ECON3350/7350 VAR Models – II
Eric Eisenstat
The University of Queensland
Lecture 10
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 11
1 / 17
VAR Models
Recommended readings
Author
Title
Chapter
Call No
Enders Verbeek
Applied Econometric Time Series, 4e
A Guide to Modern Econometrics
6 9.5
HB139 .E55 2015 HB139 .V465 2012
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 2 / 17
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 from 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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 3 / 17
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 + b212σ2)/(1 − b12b21)2.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 4 / 17
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
∞
xt = (In − A1)−1 a0 + AhB−1εt−h.
h=0
This is the VMA(∞) representation of the VAR(1).
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 5 / 17
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 shocks 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
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11
6 / 17
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 .
s=0 s=0
contrib of ε1 contrib of ε2
s=0
contrib of εn
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 .
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 7 / 17
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 form as
xt = x ̄ + Φ(L)εt, 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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 8 / 17
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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 9 / 17
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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 10 / 17
We will discuss two types of factorisations:
1 Cholesky factorisation: exact identification obtained by setting
Σ = In and C to a lower (sometimes upper) triangular matrix; this generates a recursive structure for the set of equations et = Cεt;
2 structural factorisation: possibly non-recursive and/or over-identified structures for et = Cεt obtained by setting elements of C to zero according to theoretical considerations.
Example of a Cholesky factorisation:
e1,t c11
e2,t c21
e3,t = c31
e4,t c41
0000 c22000 c32c3300 c42 c43 c44 0
ε1,t ε2,t
ε3,t . ε4,t
e5,t c51 c52 c53 c54
c55 ε5,t
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 11
11 / 17
SVAR Example with Structural Factorisation
Consider estimating a VAR with
xt = (HKRETt, SINGRETt, KOREARETt, JAPRETt, THAIRETt)′.
Identification is achieved with the structural factorisation:
e1,t c11 c12 0 0
e2,t 0 c22 0 0
0 ε1,t 0 ε2,t 0 ε3,t .
e3,t = c31 c32 c33 c34
0 ε4,t e5,t c51 c52 c53 c54 c55 ε5,t
e4,t c41 c42 0 c44
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 11
12 / 17
Useful Properties of Matrix Inverses
The following identities are often useful in working with identifying restrictions:
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
We use identities such as these to check which zero restrictions on C correspond to zero restrictions on B.
If C is lower-triangular, then B is also lower triangular.
In terms of the current SVAR example,
.
.
c11 c12 0 0 0−1 b11 b12 0 0 0
0 c22
0 0 0 c33 c34 0
0 b22 =b31 b32
0 0 0
c31 c32 c41 c42
b33 b34 0
0 c44 0
c51 c52 c53 c54 c55 b51 b52 b53 b54 b55
b41 b42
0 b44 0
Eric Eisenstat
(School of Economics) ECON3350/7350
Week 11
13 / 17
SVAR IRFs
Response of HKRET to Structural One S.D. Innovations
.016
.012 .012 .008 .008 .004 .004 .000 .000
Response of SINGRET to Structural One S.D. Innovations
Response of KOREARET to Structural One S.D. Innovations
-.004
.016 .020 .016 .012 .008 .004 .000 -.004
-.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
.016 .016
.012 .012
.008 .008
.004 .004
.000 .000
-.004 -.004
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
Shock1 Shock2 Shock3 Shock4 Shock5
Shock1 Shock2 Shock3 Shock4 Shock5
Eric Eisenstat (School of Economics)
ECON3350/7350
Week 11
14 / 17
SVAR FEVDs
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 15 / 17
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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 16 / 17
Testing for Granger Non-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,
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.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 11 17 / 17