Econometrics 2 - Lecture 5
Multi-equation Models


Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
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Multiple Dependent Variables
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Economic processes: simultaneous and interrelated development of a multiple set of variables
Examples:
nHouseholds consume a set of commodities (food, durables, etc.); the demanded quantities depend on
the prices of commodities, the household income, the number of persons living in the household,
etc.; a consumption model includes a set of dependent variables and a common set of explanatory
variables.
nThe market of a product is characterized by (a) the demanded and supplied quantity and (b) the
price of the product; a model for the market consists of equations representing the development and
interdependencies of these variables.
nAn economy consists of markets for commodities, labour, finances, etc.; a model for a sector or
the full economy contains descriptions of the development of the relevant variables and their
interactions.

Systems of Regression Equations
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Economic processes involve the simultaneous developments as well as interrelations of a set of
dependent variables
nFor modelling an economic process a system of relations, typically in the form of regression
equations: multi-equation model
Example: Two dependent variables yt1 and yt2 are modelled as
yt1 = x‘t1β1 + εt1
yt2 = x‘t2β2 + εt2
with V{εti} = σi2 for i = 1, 2, Cov{εt1, εt2} = σ12 ≠ 0
 Typical situations:
1.The set of regressors xt1 and xt2 coincide
2.The set of regressors xt1 and xt2 differ, may overlap
3.Regressors contain one or both dependent variables
4.Regressors contain lagged variables

Types of Multi-equation Models
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Multivariate regression or multivariate multi-equation model
nA set of regression equations, each explaining one of the dependent variables
qPossibly common explanatory variables
qSeemingly unrelated regression (SUR) model: each equation is a valid specification of a linear
regression, related to other equations only by the error terms
qSee cases 1 and 2 of “typical situations” (slide 4)
Simultaneous equations models
nDescribe the relations within the system of economic variables
qin form of model equations
qSee cases 3 and 4 of “typical situations” (slide 4)
Error terms: dependence structure is specified by means of second moments or as joint probability
distribution

Capital Asset Pricing Model
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Capital asset pricing (CAP) model: describes the return Ri of asset i
  Ri - Rf = βi(E{Rm} – Rf) + εi
with
qRf: return of a risk-free asset
qRm: return of the market portfolio
nβi: indicates how strong fluctuations of the returns of asset i are determined by fluctuations of
the market as a whole
nKnowledge of the return difference Ri - Rf will give information on the return difference Rj -
Rf of asset j, at least for some assets
nAnalysis of a set of assets i = 1, …, s
qThe error terms εi, i = 1, …, s, represent common factors, have a common dependence structure
qEfficient use of information: simultaneous analysis

A Model for Investment
nGrunfeld investment data [Greene, (2003), Chpt.13; Grunfeld & Griliches (1960)]: Panel data set on
gross investments Iit of firms i = 1, ..., 6 over 20 years and related data
nInvestment decisions are assumed to be determined by
n Iit = βi1 + βi2Fit + βi3Cit + εit
n with
qFit: market value of firm at the end of year t-1
qCit: value of stock of plant and equipment at the end of year t-1
nSimultaneous analysis of equations for the various firms i: efficient use of information
qError terms for the firms include common factors such as economic climate
qCoefficients may be the same for the firms
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The Hog Market
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Model equations:
  Qd = α1 + α2P + α3Y + ε1  (demand equation)
Qs = β1 + β2P + β3Z + ε2   (supply equation)
Qd = Qs (equilibrium condition)
with Qd: demanded quantity, Qs: supplied quantity, P: price, Y: income, and Z: costs of production,
or
  Q = α1 + α2P + α3Y + ε1  (demand equation)
Q = β1 + β2P + β3Z + ε2   (supply equation)
nModel describes quantity and price of the equilibrium transactions
nModel determines simultaneously Q and P, given Y and Z
nError terms
qMay be correlated: Cov{ε1, ε2} ≠ 0
nSimultaneous analysis necessary for efficient use of information

