The University of Queensland
(School of Economics) Applied Econometrics for Macro and Finance Week 12 1 / 12
ECON7350 Multivariate Processes – III
Copyright By PowCoder代写 加微信 powcoder
VAR Representations
Recall the SVAR given by
Γ(L)xt =γ0 +εt, and the VAR given by
A(L)xt =a0 +et, which are related by
A(L) = B−1Γ(L),
IfA(1)=In−A1−···−Ap isnotinvertible,then
it cannot be the case that all variables in xt are characterised by I(0) processes; at least one stochastic trend must be present in the multivariate process {xt}.
Γ(L)=B−Γ1L−···−ΓpLp,
E(εtε′t)=Σ, E(ete′t)=Ω,
A(L)=In −A1L−···−ApLp,
a0 = B−1γ0, Ω = B−1Σ B−1′ .
(School of Economics) Applied Econometrics for Macro and Finance Week 12 2 / 12
Cointegration
Recall that yt ∼ I(d) is short for “the process {yt} is integrated of order d”. Similarly for a n × 1 vector xt, the notation xt ∼ I(d) corresponds to xi,t ∼ I(d) for
i = 1,…,n.
Recall that if xt ∼ I(1), cointegration of rank r < n means there exist r vectors β1, . . . , βr
such that β′jxt = zj,t is I(0).
We call the collection of cointegrating vectors β = (β1, . . . , βr), the cointegrating matrix (of
dimension n × r).
Recall also that cointegrating vectors are not unique: (βK)′xt = K′zt is also I(0), so β = βK
for any invertible r × r matrix K, is also a cointegrating matrix of xt.
Unrestricted β cannot be estimated from the data because only the cointegrating space—spanned by cointegrating vectors β1, . . . , βr—is identified.
(School of Economics) Applied Econometrics for Macro and Finance Week 12 3 / 12
Vector Error Correction Model
The Vector Error Correction Model (VECM) is a generalization of the ECM that we use to analyze cointegrated systems.
Example: n=3,p=2,r=2:
z1,t 1 β21 β31 x1,t x1,t + β21x2,t + β31x3,t
z = β 1 β x2,t= β x +x +β x ∼I(0).
2,t 12 32 x3,t 12 1,t 2,t
xt =a0 +A1xt−1 +A2xt−2 +et, ∆xt = a0 +Πxt−1 +Ψ1∆xt−1 +et
whereΠ=−A(1)=αβ′ (αis3×2)andΨ1 =−A2.
(School of Economics) Applied Econometrics for Macro and Finance
Relationship Between VAR and VECM
Every VAR can be written as a VECM and vice versa:
xt =a0 +A1xt−1 +···+Apxt−p +et,
∆xt =a0 +Πxt−1 +Ψ1∆xt−1 +···+Ψp−1∆xt−p+1 +et.
The relationship between VAR coefficients and VECM coefficients is:
Π = −(In − A1 − · · · − Ap), Ψ1 = −(A2 + · · · + Ap),
Ψp−1 = −Ap.
In a cointegrated system with r < n the matrix Π is restricted, which corresponds to non-linear restrictions on A1, . . . , Ap.
To impose explicitly, we need to estimate the VECM.
(School of Economics) Applied Econometrics for Macro and Finance Week 12 5 / 12
Letxt ben×1andr=rankA(1). Supposethatforeachxi,t inxt,i=1,...,n,either xi,t ∼ I(1) or xi,t ∼ I(0) holds.
1 Ifr=n,thenxt ∼I(0).
2 If0
————-+————————————————————
_ce1 |
aaugval| 1 . . . . . ajpgval | -1.392016 .32956 -4.22 0.000 -2.0380 -0.7461 abdgval | 1.644154 .37565 4.38 0.000 0.9079 2.3804
aukgval | -0.322184
ausgval | -0.223884
_cons | -78.32383
-1.22 -0.77 .
0.223 0.441 .
————————————————————————–
(School of Economics) Applied Econometrics for Macro and Finance Week 12 8 / 12
Estimated Cointegrating Relation Error
(School of Economics) Applied Econometrics for Macro and Finance Week 12 9 / 12
Specifying the Cointegration Rank
In the previous example, a VECM with n = 5 and r = 1 was estimated.
In general, 0 ≤ r ≤ n and is unknown: choice of r presents another dimension to model
specification and uncertainty.
When r = n, the VAR and VECM are equivalent, but when r < n, the VECM is a VAR with
non-linear restrictions on the coefficients.
advantage: more parsimonious model leading to more precise inference; disadvantage: potentially mis-specified model leading to inconsistent inference.
To the extent that the cointegration space is identified, data is (in theory) informative about r.
(School of Economics) Applied Econometrics for Macro and Finance Week 12 10 / 12
Estimating the Co-integration Rank
When xt ∼ I(1), r = rank Π = rank A(1) is the co-integration rank of xt.
xt ∼ I(1) is often an assumption taken for granted in methodologies developed for
estimating VECMs, but not always innocuous in empirical work.
One method for estimating r, which does not assume xt ∼ I(1), is the Johansen cointegration
1 Estimate a VAR and compute canonical correlations from parameters.
2 Tracetest: H0 :r=r ̄,H1 :r>r ̄;
3 Maxeigenvaluetest: H0 :r=r ̄,H1 :r=r ̄+1;
Statistics for both tests are simple but have non-standard distributions.
Estimation of r typically entails substantial uncertainty, but many popular methods do not
account for estimation uncertainty in r.
(School of Economics) Applied Econometrics for Macro and Finance Week 12 11 / 12
Identifying Restrictions
To estimate a VECM with r < n, restrictions on β are needed for identification.
When r = 1 (as in the example), this is straightforward: set one element of β to 1.
When 1 < r < n, a common approach is to set the first r rows and r columns of β to Ir,
Ir β = β
and β is (n−r)×r.
Restrictions above imply the ordering of xt is again important: each of the first r
variables must enter exactly one equilibrium relationship.
A VECM can always be transformed to a corresponding VAR.
This means we can also derive a SVAR from an estimated VECM.
However, when 1 < r < n, identifying restrictions on β are not always straightforward to
combine with restrictions on B (or C).
(School of Economics) Applied Econometrics for Macro and Finance Week 12 12 / 12
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com