The unrestricted VAR model
Katarina Juselius
August 7, 2012
Katarina Juselius ()
The unrestricted VAR model
August 7, 2012
1 / 10
The Unrestricted VAR
(Chapter head:)
Assume that we have derived an estimator under the assumption of
multivariate normality and take the model to the data
If the multivariate normality assumption is correct, the residuals
should not deviate signi…cantly from the assumption εt
INp (0, ·).
If they do not pass the tests, for example, because they are
autocorrelated or heteroscedastic, or because the distribution is
skewed or leptocurtic, then the estimates may no longer have optimal
properties and cannot be considered full information maximum
likelihood (FIML) estimates.
The obtained parameter estimates (based on an incorrectly derived
estimator) may not have any meaning, and since we do not know
their “true” properties, inference is likely to be hazardous.
Some assumptions are more crucial for the properties of the estimates
than others and we will discuss robustness properties against
moderate violations of the assumptions.
Katarina Juselius ()
The unrestricted VAR model
August 7, 2012
2 / 10
A VAR(2) for the Danish data
= Π1 xt 1 + Π2 xt 2 + ΦDt + εt
t = 1, ..., T εt
INp (0, Ω)
xt
(1)
where Dt = [Dq1t , Dq2t , Dq3t , µ0 ] contains three centered seasonal
dummies and a constant.
The estimated model has at least one characteristic root very close to
the unit circle and xt is not likely to be stationary.
The ‘t’-ratios are more likely to be distributed as the Dickey-Fuller’s τ
and not as Student’s t.
Katarina Juselius ()
The unrestricted VAR model
August 7, 2012
3 / 10
2
6
6
6
6
4
mtr
ytr
∆pt
Rm,t
Rb,t
2
6
6
+ 6
6
4
2
6
6
^· = 6
6
4
3
2
7 6
7 6
7=6
7 6
5 4
0.19
0.03
0.02
0.00
0.01
1.0
0.09
0.34
0.16
0.17
Katarina Juselius ()
0.55
0.01
0.01
0.00
0.01
0.12
0.16
0.10
0.00
0.04
0.12
1.04
0.11
0.00
0.03
0.55
0.10
0.21
0.02
0.00
0.93
0.23
0.02
0.03
0.03
1.06
0.43
1.82
0.16
0.06
1.0
0.12 1.0
0.05 0.22 1.0
0.09 0.25 0.33 1.0
3.97
1.69
2.46
1.05
0.07
3.06
2.69
1.55
0.26
0.37
3
8.05
2.18
1.79
0.30
1.27
32
76
76
76
76
54
2
7
6
7
6
7, σ
6
7 ^ε = 6
5
4
The unrestricted VAR model
32
76
76
76
76
54
mtr 2
ytr 2
∆pt 2
Rm,t 2
Rb,t 2
0.0369
0.0147
0.0087
0.0012
0.0016
mtr 1
ytr 1
∆pt 1
Rm,t 1
Rb,t 1
3
3
7
7
7
7
5
7
7
7 + ΦDt + εt
7
5
3
7
7
7
7
5
August 7, 2012
4 / 10
Discussing the results
Some statistics
Log(Lmax ) = 2241.28, log Ω̂ = 51.9, R 2 (LR ) = 0.99
F-tests on the signi…cance of single regressors distributed as F(5, 101):
mtr 1 ytr 1 ∆pt 1 Rm,t 1 Rb,t 1
6.6 24.4 2.22
30.2
38.7
Katarina Juselius ()
mtr 2 ytr 2 ∆pt 2 Rm,t
2.2
4.7
1.5
2.1
The unrestricted VAR model
August 7, 2012
2
5 / 10
R
Di¤erent ECM-representations
The so-called Vector Equilibrium Correction Model (hereafter VECM)
gives a convenient reformulation of the unrestricted VAR without imposing
any binding restrictions on the model parameters:
(m )
∆xt = Γ1 ∆xt
1
(m )
+ Γ2 ∆xt
2
+
(m )
1 ∆xt k +1 + Πxt m
+ Γk
+ ΦDt + εt
where m is an integer between 1 and k de…ning the lag placement of the
ECM term.
1
2
3
4
Multicollinearity is signi…cantly reduced in the error-correction form.
Di¤erences are much more ‘orthogonal’than the levels of variables.
All information about long-run e¤ects is summarized in the levels
matrix ½ .
The interpretation of the estimates can be naturally classi…ed into
short-run and long-run e¤ects.
The VECM-formulation gives a direct answer to the question ’why the
in‡ation rate, say, changed from the previous to the present period’.
Katarina Juselius ()
The unrestricted VAR model
August 7, 2012
6 / 10
The ECM formulation with m=1
(1 )
∆xt = Γ1 ∆xt
where Π =
2
6
6
6
6
4
∆mtr
∆ytr
∆pt2
∆Rm,t
∆Rb,t
2
6
6
+6
6
4
3
Π1
(I
2
7 6
7 6
7=6
7 6
5 4
0.26
0.02
0.02
0.00
0.00
Katarina Juselius ()
0.19
0.03
0.02
0.00
0.01
0.24
0.12
0.01
0.00
0.00
1
Π2 ), and
0.12
0.16
0.10
0.00
0.04
1.48
0.33
0.77
0.01
0.03
+ Πxt
(1 )
Γ1
1
+ ΦDt + εt
Π2 .