Klein‘s Model I
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1.Ct = α1 + α2Pt + α3Pt-1 + α4(Wtp+ Wtg) + εt1 (consumption)
2.It = β1 + β2Pt + β3Pt-1 + β4Kt-1 + εt2   (investment)
3.Wtp = γ1 + γ2Xt + γ3Xt-1 + γ4t + εt3 (wages)
4.Xt = Ct + It + Gt
5.Kt = It + Kt-1
6.Pt = Xt – Wtp – Tt
with C (consumption), P (profits), Wp (private wages), Wg (governmental wages), I (investment),
K-1 (capital stock), X (national product), G (governmental demand), T (taxes) and t [time
(year-1936)]
nModel determines simultaneously C, I, Wp, X, K, and P
nSimultaneous analysis necessary in order to take dependence structure of error terms into account:
efficient use of information
n

Examples of Multi-equation Models
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Multivariate regression models
nCapital asset pricing (CAP) model: for all assets, return Ri is a function of E{Rm} – Rf;
dependence structure of the error terms caused by common variables
nModel for investment: firm-specific regressors, dependence structure of the error terms like in
CAP model
nSeemingly unrelated regression (SUR) models
Simultaneous equations models
nHog market model: endogenous regressors, dependence structure of error terms
nKlein’s model I: endogenous regressors, dynamic model, dependence of error terms from different
equations and possibly  over time

Single- vs. Multi-equation Models
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Complications for estimation of parameters of multi-equation models:
nDependence structure of error terms
nViolation of exogeneity of regressors
Example: Hog market model, demand equation
Q = α1 + α2P + α3Y + ε1
nCovariance matrix of ε = (ε1, ε2)’
n
n
nP is not exogenous: Cov{P,ε1} = (σ12 - σ12)/(β2 - α2) ≠ 0
Statistical analysis of multi-equation models requires methods adapted to these features

Analysis of Multi-equation Models
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Issues of interest:
nEstimation of parameters
nInterpretation of model characteristics, prediction, etc.
Estimation procedures
nMultivariate regression models
qGLS , FGLS, ML
nSimultaneous equations models
qSingle equation methods: indirect least squares (ILS), two stage least squares (TSLS), limited
information ML (LIML)
qSystem methods of estimation: three stage least squares (3SLS), full information ML (FIML)
qDynamic models: estimation methods for vector autoregressive (VAR) and vector error correction
(VEC) models
n

Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
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Example: Income and Consumption
nModel for income (Y) and consumption (C)
n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t
n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t
n with (possibly correlated) white noises ε1t and ε2t
nNotation: Zt = (Yt, Ct)‘, 2-vectors δ and ε, and (2x2)-matrix Θ = (θij), the model is
n
n
n
n in matrix notation
n Zt = δ + ΘZt-1 + εt
nRepresents each component of Z as a linear combination of lagged variables
nExtension of the AR-model to the 2-vector Zt: vector autoregressive model of order 1, VAR(1) model
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The VAR(p) Model
nVAR(p) model: generalization of the AR(p) model for k-vectors Yt
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt
n with k-vectors Yt, δ, and εt and kxk-matrices Θ1, …, Θp
nUsing the lag-operator L:
n Θ(L)Yt = δ + εt
n with matrix lag polynomial Θ(L) = I – Θ1L - … - ΘpLp
qΘ(L) is a kxk-matrix
qEach matrix element of Θ(L) is a lag polynomial of order p
nError terms εt
qhave covariance matrix Σ (for all t); allows for contemporaneous correlation
qare independent of Yt-j, j > 0, i.e., of the past of the components of Yt
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The VAR(p) Model, cont’d
nVAR(p) model for the k-vector Yt
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt
nVector of expectations of Yt: assuming stationarity
n E{Yt} = δ + Θ1 E{Yt} + … + Θp E{Yt}
n gives
n E{Yt} = μ = (Ik – Θ1 - … - Θp)-1δ = Θ(1)-1δ
n i.e., stationarity requires that the kxk-matrix Θ(1) is invertible
nIn deviations yt = Yt – μ, the VAR(p) model is
n Θ(L)yt = εt
nMA representation of the VAR(p) model, given that Θ(L) is invertible
n Yt = μ + Θ(L)-1εt  = μ + εt + A1εt-1 + A2εt-2 + …
nVARMA(p,q) Model: Extension of the VAR(p) model by multiplying εt (from the left) with a matrix
lag polynomial A(L) of order q
nVAR(p) model with m-vector Xt  of exogenous variables, kxm-matrix Γ
n Yt = Θ1Yt-1 + … + ΘpYt-p + ΓXt + εt
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Reasons for Using a VAR Model
nVAR model represents a set of univariate ARMA models, one for each component
nReformulation of simultaneous equations models as dynamic models
nTo be used instead of simultaneous equations models:
qNo need to distinct a priori endogenous and exogenous variables
qNo need for a priori identifying restrictions on model parameters
nSimultaneous analysis of the components: More parsimonious, fewer lags, simultaneous consideration
of the history of all included variables
nAllows for non-stationarity and cointegration
nAttention: the number of parameters to be estimated increases with k and p
n Number of parameters
n    in Θ(L)
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p
1
2
3
k=2
4
8
12
k=4
16
32
48

Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
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Example: Income and Consumption
nModel for income (Yt) and consumption (Ct)
n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t
n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t
n with (possibly correlated) white noises ε1t and ε2t
nMatrix form of the simultaneous equations model:
n A (Yt, Ct)‘ = Γ (1, Yt-1, Ct-1)‘ + (ε1t, ε2t)’
n with
n
n
n
nVAR(1) form: Zt = δ + ΘZt-1 + εt or
n
n
n
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Simultaneous Equations Models in VAR Form
nModel with m endogenous variables (and equations), K regressors
n Ayt = Γzt + εt = Γ1 yt-1 + Γ2 xt + εt
n with m-vectors yt and εt, K-vector zt, (mxm)-matrix A, (mxK)-matrix Γ, and (mxm)-matrix Σ =
V{εt};
nzt contains lagged endogenous variables yt-1 and exogenous variables xt
nRearranging gives
n yt = Θ yt-1 + δt + vt
n with Θ = A-1 Γ1, δt = A-1 Γ2 xt, and vt = A-1 εt
nExtension of yt by regressors xt: the matrix δt becomes a vector of deterministic components
(intercepts)
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VAR Model: Estimation
nVAR(p) model for the k-vector Yt
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt, V{εt} = Σ
nComponents of Yt: linear combinations of lagged variables
nError terms: Possibly contemporaneously correlated, covariance matrix Σ, uncorrelated over time
nSUR model
nEstimation, given the order p of the VAR model
nOLS estimates of parameters in Θ(L) are consistent
nEstimation of Σ based on residual vectors et = (e1t, …, ekt)’:
n
n
nGLS estimator coincides with OLS estimator: same explanatory variables for all equations
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VAR Model: Estimation, cont’d
nChoice of the order p of the VAR model
nEstimation of VAR models for various orders p
nChoice of p based on Akaike or Schwarz information criterion
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Income and Consumption
nAWM data base, 1971:1-2003:4: PCR (real private consumption), PYR (real disposable income of
households); respective annual growth rates of logarithms: C, Y
nFitting Zt = δ + ΘZt-1 + εt with Z = (Y, C)‘ gives
n
n
n
n
n
n
n
n
n with AIC = -14.60; for the VAR(2) model: AIC = -14.55
nIn GRETL: OLS equation-wise, VAR estimation, SUR estimation give very similar results
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δ
Y-1
C-1
adj.R2
Y
θij
0.001
0.815
0.106
0.82
t(θij)
0.39
11.33
1.30
C
Θij
0.003
0.085
0.796
0.78
t(θij)
2.52
1.23
10.16

