The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 11 1 / 16
ECON7350 Multivariate Processes – II
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SVARs and Reduced Form VARs
Recall that combining two or more ARDL specifications leads to the structural VAR. For the p = 1 case, this is given by:
Bxt = γ0 + Γ1xt−1 + εt, E(εtε′t) = Σ,
where B is invertible with ones on the diagonal and Σ is diagonal. The SVAR always has a
corresponding reduced form VAR given by (in the p = 1 case): xt = B−1γ0 + B−1Γ1xt−1 + B−1εt,
xt =a0 +A1xt−1 +et, E(ete′t)=Ω≡B−1ΣB−1′, where Ω is positive definite, symmetric, but not necessarily diagonal.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 2 / 16
SVARs and Reduced Form VARs
In the bivariate case, the structural shocks εt are related to reduced form errors et by: e1,t = (ε1,t − b12ε2,t)/(1 − b12b21), e2,t = (ε2,t − b21ε1,t)/(1 − b12b21),
and the variances / covariances are related by:
ω12 = (σ12 + b212σ2)/(1 − b12b21)2, ω2 = (σ2 + b21σ12)/(1 − b12b21)2, ω12 = −(b21σ12 + b12σ2)/(1 − b12b21)2.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 3 / 16
VMA Representation of the VAR(1)
By repeated backwards substitution of the SVAR(1), we obtain xt =In +B−1Γ1 +···+B−1Γ1hB−1γ0
+ B−1εt + B−1Γ1B−1εt−1 + · · · + B−1Γ1h B−1εt−h
+ B−1Γ1h+1 xt−h.
Recall that the VAR(1) is stable as long as all eigenvalues of A1 ≡ B−1Γ1 are less than one in absolute value. In this case, Ah1 ≡ B−1Γ1h −→ 0 as h −→ ∞ and we can write
x t = ( I n − A 1 ) − 1 a 0 + A h1 B − 1 ε t − h .
This is the VMA(∞) representation of the VAR(1).
(School of Economics) Applied Econometrics for Macro and Finance Week 11 4 / 16
Impulse Response Functions
The n × n matrix Φh = Ah1 B−1 contains all the impulse responses at horizon h.
Specifically, the (i,j)th element of Φh, denoted φij,h, is the impulse response of variable i to
shock j at h periods after impact.
The impulse response function (IRF), defined for variable i to a given shock j, plots impulse
responses against the time horizon 0, 1, . . . , h periods after impact.
In the bivariate case, we might obtain the IRF for x1 to a shock in ε2 from
x1,t x ̄1 φ11,0 φ12,0 ε1,t x = x ̄ + φ φ ε
2,t 2 21,0 22,0 2,t φ11,1
φ12,1 ε1,t−1 φ2,11 φ12,2 ε1,t−2 +φφε+φφε+···
21,1 22,1 2,t−1 21,2 22,2 2,t−2 (School of Economics) Applied Econometrics for Macro and Finance Week 11
Forecast Error Variance Decompositions
The variance of the h-step ahead forecast error for variable i is
nnn Var(xi,t+h − x ̄i) = φ2ij,0σj2 + φ2ij,1σj2 + · · · + φ2ij,hσj2,
j=1 j=1 j=1 hhh
= φ2i1,sσ12 + φ2i2,sσ2 + · · · + φ2in,sσn2 .
contrib of ε1 contrib of ε2
We can use this to evaluate how much of the forecast error variance of variable i at horizon h
is explained by shock j, relative to all other shocks:
hs=0 φ2ij,sσj2
FEVij,h = hs=0 φ2i1,sσ12 + hs=0 φ2i2,sσ2 + · · · + hs=0 φ2in,sσn2 .
(School of Economics) Applied Econometrics for Macro and Finance Week 11 6 / 16
contrib of εn
Identification of IRFs and FEVDs
Both IRFs and FEVDs measure the dynamic effect of a structural shock εj on a variable xi. Both can be computed easily from the VMA(∞) representation, written in lag-polynomial
where Φ(L) = Φ0 + Φ1L + · · · and Φh = Ah1 B−1.
Note that this formulation is valid for VARs with any lag length p—just rewrite a VAR(p) as a VAR(1) using the companion form.
From data, we can easily estimate A1 using the reduced form VAR (and, e.g., least squares methods). The trouble is with estimating B.
xt = x ̄ + Φ(L)εt,
(School of Economics) Applied Econometrics for Macro and Finance Week 11 7 / 16
Identification of IRFs and FEVDs
Two ways to estimate B:
1 estimate the SVAR directly (OLS may no longer work, but can use more complex
methods such as MLE);
2 estimate the reduced form VAR, then compute Σ, B, Γ1 from estimates of Ω, A1.
Either approach requires identifying restrictions. We will focus on the second.
As previously discussed, the key issue is that there are many SVARs that correspond to any
given VAR.
