Univariate time series Multivariate time series Panels

Advanced Econometrics
Based on the textbook by Verbeek:
A Guide to Modern Econometrics

Robert M. Kunst
robert.kunst@univie.ac.at

University of Vienna
and
Institute for Advanced Studies Vienna

May 2, 2013

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Univariate time series Multivariate time series Panels

Outline

Univariate time series

Multivariate time series

General concepts
Cointegration
Vector autoregressions
Multivariate cointegration

Panels

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Univariate time series Multivariate time series Panels

Advanced Econometrics University of Vienna and Institute for Advanced Studies Vienna
Univariate time series Multivariate time series Panels

General concepts

Issues with multivariate time series

◮ Regressing integrated variables on other integrated variables

often yields apparently significant coefficients (even if there is
no relation) and integrated residuals: spurious regression;
◮ If residuals are stationary, the relation may be an important
dynamic equilibrium: cointegration;
◮ Cointegration is closely related to the dynamic mechanism
that is directed toward the equilibrium: error correction;
◮ The multivariate generalization of the autoregressive model is
the vector autoregression (VAR), which sets a convenient
frame for cointegration, causality etc.

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General concepts

A simple dynamic model

Consider two stationary variables Y and X , with X regarded as

exogenous, related by an autoregressive distributed lags (ARDL)
model
Yt = δ + θYt−1 + φ0 Xt + φ1 Xt−1 + εt ,
with white-noise εt and |θ| < 1. The entity

∂Yt
= φ0
∂Xt
is called the impact multiplier.

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General concepts

The equilibrium multiplier

Straightforward evaluation yields
∂Yt+1
= θφ0 + φ1
∂Xt
and
∂Yt+2 ∂Yt+1
=θ = θ(θφ0 + φ1 )
∂Xt ∂Xt

The sum of all these multipliers describes the long-run effect

∞
X ∂Yt+j θφ0 + φ1 φ0 + φ1
= φ0 + = ,
∂Xt 1−θ 1−θ
j=0

the long-run multiplier or equilibrium multiplier β.

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General concepts

Error correction

X and Y are stationary, for this reason

δ
E(Yt ) = + βE(Xt ) = α + βE(Xt )
1−θ
for some constant α. Some manipulation yields

∆Yt = φ0 ∆Xt − (1 − θ)(Yt−1 − α − βXt−1 ) + εt ,

an error-correction model: Y adjusts to deviations from

equilibrium, with 1 − θ representing the intensity of adjustment.

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General concepts

Generalizations to higher lag orders

Assume the general form

θ(L)Yt = δ + φ(L)Xt + εt ,

with lag polynomials of orders p and q and invertible θ(z).

Applying the inverse operator θ −1 (L) yields

Yt = θ −1 (L)φ(L)Xt + θ −1 (L)εt ,

and the long-run multiplier becomes

φ0 + φ1 + . . . + φq
θ −1 (1)φ(1) = ,
1 − θ1 − . . . − θp

with the denominator positive because of stability.

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Cointegration

The basic experiment of spurious regression

Presume Xt and Yt are independent random walks. The regression

Yt = α + βXt + error

tends to produce significant coefficients and non-zero R 2 . Flanking

diagnostics such as the Durbin-Watson statistic tend to indicate
that something is wrong. In fact, the residual is I(1) by
construction.
To avoid spurious regression, first subject Y and X to unit-root
tests. If these tests support the null of I(1), test residuals. If
unit-root tests on residuals fail to reject, we have a spurious
regression problem. Note, however, that the usual Dickey-Fuller
significance points are invalid for residual unit-root tests.

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Cointegration

Non-spurious regression with integrated variables

Consider two I(1) variables Yt and Xt , with Yt − βXt stationary

I(0). Then, Y and X are said to be cointegrated. The regression

Yt = α + βXt + error

yields a coefficient estimate β̂ that is super-consistent for β, i.e.

√
T (β̂ − β) → 0, T (β̂ − β) ⇒ D,

with a non-standard limit distribution D. Note that errors are I(0)

but they are usually not white noise here. This regression is called
a cointegrating regression.

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Cointegration

Cointegration with two variables

Definition
If Yt and Xt are both I(1) and there exists a β such that
Zt = Yt − βXt is I(0), then X and Y are cointegrated and
(1, −β)′ is the cointegrating vector.