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Impulse-response Function
nMA representation of the VAR(p) model
n Yt = Θ(1)-1δ + εt + A1εt-1 + A2εt-2 + …
nInterpretation of As: the (i,j)-element of As represents the effect of a one unit increase of
εjt upon the i-th variable Yi,t+s in Yt+s
nDynamic effects of a one unit increase of εjt upon the i-th component of Yt are corresponding to
the (i,j)-th elements of Ik, A1, A2, …
nThe plot of these elements over s represents the impulse-response function of the i-th variable in
Yt+s on a unit shock to εjt
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Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
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Stationarity and Non-stationarity
nAR(1) process Yt = θYt-1 + εt
nis stationary, if the root z of the characteristic polynomial
n Θ(z) = 1 - θz = 0
n fulfils |z| > 1, i.e., |θ| < 1;
qΘ(z) is invertible, i.e., Θ(z)-1 can be derived such that Θ(z)-1Θ(z) = 1
qYt can be represented by a MA(∞) process: Yt = Θ(z)-1εt
nis non-stationary, if
n z = 1 or θ = 1
n i.e.,Yt ~ I(1), Yt has a stochastic trend
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VAR Models, Non-stationarity, and Cointegration
nVAR(1) model for the k-vector Yt
n Yt = δ + Θ1Yt-1 + εt
nIf Θ(L) = I – Θ1L is invertible,
n Yt = Θ(1)-1δ + Θ(L)-1εt  = μ + εt + A1εt-1 + A2εt-2 + …
n i.e., each variable in Yt is a linear combination of white noises, is a stationary I(0) variable
nIf Θ(L) is not invertible, not all variables in Yt can be stationary I(0) variables: at least one
variable must have a stochastic trend
qIf all k variables have independent stochastic trends, all k variables are I(1) and no
cointegrating relation exists; e.g., for k = 2:
q
q
q i.e., θ11 = θ22 = 1, θ12 = θ21 = 0
qThe more interesting case: at least one cointegrating relation; number of cointegrating relations
equals the rank r{Θ(1)} of matrix Θ(1)
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Example: A VAR(1) Model
nVAR(1) model for k-vector Y in differences with Θ(L) = I - Θ1L
n ΔYt = - Θ(1)Yt-1 + δ + εt
n r = r{Θ(1)}: rank of (kxk) matrix Θ(1) = Ik - Θ1
1.r = 0: then ΔYt = δ + εt, i.e., Y is a k-dimensional random walk, each component is I(1), no
cointegrating relationship
2.r < k: (k - r)-fold unit root, (kxr)-matrices γ and β can be found, both of rank r, with
Θ(1) = γβ'
the r columns of β are the cointegrating vectors of r cointegrating relations (β in normalized
form, i.e., the main diagonal elements of β being ones)
3.r = k: VAR(1) process is stationary, all components of Y are I(0)
n
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Cointegrating Space
nYt: k-vector, each component I(1)
nCointegrating space:
nAmong the k variables, r ≤ k-1 independent linear relations βj‘Yt, j = 1, …, r, are possible so
that βj‘Yt ~ I(0)
nIndividual relations can be combined with others and these are again I(0), i.e., not the
individual cointegrating relations are identified but only the r-dimensional space
nCointegrating relations should have an economic interpretation
nCointegrating matrix β:
nThe kxr matrix β = (β1, …, βr) of vectors βj that state the cointegrating relations βj‘Yt ~ I(0),
j = 1, …, r
nCointegrating rank: the rank of matrix β: r{β} = r
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Granger‘s Representation Theorem
nGranger’s Representation Theorem (Engle & Granger, 1987): If a set of I(1) variables is
cointegrated, then an error-correction (EC) relation of the variables exists
nExtends to VAR models: if the I(1) variables of the k-vector Yt are cointegrated, then an
error-correction (EC) relation of the variables exists
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Granger‘s Representation Theorem for VAR Models
nVAR(p) model for the k-vector Yt
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt
n transformed into
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt  (A)
