1 +
=0︷︸︸︷b12 zt + γ11yt−1 + γ12zt−1 + ε1t
zt = δ2 + b21yt + γ21yt−1 + γ22zt−1 + ε2t
This is a kind of exogeneity: we assume that there is nocontemporaneous impact (or Wold causality)of zt to yt
This operation is called ordering of the variables and is made witheconomic arguments.
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IRF
With the Choleski decomposition we can now compute the ImpulseResponse Function and interpret it as the impact of shocks of xi to xj .
Problem
The results differ when the ordering is changed!
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Choleski
Remember the relation between the SVAR residuals ut and the VARresiduals εt :
εt = But
u1t = (ε1t −=0︷︸︸︷b12 ε2t)/(1−
=0︷︸︸︷b12 b21) = ε1t
u2t = (ε2t − b21ε1t)(1−=0︷︸︸︷b12 b21) = ε2t − b21ε1t
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Other restrictions:
Coefficient: b12 = 0
Variance: Var(ε1t) = 1
Symmetry: b12 = b21
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Cho decomposition
How do we obtain the B?
Var(ut) = Ω from reduced VAR (estimated)
Define Var(uεt) ≡ Σ in the SVAR (not observed)
Theoretical relationship: ut = B−1εt
So we obtain: Ω = B−1ΣB−1′
If we make further the assumption that Σ = I we have:Ω = B−1B−1′
Under the assumption that P = B−1 is lower triangular matrix, we havethe unique decomposition (Choleski): Ω = PP
′
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SVAR
B0yt = c0 + B1yt−1 + B2yt−2 + · · ·+ Bpyt−p + εt[1 B0;1,2
B0;2,1 1
] [y1,t
y2,t
]=
[c0;1
c0;2
]+
[B1;1,1 B1;1,2
B1;2,1 B1;2,2
] [y1,t−1
y2,t−1
]+
[ε1,t
ε2,t
]
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Problem: results depend on the ordering of the variables! See sensitivityanalysis during conference: tested for many different variables and obtain
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same result. Here same variables but different output.
Outline
1 Lectures
2 Impulse response function
3 Structural VAR modelsStuctural vector autoregressive model (SVAR)
Choleski decompositionBlanchard-Quah decomposition
Impulse-response functionForecast variance decomposition
4 Impulse response functionStable VAR case
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Outline
1 Lectures
2 Impulse response function
3 Structural VAR modelsStuctural vector autoregressive model (SVAR)
Choleski decompositionBlanchard-Quah decomposition
Impulse-response functionForecast variance decomposition
4 Impulse response functionStable VAR case
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Forecast of a VAR(1)
Take the VAR(1):
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Outline
1 Lectures
2 Impulse response function
3 Structural VAR modelsStuctural vector autoregressive model (SVAR)
Choleski decompositionBlanchard-Quah decomposition
Impulse-response functionForecast variance decomposition
4 Impulse response functionStable VAR case
Matthieu Stigler [email protected] () Structural VAR models December 9, 2008 33 / 33