The cointegrating relation can be interpreted as a long-run

equilibrium in a non-stationary dynamic model, and the model can
also be written as an error-correction model (see below).

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Cointegration

Testing for cointegration in a simple regression

In order to determine whether the regression is spurious or
cointegrating, it is recommended to first test X and Y for unit
roots. If the unit-root tests fail to reject, run the regression (no
lags!) and apply unit-root tests to residuals. Use different
significance points from the usual DF test. Appropriate significance
points have been calculated by Phillips & Ouliaris, for
example.

◮ If the residual unit-root test rejects, there is evidence on

cointegration;
◮ If the residual unit-root test fails to reject, the regression is
spurious.

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Cointegration

Cointegration and error correction

Engle & Granger have shown that, if X and Y are

cointegrated, then there exists an error-correction representation

θ(L)∆Yt = δ + φ(L)∆Xt−1 − γZt−1 + α(L)εt ,

with Zt = Yt − βXt . If Y deviates positively from the equilibrium

at t − 1, ∆Y corrects back to it at t by being smaller than usual,
and similarly for negative deviations. X could also adjust, but this
can only be captured in a full multivariate model.

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Vector autoregressions

Vector autoregressions: the concept

The vector autoregressive model (VAR) is a multivariate
generalization of the univariate AR model: variables are vectors,
coefficients are matrices:

Yt = δ + Θ1 Yt−1 + . . . + Θp Yt−p + εt ,

with Yt , Yt−1 , . . . , Yt−p , εt , δ k–vectors and Θj , j = 1, . . . , p

k × k–matrices. The lag polynomial is now a matrix polynomial

Θ(L) = Ik − Θ1 L − . . . − Θp Lp ,

with Ik denoting the k–dimensional identity matrix. In short, we

write
Θ(L)Yt = εt .

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Vector autoregressions

Stationarity of multivariate processes

In analogy to the univariate case, the vector variable Yt is said to

be (covariance) stationary, iff
◮ EYt = µ = (µ1 , . . . , µk )′ ;
◮ E(Yt − µ)(Yt − µ)′ = varYt = ΣY ;
◮ E(Yt − µ)(Yt−h − µ)′ = cov(Yt , Yt−h ) = Γ(h).

The matrix-valued function Γ(h) is the cross-covariance function,

its standardized version is the cross-correlation function (CCF). It
is skew symmetric, i.e. Γ(h)′ = Γ(−h). Also note that Γ(0) = ΣY .

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Vector autoregressions

Multivariate white noise

A k–dimensional variable (εt ) is said to be white noise, iff

◮ εt is stationary;
◮ Eεt = (0, . . . , 0)′ ;
◮ Γ(h) = 0 for h 6= 0.

Note that component j is uncorrelated with k at all leads and lags,

but typically correlated simultaneously, i.e. Σε is not usually
diagonal.

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Vector autoregressions

Stability of the VAR

O.c.s. that the VAR is stable iff all solutions (roots) ζ of the
(scalar) determinantal polynomial equation

det Θ(L) = 0

fulfill the condition |ζ| > 1. Then, it makes sense to consider the
infinite-order MA representation

Yt = Θ(1)−1 δ + Θ(L)−1 εt = µ + A(L)εt .

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Vector autoregressions

The impulse response function

The coefficient matrices of the MA representation of a VAR can be
interpreted as derivatives
∂Yt+h
Ah = ,
∂ε′t

and thus as indicating the response in the components of Y to

‘shocks’ in the components of the error process ε. For this reason,
the k 2 components of the function A(h) are called the
impulse-response function.
A(h) does not really indicate the response to shocks in the
components of past variables Y , due to dynamic feedback. A(h)
also fails to reflect the immediate response of other components in
ε, as Σε is not generally diagonal.
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Multivariate cointegration

Multivariate cointegration: the concept

Among k I(1) variables, there may be up to k − 1 linearly

independent cointegrating vectors. The cointegrating rank r may
be any value in {0, 1, . . . , k − 1}. Regression methods are an
unreliable device for the estimation of the cointegrating vectors
that may be summarized in a k × r –dimensional cointegrating
matrix of rank r , whose columns are cointegrating vectors.

Multivariate cointegration is best described in the framework of a

vector autoregression. Assume that all components of the VAR are
either I(0) or I(1).