qΠ = – Θ(1) = – (Ik – Θ1 – … – Θp): „long-run matrix“, kxk, determines the long-run dynamics of Yt
qΓ1, …, Γp-1 (kxk)-matrices, functions of Θ1,…, Θp
nΠYt-1 is stationary: ΔYt and εt are I(0)
nThree cases
1.r{Π} = r with 0 < r < k: there exist r stationary linear combinations of Yt, i.e., r
cointegrating relations
2.r{Π} = 0: Π = 0, no cointegrating relation, equation (A) is a VAR(p) model for stationary
variables ΔYt
3.r{Π} = k: all variables in Yt are stationary, Π = - Θ(1) is invertible
n
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Vector Error-Correction Model
nVAR(p) model for the k-vector Yt
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt
n transformed into
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt
n with r{Π} = r and Π = γβ' gives
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt  (B)
nr cointegrating relations β'Yt-1
nAdaptation parameters γ measure the portion or speed of adaptation of Yt in compensation of the
equilibrium error Zt-1 = β'Yt-1
nEquation (B) is called the vector error-correction (VEC) model
n
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Example: Bivariate VAR Model
nVAR(1) model for the 2-vector Yt = (Y1t, Y2t)’
n Yt = ΘYt-1 + εt ; and ΔYt = ΠYt-1 + εt
nLong-run matrix
n
n
n
nΠ = 0, if θ11 = θ22 = 1, θ12 = θ21 = 0, i.e., Y1t, Y2t are random walks
nr{Π} < 2, if (θ11 – 1)(θ22 – 1) – θ12 θ21 = 0; cointegrating vector: β‘ = (θ11 – 1, θ12), long-run
matrix
n
n
n
nThe error-correction form is
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Deterministic Component
nVEC(p) model for the k-vector Yt
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt  (B)
nThe deterministic component (intercept) δ:
1.E{ΔYt} = 0, i.e., no deterministic trend in any component of Yt: given that Γ = Ik – Γ1 – … –
Γp-1 has full rank:
qΓ E{ΔYt} = δ + γE{Zt-1} = 0 with equilibrium error Zt-1 = β'Yt-1
qE{Zt-1} corresponds to the intercepts of the cointegrating relations; with r-dimensional vector
E{Zt-1} = α (and hence δ = - γα)
q ΔYt = Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γ(- α + β'Yt-1) + εt  (C)
qIntercepts only in the cointegrating relations, i.e., no deterministic  trend in the model
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Deterministic Component, cont’d
nVEC(p) model for the k-vector Yt
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt  (B)
nThe deterministic component (intercept) δ:
2.Addition of a k-vector λ with identical components to (C)
n ΔYt = λ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γ(- α + β'Yt-1) + εt
qLong-run equilibrium: steady state growth with growth rate E{ΔYt} = Γ-1λ
qDeterministic trends cancel out in the long run, so that no deterministic trend in the
error-correction term; cf. (B)
qAddition of k-vector λ can be repeated: up to k-r separate deterministic trends can cancel out in
the error-correction term
qThe general notation is equation (B) with δ containing r intercepts of the long-run relations and
k-r deterministic trends in the variables of Yt
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The Five Cases
nBased on empirical observation and economic reasoning, choice between:
1)Unrestricted constant: variables show deterministic linear trends
2)Restricted constant: variables not trended but mean distance between them not zero; intercept in
the error-correction term
3)No constant
n Generalization: deterministic component contains intercept and trend
4)Constant + restricted trend: cointegrating relations include a trend but the first differences of
the variables in question do not
5)Constant + unrestricted trend: trend in both the cointegrating relations and the first
differences, corresponding to a quadratic trend in the variables (in levels)
n
n
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Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
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Choice of the Cointegrating Rank