This means that times series data alone is not enough to identify all the parameters of an SVAR; we need additional information.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 8 / 16
Identification Using Exclusion Restrictions
The common approach we will consider is to compute:
1 B−1ΣB−1′ =Ω,
2 Γ1 = BA1.
To implement this, we need to impose restrictions on B and Σ such that the decomposition
B−1Σ B−1′ = Ω is unique.
Many approaches have been developed to achieve this in various contexts. The most common of these amount to imposing zero restrictions (or exclusion restrictions) on either the elements of B or the elements of C = B−1.
The basic idea is that economic theory often elicits strong implications for the relationships between reduced form errors and structural shocks, i.e. et = Cεt.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 9 / 16
The original and most widely used approach to obtain B and Σ from Ω is to use the Cholesky factorisation.
This corresponds to exact identification obtained by setting Σ = In and C = B−1 to a lower (sometimes upper) triangular matrix.
The result is a recursive structure for the set of equations et = Cεt. Example of a Cholesky factorisation:
0000 ε1,t
c22000 ε2,t
c32c3300 ε3,t .
e4,t c41 c42 c43 c44 0 ε4,t
e1,t c11 e2,t c21 e3,t = c31
e5,t c51 c52 c53 c54 c55 Applied Econometrics for Macro and Finance
(School of Economics)
Useful Properties of Matrix Inverses
If C is lower triangular, then B = C−1 is also lower triangular.
This property makes a Cholesky decomposition very convenient in practice.
More generally, the following identities are often useful:
A 0−1 A−1 0 A B−1 A−1 −A−1BD−1 C D = −D−1CA−1 D−1 , 0 D = 0 D−1 .
Using these identities, we can also implement Cholesky-like decompositions: c11 c12 0 0 0−1 b11 b12 0 0 0
0 b22 =b31 b32
0 0 0 b33 b34 0 .
c33 c34 0
c31 c32 c41 c42
0 b44 0 c51 c52 c53 c54 c55 b51 b52 b53 b54 b55
The matrix inverse identities tell us how zeros in C correspond to zeros in B.
(School of Economics) Applied Econometrics for Macro and Finance Week 11
SVAR Example with a Cholesky-like Factorisation
Consider estimating a VAR with
xt = (HKRETt, SINGRETt, KOREARETt, JAPRETt, THAIRETt)′.
Identification is achieved with the Cholesky-like factorisation:
e1,t e2,t
e3,t = e4,t
c11 c12 0 c22
c31 c32 c41 c42 c51 c52
c33 c34 0 c44 c53 c54
0 ε1,t 0 ε2,t
0 ε3,t. 0 ε4,t
(School of Economics)
Applied Econometrics for Macro and Finance
IRFs for One Realisation of the SVAR
.016 .012 .008 .004 .000 -.004
.012 .008 .004 .000
.016 .012 .008 .004 .000
Response of HKRET to Structural One S.D. Innovations
Response of SINGRET to Structural One S.D. Innovations
Response of KOREARET to Structural One S.D. Innovations
.016 .012 .008 .004 .000 -.004
.016 .012 .008 .004 .000 -.004
1 2 3 4 5 6 7 8 9 10
Response of JAPANRET to Structural One S.D. Innovations
1 2 3 4 5 6 7 8 9 10
Response of THAIRET to Structural One S.D. Innovations
1 2 3 4 5 6 7 8 9 10
Shock1 Shock2 Shock3 Shock4 Shock5
-.004 -.004
Shock1 Shock2 Shock3 Shock4 Shock5
Shock1 Shock2 Shock3 Shock4 Shock5
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
Shock1 Shock2 Shock3 Shock4 Shock5
Shock1 Shock2 Shock3 Shock4 Shock5
(School of Economics)
Applied Econometrics for Macro and Finance
FEVDs for One Realisation of the SVAR
(School of Economics) Applied Econometrics for Macro and Finance Week 11 14 / 16
Granger (Non-)Causality
Granger causality is a statement about whether past information on some variable, say zt, improves the forecast of another variable, say yt, where both zt and yt are endogenous variables in a VAR.
Formally, if
E(yt+1 |zt,…,z0,yt,…,y0) = E(yt+1 |yt,…,y0), then zt does not Granger-cause yt.
Unlike IRFs and FEVs, Granger (non-)causality is derived entirely from the reduced form VAR (so no additional restrictions are necessary).
Two things to beware of:
1 data-based tests can confirm Granger causality but they cannot confirm Granger
non-causality;
2 Granger non-causality does not imply exogeneity because yt can still be
contemporaneously correlated with zt.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 15 / 16
Testing for -Causality
If the VAR
xt =a0 +A1xt−1 +···+Apxt−p +et is stable, then we only need to do an F -test.
To test that xjt does not Granger-cause xit, specify
H0 :aij,1 =aij,2 =···=aij,p =0,
where aij,l is the (i,j)th element of Al, then implement a standard F-test on the restrictions. For the non-stationary case, an MWALD test can be used instead.
(School of Economics) Applied Econometrics for Macro and Finance Week 11 16 / 16
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