Program
Introduction VAR Modelling
Econometrics II – Chapter 7.6
Vector Autoregressive (VAR) model VAR(1),
assumptions
Vector AutoRegressive models
AR(2) → VAR(1), VAR(1) → ARMA(m,m − 1)
Unit root tests DF and “unit root tests” VAR(1)
Marius Ooms
VAR(2) and VECM
Estimation and Inference in VAR(p)
Tinbergen Institute Amsterdam
Testing for VAR order p
Testing for cointegrating rank, r.
Chapter 7.6 – p. 1/24
Introduction to VAR Models
Chapter 7.6 – p. 2/24
VAR Introduction, continued
Multivariate Time Series Modelling.
Model linear dynamic relationships within a set of m
variables based on n vector observations, modelling
(only)
m means (possibly time-varying)
m (forecast error) variances,
0.5 · m · (m − 1) possible covariances,
(n − 1) m × m autocovariance matrices.
Do not make a priori incredible exogeneity or (Granger
non-) causality assumptions.
Chapter 7.6 – p. 3/24
Point and Interval prediction (conditional mean and
(co)variances): One-step, Multistep.
Time Series Decomposition of mean and (forecast
error) variance: Trend, Stationary Components.
Test for (unit-root) (non)-stationarity of (linear
combinations) of variables: testing for the number of
I(0) equilibrium relationships, testing for the
number of common I(1) trends.
In Advanced Courses: Sims’ economic intepretations
of VMA(∞) form of VAR: Impulse responses, Forecast
error variance decompositions, Structural (Identified)
VARs.
Chapter 7.6 – p. 4/24
Eviews VAR(1) with ’trend in VAR’
VAR(1) plus DT, ”DGP form”
The corresponding VAR(1) DGP (Data Generating
Process) is given by
The VAR(1) model in ”regression form” is given by
Yt = γ1 + γ2 t + ΦYt−1 + εt ,
Yt = µ1 + µ2 t + ut ,
t = 1, . . . , n
where
Yt is an m vector of “endogenous” variables.
Φ is the m × m matrix of coefficients.
The m−vector of disturbances εt has mean zero and
variance matrix Ωε .
Φ(L)ut = εt .
(DGP)
µ1 + µ2 t is interpreted as E(Yt ): the vector of expected
values of the variables in Yt at time t.
The relationship between γ1 , γ2 (intercept and trend vector
in VAR regression form) and µ1 , µ2 (constant and trend in
Data Generating Process form) is a bit involved, see Heij et
al. §7.6.1 (I(0) cases) and §7.6.3. (I(1) cases).
Note: Heij adjusts the notation: γ1 = α in §7.6.1 (7.35), γ1 = γ in §7.6.3 (7.39), µ1 = µ
in §7.6.1 (7.38) and B ′ µ2 = δ in §7.6.3 on p. 424. See als comparison with DGP and
regression model D-F test below.
The disturbances are uncorrelated at different time
periods: εt ∼ W N .
Eviews (VAR specification->Cointegration) calls γ2 t: ”trend in VAR”
Note: Eviews calls µ2 t ”Linear trend in data”.
Chapter 7.6 – p. 6/24
Chapter 7.6 – p. 5/24
AR(2) → VAR(1), ”companion form”
VAR Assumptions Eviews
VAR assumptions (in macroeconomic time series analysis):
Each variable yi,t is I(0) (DT) or I(1) (ST). In particular
we exclude:
I(2) processes (cf. Heij page 599, implicitly
assumed in §7.6.3)
I(−1) processes (can never be approximated by a
VAR → why not?)
εt has a multivariate normal distribution.
Connection AR(2) – Bivariate VAR(1) helps understanding
stability conditions:
y1,t = φ1 y1,t−1 + φ2 y2,t−1 + εt
y2,t = y1,t−1 +
0
This example shows how a (V)AR(p) process can be
written in a VAR(1) “companion form”.
Exercise (1): Check stability conditions (on λ1 , λ2 ) for this
example and show that they correspond to the stability
condition for a univariate AR(2) process.
Hint: See also Examples 17 and 21 in chapter 14 of
Binmore and Davies (2001).