=
0.55
0.10
0.21
0.02
0.00
5.04
2.11
0.64
0.11
0.01
(2)
1.06
0.43
1.82
0.16
0.06
4.99
0.51
0.24
0.04
0.09
The unrestricted VAR model
32
76
76
76
76
54
3.06
2.69
1.55
0.26
0.37
mtr 1
ytr 1
∆pt 1
Rm,t 1
Rb,t 1
32
76
76
76
76
54
∆mtr 1
∆ytr 1
∆pt2 1
∆Rm,t 1
∆Rb,t 1
3
7
7
7 + ΦDt + εt
7
5
August 7, 2012
7 / 10
3
7
7
7
7
5
The ECM formulation with m=2
(2 )
∆xt = Γ1 ∆xt
with Π = (I
Π1
remains unchanged, but
2
6
6
6
6
4
∆mtr
∆ytr
∆ 2 pt
∆Rm,t
∆Rb,t
2
6
6
+6
6
4
3
2
7 6
7 6
7=6
7 6
5 4
0.26
0.02
0.02
0.00
Katarina Juselius ()
2 + ΦDt + εt
(2 )
Π2 ) and Γ1 = (I Π1 ). The
(2 )
not the Γ1 matrix.
0.45
0.01
0.01
0.00
0.01
0.24
0.12
0.01
0.00
1
0.12
0.04
0.11
0.00
0.03
1.48
0.33
0.77
0.01
+ Πxt
0.93
0.23
0.98
0.03
0.03
5.04
2.11
0.64
0.11
3.97
1.69
2.46
0.05
0.07
4.99
0.51
0.24
0.04
The unrestricted VAR model
32
76
76
76
76
54
(3)
Π matrix
8.05
2.18
1.79
0.30
0.27
mtr 2
ytr 2
∆pt 2
Rm,t 2
32
76
76
76
76
54
∆mtr 1
∆ytr 1
∆ 2 pt 1
∆Rm,t 1
∆Rb,t 1
3
7
7
7 + ΦDt + εt
7
5
August 7, 2012
8 / 10
3
7
7
7
7
5
The VAR(2) model in acceleration rates, changes and levels
∆2 xt = Γ∆xt
where Γ =
2
6
6
6
6
4
∆2 mtr
∆2 ytr
∆∆2 pt
∆2 Rm,t
∆2 Rb,t
2
6
6
+6
6
4
(I
3
(1 )
Γ1 )
2
7 6
7 6
7=6
7 6
5 4
0.26
0.02
0.02
0.00
0.00
Katarina Juselius ()
0.24
0.12
0.01
0.00
0.00
1
+ Πxt
2
+ ΦDt + εt
(I + Π2 ), and Π =
=
1.45
0.01
0.01
0.00
0.01
0.12
0.96
0.11
0.00
0.03
1.48
0.33
0.77
0.01
0.03
5.04
2.11
0.64
0.11
0.01
0.93
0.23
1.98
0.03
0.03
4.99
0.51
0.24
0.04
0.09
The unrestricted VAR model
(I
3.97
1.69
2.46
0.95
0.07
32
76
76
76
76
54
(4)
Π1
Π2 )
8.05
2.18
1.79
0.30
0.73
mtr 2
ytr 2
∆pt 2
Rm,t 2
Rb,t 2
32
∆mtr 1
7 6 ∆ytr 1
76 2
7 6 ∆ pt 1
76
5 4 ∆Rm,t 1
∆Rb,t 1
3
7
7
7 + ΦDt + εt
7
5
August 7, 2012
9 / 10
Comparing the F-tests in single regressors
The VAR in levels
mtr 1 ytr 1 ∆pt 1 Rm,t 1 Rb,t 1
6.6 24.4 2.22
30.2
38.7
mtr 2 ytr 2 ∆pt 2 Rm,t
2.2
4.7
1.5
2.1
2
Rb
5
The VECM with m=1
∆mtr
2.2
1
∆ytr 1 ∆pt2 1 ∆Rm,t
4.7
1.5
2.1
∆Rb,t
5.6
1
mtr 1 ytr 1 ∆pt 1 Rm,t
5.8
3.9 10.3
4.6
1
1
The VECM with m=2
∆mtr
6.0
1
∆ytr 1 ∆2 pt 1 ∆Rm,t
3.7
30.9
3.6
1
∆Rb,t
6.9
1
mtr 2 ytr 2 ∆pt 2 Rm,t
5.8
3.9 10.3
4.6
2
The VECM with acceleration rates, changes and levels
∆mtr 1 ∆ytr 1 ∆2 pt 1 ∆Rm,t 1 ∆Rb,t 1 mtr 2 ytr 2 ∆pt 2 Rm,t
48.8
27.2 () 106.9
23.4
27.5
5.8
3.9August10.3
Katarina Juselius
The
unrestricted VAR
model
7, 2012
104.6
/ 10
2