nBased on k-vector Yt ~ I(1)
nEstimation procedure needs as input the cointegrating rank r
nTesting for cointegration
nEngle-Granger approach
nJohansen‘s R3 method
n
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The Engle-Granger Approach
nNon-stationary processes Yt ~ I(1), Xt ~ I(1); the model is
n Yt = α + βXt + εt
nStep 1: OLS-fitting
nTest for cointegration based on residuals, e.g., DF test with special critical values; H0:
residuals are I(1), no cointegration
nIf H0 is rejected,
qOLS fitting in step 1 gives consistent estimate of the cointegrating vector
qStep 2: OLS estimation of the EC model based on the cointegrating vector from step 1
nCan be extended to k-vector Yt = (Y1t, ..., Ykt)’:
nStep 1 applied to Y1t = α + β1Y2t + ... + βkYkt + εt
nDF test of H0: residuals are I(1), no cointegration
n
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Engle-Granger Cointegration Test: Problems
nResidual based cointegration tests can be misleading
nTest results depend on specification
qWhich variables are included
qNormalization of the cointegrating vector, which variable on left hand side
nTest may be inappropriate due to wrong specification of cointegrating  relation
nTest power suffers from inefficient use of information (dynamic interactions not taken into
account)
nTest gives no information about the rank r
n
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Johansen‘s R3 Method
nReduced rank regression or R3 method: an iterative method for specifying the cointegrating rank r
nAlso called Johansen's test
nThe test is based on the k eigenvalues λi (λ1> λ2>…> λk) of
n Y1‘Y1 – Y1‘ΔY(ΔY‘ΔY)-1ΔY‘Y1,
n with ΔY: (Txk) matrix of differences ΔYt, Y1: (Txk) matrix of Yt-1
qeigenvalues λi fulfil 0 ≤ λi < 1
qif r{Θ(1)} = r, the k-r smallest eigenvalues obey
q log(1- λj) = λj  = 0,  j = r+1, …, k
nJohansen’s iterative test procedures
qTrace test
qMaximum eigenvalue test or max test
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Hackl, Econometrics 2, Lecture 5
42
Trace and Max Test: The Procedures
nLR tests, based on the assumption of normally distributed errors
nTrace test: for r0 = 0, 1, …, test of H0: r ≤ r0 (r0 or fewer cointegrating relations) against H1:
r0 < r ≤ k
n λtrace(r0) = - T Σkj=r0+1log(1- Îj)
qÎj: estimator of λj
qH0 is rejected for large values of λtrace(r0)
qStops when H0 is not rejected for the first time
qCritical values from simulations
nMax test: tests for r0 = 0, 1, …: H0: r ≤ r0 (the eigenvalue λr0+1 is different from zero) against
H1: r = r0+1
n λmax(r0) = - T log(1 - Îr0+1)
qStops when H0 is not rejected for the first time
qCritical values from simulations
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Hackl, Econometrics 2, Lecture 5
43
Trace and Max Test: Critical Limits
nCritical limits are shown in Verbeek’s Table 9.9 for both tests
nDepend on presence of trends and intercepts
qCase 1: no deterministic trends, intercepts in cointegrating relations
qCase 2: k unrestricted intercepts in the VAR model, i.e., k - r deterministic trends, r intercepts
in cointegrating relations
nDepend on k – r
nNeed small sample correction, e.g., factor (T-pk)/T  for the test statistic: avoids too large
values of r
n
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Hackl, Econometrics 2, Lecture 5
44
Example: Purchasing Power Parity
nVerbeek’s dataset ppp: Price indices and exchange rates for France and Italy, T = 186
(1/1981-6/1996)
nVariables: LNIT (log price index Italy), LNFR (log price index France), LNX (log exchange rate
France/Italy)
nPurchasing power parity (PPP): exchange rate between the currencies (Franc, Lira) equals the ratio
of price levels of the countries
nRelative PPP: equality fulfilled only in the long run; equilibrium or cointegrating relation
n LNXt = α + β LNPt + εt
n with LNPt = LNITt – LNFRt, i.e., the log of the price index ratio France/Italy
nGeneralization:
n LNXt = α + β1 LNITt – β2 LNFRt + εt
April 9, 2013