All characteristic roots: |λi | ≤ 1, VAR only has unit
roots λi = 1 and stable roots (eigenvalues of Φ).
Exclude explosive processes.
Chapter 7.6 – p. 7/24
Chapter 7.6 – p. 8/24
VAR(1) → ARMA(m, m − 1)
VAR(1) → ARMA(m, m − 1), continued
“Joint” multivariate VAR(1) implies “marginal” univariate
ARMAs! VAR(1) → ARMA(m, m − 1)
Consider the first VAR equation
In the absence of unit roots |λi | < 1, i = 1, . . . , m, we
showed that y1,t follows an ARMA(m, m − 1) process.
In the presence of unit roots, λi = 1, y1,t follows an
ARIMA process, which is not straightforward to derive,
without further assumptions.
y1,t = |Φ(L)|−1 C1· (L)εt ,
with d(L) = |Φ(L)| the scalar determinant of Φ(L) and
C1· (L) the first row of the m × m matrix of cofactors of
Φ(L). Zellner and Palm (1974) used this argument to
explain ARMA behaviour in macroeconomic series.
The characteristic roots or “eigenvalues of Φ” λi , i.e. the
solutions to |Φ(λ−1 )| = 0, determine stability of the VAR.
We only consider cases where y1,t is ARIMA(p, 0, q) or
ARIMA(p, 1, q).
Exercise (2): Check Illustration for VAR(1) on Heij et al.
page 659.
|Im − λ−1 Φ| = 0 ⇔ |λIm − Φ| = 0.
Chapter 7.6 – p. 9/24
Recap univariate Unit root tests DF
Chapter 7.6 – p. 10/24
“Unit root tests” VAR(1), cf. previous slide!
VAR unit root test regressions trending data: §7.6.3.
In a VAR we have more than two hypotheses!
Connection stability test VAR(1) with univariate
DF-regression
Yt = µ1 + µ2 t + ut ,
ut = Φut−1 + εt
DGP
∆Yt = γ1 + γ2 t + (Φ − I)Yt−1 + εt
test regression
In DF, AR unit root regression test for trending data, §7.3.3:
H0 : ST, φ − 1 = ρ = 0: unit root
H1 : DT, root(s) |λi | < 1 of Φ(λ−1
i ) = 0
Deterministic terms H0 : γ2 = 0: no quadratic trend in
DGP.
Deterministic terms H1 : γ2 unrestricted, allows for
deterministic trend in I(0) case.
H0 , Φ − I = Π = 0, m separate random walks. All m
λi = 1. All linear combinations of yi ST. m unit roots.
No ECM. Model only in differences of Yt .
Extra possibility: H1 , Π = αβ ′ has rank 1. ∆yt
“corrected” by 1 “deviation from equilibrium”, or
cointegrating vector β ′ Yt−1 where β ′ yt ∼ I(0). m − 1
unit roots. 1 ECM. Model in differences and one
combination of levels ∆Yt = αβ ′ Yt−1 + εt
Hm , Π has full rank. All linear combinations of Yt are
I(0). All m eigenvalues of Φ: λi < 1: No unit roots. All
m eigenvalues of Π unequal to zero. Model in levels.
see next slide for deterministic terms!
Chapter 7.6 – p. 11/24
Chapter 7.6 – p. 12/24
“Unit root tests” VAR(1), deterministic terms
Deterministic terms in effective multivariate unit root test
regressions: H1 vs Hm .
VAR(2) and (Eviews) VECM(1)
VAR(2) and VECM(1) are natural extensions of VAR(1) and
ECM models:
H0 : Trend in regression restricted: γ2 = 0, so that we
do not get quadratic trend (µ3 t2 = 0) in data DGP form
of VAR. Remember: we have m unit roots.
H1 Trend coefficients in regression, γ2 , restricted so
that only the cointegrating I(0) combination β ′ yt has a
deterministic trend: “restricted” trend term in test.
Remember: we have m − 1 unit roots. Eviews: trend in CE, no
trend in VAR.