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Multivariate cointegration

Error-correction representation of VAR

Every VAR of order p

Yt = δ + Θ1 Yt−1 + . . . + Θp Yt−p + εt

can be re-written as a vector error-correction model (VECM)

∆Yt = δ + Γ1 ∆Yt−1 + . . . + Γp−1 ∆Yt−p+1 + ΠYt−1 + εt ,

with matrices Γj and Π being functions of Θ1 , . . . , Θp , in particular

Π = −Ik + Θ1 + . . . + Θp = −Θ(1)

for the long-run matrix Π, which represents all cointegration

features in the system.

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Multivariate cointegration

The long-run matrix Π

All terms in the model ∆Yt−j , j = 0, . . . , p − 1, εt ,δ are stationary,
so ΠYt−1 must also be stationary. Three cases are of interest:
◮ Π = 0 and rkΠ = 0, the model is a VAR in ∆Y , and there is
no cointegration (k unit roots);
◮ rkΠ = k, any vector β yields β ′ Y stationary, hence Y is
already I(0) and stationary (0 unit roots);
◮ 0 < rkΠ = r < k, we can represent Π = γβ ′ with
k × r –matrices γ and β (k − r unit roots). r is the
cointegrating rank, and all columns of β form a basis of the
cointegrating space. Π and r are unique, though α and β are
not.

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Multivariate cointegration

No drift in the system

Take expectations on the VECM form (∆Y is stationary):

(I − Γ1 − . . . − Γp )E∆Yt = δ + γE(β ′ Yt−1 ) + Eεt

If there is no linear trend in any Y component, the l.h.s. is 0, and

so is the r.h.s., which implies that δ is proportional to γ or

δ = −γα,

which yields the VECM with restricted intercepts

∆Yt = γ(−α + β ′ Yt−1 ) + Γ1 ∆Yt−1 + . . . + Γp−1 ∆Yt−p+1 + εt .

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Multivariate cointegration

Example: cointegration in a bivariate VAR

Consider
      
Yt θ11 θ12 Yt−1 ε1.t
= + ,
Xt θ21 θ22 Xt−1 ε2.t

with  
θ11 − 1 θ12
Π = −Θ(1) = .
θ21 θ22 − 1
Cointegration implies that rkΠ = 1 and that the determinant is 0,
i.e. (θ11 − 1)(θ22 − 1) − θ12 θ21 = 0 or
 
θ11 − 1 θ12
Π= θ12 θ21 .
θ21 θ11 −1

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Multivariate cointegration

Example: the bivariate VECM

For the bivariate VECM, we obtain

     
∆Yt 1 ε1.t
= θ21 {(θ 11 − 1)Y t−1 + θ X
12 t−1 } + ,
∆Xt θ11 −1 ε2.t

with the recognizable cointegrating vector (θ11 − 1, θ12 ) = β ′ . One

can check that (θ11 − 1)Yt + θ12 Xt is stationary. Note that
γ = (θ11 − 1, θ21 )′ and β = (1, θ11θ12−1 )′ would also work.

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Multivariate cointegration

Testing for cointegration in a VAR

For k > 2, regression-based testing for cointegration using
residuals from cointegrating regressions becomes unreliable. The
likelihood-ratio test statistic on the cointegrating rank
H0 : r ≤ r0 vs. HA : r > r0
uses the smallest canonical correlations between variables and their
first differences (eigenvalues of specific matrices)
λ̂1 ≥ λ̂2 ≥ . . . ≥ λ̂k with 0 < λ̂j < 1. The ‘trace’ statistic
k
X
λ(r0 ) = −T log(1 − λ̂j )
j=r0 +1

has a non-standard distribution under H0 . Tabulated significance

points must be used. These differ for the case of a no-drift
restriction.
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Multivariate cointegration

Recommended steps for the ‘Johansen procedure’

1. Make sure that all component variables are either I(0) or I(1).
The procedure works if some variables are stationary.
2. Determine a lag order p for the VAR, e.g. by AIC.
3. Decide whether to use the no-trend restriction.
4. Determine the cointegrating rank by the trace test sequence,
using a VAR with p lags.
5. Use the estimates for all coefficients from a VECM estimation
that uses the rank r and p − 1 augmenting lags. This step
yields γ̂ and the cointegrating vector(s) β̂, among others.
Why not estimate the cointegrating vector from a regression after
step 4? This is much less efficient.

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