Hm : Deterministic terms are unrestricted under Hm , γ2
unrestricted. Trend term in ”regression form” leads to
same order deterministic trend in ”DGP form” of VAR
as there are no unit roots. Eviews: Unrestricted VAR.
Yt
= γ1 + Φ1 Yt−1 + Φ2 Yt−2 + εt ,
∆Yt = γ1 − Φ2 ∆Yt−1 + ΠYt−1 + εt ,
where
Π = Φ1 + Φ2 − I = −Φ(1).
(and γ2 = 0 for simplicity).
As in the VAR(1) one can show that a full rank of Π is a
necessary condition for stationarity and a reduced rank of
Π is necessary for cointegration.
Compare the VECM with the Augmented Dickey Fuller test
regression: The matrix −Π replaces the ρ of the ADF.
this VECM is called VECM(1) in Eviews.
Chapter 7.6 – p. 14/24
Chapter 7.6 – p. 13/24
More cointegrating vectors, m = 3, r = 2
Estimation of VARs and VECMs, UVAR
Consider a VAR(1) with m = 3 with eigenvalues of Φ:
λ1 = 1 and |λ2 |, |λ3 | < 1. There is one eigenvalue of
Π = Φ − I equal to zero: Π has (reduced) rank of 2.
Two approaches:
There are 2 independent linear combinations of yt
(eigenvectors of Φ corresponding to λ2 and λ2 ), which
produce stationary I(0) series: 2 cointegrating vectors.
One linear combination of yt (corresponding to λ1 ) is I(1):
1 common trend.
The VECM form for m = 3 produces a Π = AB ′ . A, the
matrix of adjustment coefficients, and B, the matrix of
cointegrating vectors, are 3 × 2 with rank 2.
In economic applications A and/or B have a number of
fixed ones and zeros, facilitating interpretation. Ex. 7.28.
Chapter 7.6 – p. 15/24
(i) Direct estimation and standard asymptotic inference of
the system in unrestricted VAR (UVAR) (put
γ1 = γ2 = 0 for simplicity):
Yt = Φ1 Yt−1 + Φ2 Yt−2 + · · · + Φp Yt−p + εt
∆Yt = Γ1 ∆Yt−1 + · · · + Γp−1 ∆Yt−p+1 + ΠYt−1 + εt
→ only appropriate if all the eigenvalues of Π are
larger than zero. Deterministic terms, γ1 + γ2 t, can be
added unrestrictedly.
(ii) see next slide
Chapter 7.6 – p. 16/24
Test of p, standard inference in a VAR I
Estimation and inference in VECMs
(ii) In case of possible I(1) components: Determine the
number r, 0 ≤ r ≤ m of possible cointegrating vectors
and then estimate
∆Yt = Γ1 ∆Yt−1 + · · · + Γp−1 ∆yt−p+1 + AB ′ Yt−1 + εt
Conditional on a choice for r cointegrating variables
B ′ Yt−1 , A is estimated jointly with Γ(L). Deterministic
terms can be added in two ways, unrestricted as
intercept or trend, or restricted as intercept or trend in
cointegrating vectors only. Conditional on known and
correct B ′ Yt−1 standard asymptotic inference applies.
This method is appropriate when (some of) the Y
variables are I(1).
Testing the Order of VAR using Likelihood Ratio tests
Suppose that a VAR(p1 ) is estimated and we want to test
the hypothesis that the order is p0 < p1 . The maximised
log-likelihood when a VAR with m variables is fitted to n
observation points is
\ = constant + n log |Ω
ˆ −1 |
log(L)
2
ˆ is the variance-covariance matrix of the residuals
where Ω
from the VAR equations. Note that the ML method
minimises the (log) determinant of residual variance
matrix. Model selection using AIC and SIC minimises this
term corrected for a penalty function.
Chapter 7.6 – p. 17/24
Test on p, standard VAR inference II
Chapter 7.6 – p. 18/24
Tests on r, nonstandard inference in VAR
When p0 lags are used, the maximised log-likelihood is
Testing the cointegrating rank r using LR tests The
popular likelihood ratio test of Søren Johansen, most
cited author in economics in 1990-2000, compares ML
estimates under the different rank restrictions implied by
(non-)cointegration. Also known as trace test.
\0 = constant + n log |Ω
ˆ −1
log(L)
0 |
2
and when p1 lags are used,
Remember conditions for application!:
\1 = constant + n log |Ω
ˆ −1 |.
log(L)
1
2
All variables yi,t are I(0) or I(1).
The VAR does not contain characteristic roots outside
the unit circle.
The likelihood ratio test is then
h
i
a
\
\
ˆ
ˆ
LR = −2(log(L)0 − log(L)1 ) = n log |Ω0 | − log |Ω1 | ∼ χ2q
with q = m2 (p1 − p0 ): the number of restrictions imposed
under H0 .
Chapter 7.6 – p. 19/24
The VAR is not misspecified
The different hypotheses employed in LR testing in a
VAR(p) are extensions of the hypotheses in a VAR(1).
Chapter 7.6 – p. 20/24
Hypotheses in testing rank r in a VAR(p)
Hypotheses on deterministics in testing for r
Hypotheses used in testing:
H0 , −Φ(1) = Π = 0, m separate random walk
components. All linear combinations of yi have a ST. m
unit roots in the VAR. No partial correlation between
growth rates and lagged levels.
′
H1 , Π = αβ has rank 1. ∆yi,t partially “corrected” by 1
cointegrating vector β ′ Yt−1 . m − 1 common trends.
H2 , Π rank 2, . . .
Hm−1 , Π cointegrating rank m − 1, 1 common trend
Hm , Π has full rank. All linear combinations of Yt are
I(0). All pm eigenvalues of Φ: λi < 1. No unit roots.
If data are trending (µ2 6= 0):
Deterministics under H0 : γ2 = 0 no trends in regression
form of VAR, to avoid quadratic trend in the DGP.
Deterministics under H1 : γ2 in regression form of VAR
restricted such that only the cointegrating combination β ′ Yt
has trend β ′ µ2 t. Other linear combinations don’t have a
trend. In other words: the deterministic trend is part of
the cointegrating combination.
Deterministics Hm : γ2 is unrestricted in the regression
form of the model as there are no unit roots. All linear
combinations of Yt are I(0).
See next slide for deterministic terms.
Chapter 7.6 – p. 22/24
Chapter 7.6 – p. 21/24
Distribution and interpretation of LR tests for r
Summary of testing for r
Assume that all series yi,t of the VAR process
Φ(L)yt = γ1 + γ2 t are I(0) or I(1).
LR / trace test H0 vs. Hm , nonstandard distribution
LR / trace test H1 vs. Hm , . . . ,
LR / trace test Hm−1 vs. Hm
All test statistic are determined by (sums of) m ρˆ2j s:
Squared Partial ’Canonical’ Correlations, ρˆ2j ,
between current growth rates ∆Yt and lagged levels
Yt−1 . No. of significant ρ2j s = r = rank of Π = no. of
cointegrating vectors. m − r = no. of common trends.
Apply tests in order above. Stop if a test does not reject.
Reject Hj against Hm for large values of the outcome of the
test. Critical values are in Exhibit 7.31. p-values in Eviews.
Use LR/trace tests for the cointegrating rank r of
Π = AB ′ = −Φ(1), where the order of the
deterministic trend in the cointegrating vector(s)
must exceed the order of the trend in the VAR by 1.
This implies that restrictions must be imposed on γ1
and/or γ2 in hypotheses with r < m.
Always allow for trend in cointegrating vector if DGP yt
has a trend (µ2 6= 0).
Start with H0 : r = 0, continue with r = 1, . . .. Stop (and
accept r = j) when Hj is not rejected against Hm .
Eviews (Johansen) uses λj for ρ2j : ”eigenvalues”, cf Example 7.28. They are not simply
the eigenvalues of Π. Eg.: one always has 0 ≤ ρ2j ≤ 1, whereas eigenvalue of Π can < 0
Chapter 7.6 – p. 23/24
If r is known, you can apply standard Wald test on the
trend coeffcient(s) in the CI vector(s), see page 673.
Chapter 7.6 – p. 24/24