PPP: Cointegrating Rank r
nAs discussed by Verbeek: Johansen test for k = 3 variables, maximal lag order p = 3
n
n
n
n
n
nH0 not rejected that smallest eigenvalue equals zero: series are non-stationary
nBoth the trace and the max test suggest r = 2
April 9, 2013
Hackl, Econometrics 2, Lecture 5
45
H0
H1
eigen-value
λtr(r0)
p-value
H1
λmax(r0)
p-value
r = 0
r ≥ 1
0.301
93.9
0.0000
r = 1
65.5
0.0000
r  ≤ 1
r ≥ 2
0.113
28.4
0.0023
r = 2
22.0
0.0035
r  ≤ 2
r = 3
0.034
6.4
0.169
r = 3
6.4
0.1690

Contents
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Model: Cointegration Tests
nVEC Model: Specification and Estimation
n
n
n
n
n
April 9, 2013
Hackl, Econometrics 2, Lecture 5
46

Hackl, Econometrics 2, Lecture 5
47
Estimation of VEC Models
nEstimation of
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt
n requires finding (kxr)-matrices α and β with Π = αβ‘
qβ: matrix of cointegrating vectors
qα: matrix of  adjustment coefficients
nIdentification problem: linear combinations of cointegrating vectors are also cointegrating
vectors
nUnique solutions for α and β require restrictions
nMinimum number of restrictions which guarantee identification is r2
nNormalization
qPhillips normalization
qManual normalization
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Phillips Normalization
nCointegrating vector
n β’ = (β1’, β2’)
n β1: (rxr)-matrix with rank r, β2: [(k-r)xr]-matrix
nNormalization consists in transforming β into
n
n
n with matrix B of unrestricted coefficients
nThe r cointegrating relations express the first r variables as functions of the remaining k - r
variables
nFulfils the condition that at least r2 restrictions are needed to guarantee identification
nResulting equilibrium relations may be difficult to interpret
nAlternative: manual normalization
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49
Example: Money Demand
nVerbeek’s data set “money”: US data 1:54 – 12:1994 (T=164)
nm: log of real M1 money stock
ninfl: quarterly inflation rate (change in log prices, % per year)
ncpr: commercial paper rate (% per year)
ny: log real GDP (billions of 1987 dollars)
ntbr: treasury bill rate
n
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50
Money Demand: Cointegrating Relations
nIntuitive choice of long-run behaviour relations
nMoney demand
n mt = α1 + β14 yt + β15 trbt + ε1t
n Expected: β14 ≈ 1, β15 < 0
nFisher equation
n inflt = α2 + β25 trbt + ε2t
n Expected: β25 ≈ 1
nStationary risk premium
n cprt = α3 + β35 trbt + ε3t
n Stationarity of difference between cpr and trb; expected: β35 ≈ 1
n
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Money Demand: Cointegrating Vectors
nML estimates, lag order p = 6, cointegration rank r = 2, restricted constant
nCointegrating vectors β1 and β2 and standard errors (s.e.), Phillips normalization
n
n
n
April 9, 2013
m
infl
cpr
y
tbr
const
β1
1.00
0.00
0.61
-0.35
-0.60
-4.27
(s.e.)
(0.00)
(0.00)
(0.12)
(0.12)
(0.12)
(0.91)
β2
0.00
1.00
-26.95
-3.28
-27.44
39.25
(s.e.)
(0.00)
(0.00)
(4.66)
(4.61)
(4.80)
(35.5)

Hackl, Econometrics 2, Lecture 5
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Estimation of VEC Models: k=2
nEstimation procedure consists of the following steps
1.Test the variables in the 2-vector Yt for stationarity using the usual ADF tests; VEC models need
I(1) variables
2.Determine the order p
3.Specification of
qdeterministic trends of the variables in Yt
qintercept in the cointegrating relation
4.Cointegration test
5.Estimation of cointegrating relation, normalization
6.Estimation of the VEC model
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Example: Income and Consumption
nModel:
n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t
n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t
nWith Z = (Y, C)‘, 2-vectors δ and ε, and (2x2)-matrix Θ, the VAR(1) model is
n Zt = δ + ΘZt-1 + εt
nRepresents each component of Z as a linear combination of lagged variables
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54
Income and Consumption: VEC(1) Model
nAWM data base: PCR (real private consumption), PYR (real disposable income of households);
logarithms: C, Y
1.Check whether C and Y are non-stationary:
n C ~ I(1), Y ~ I(1)
2.Johansen test for cointegration: given that C and Y have no trends and the cointegrating
relationship has an intercept:
n r = 1 (p < 0.05)
n the cointegrating relationship is
n C = 8.55 – 1.61Y
n with t(Y) = 18.2
n
n
n
n
n
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Hackl, Econometrics 2, Lecture 5
55
Income and Consumption: VEC(1) Model, cont’d
3.VEC(1) model (same specification as in 2.) with Z = (Y, C)’
n DZt = - γ(β‘Zt-1 + δ) + ΓDZt-1 + εt
n
n
n
n
n
n
n
nThe model explains growth rates of PCR and PYR; AIC = -15.41 is smaller than that of the
VAR(1)-Modell (AIC = -14.45)
April 9, 2013
coint
DY-1
DC-1
adj.R2
AIC
DY
γij
0.029
0.167
0.059
0.14
-7.42
t(γij)
5.02
1.59
0.49
DC
γij
0.047
0.226
-0.148
0.18
-7.59
t(γij)
2.36
2.34
1.35

Hackl, Econometrics 2, Lecture 5
56
Estimation of VEC Models
nEstimation procedure consists of the following steps
1.Test of the k variables in Yt for stationarity: ADF test
2.Determination of the number p of lags in the cointegration test (order of VAR): AIC or BIC
3.Specification of
qdeterministic trends of the variables in Yt
qintercept in the cointegrating relations
4.Determination of the number r of cointegrating relations: trace and/or max test
5.Estimation of the coefficients β of the cointegrating relations and the adjustment α
coefficients; normalization; assessment of the cointegrating relations
6.Estimation of the VEC model
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VEC Models in GRETL
nModel > Time Series > VAR lag selection…
nCalculates information criteria like AIC and BIC from VARs of order 1 to the chosen maximum order
of the VAR
nModel > Time Series > Cointegration test > Johansen…
nCalculates eigenvalues, test statistics for the trace and max tests, and estimates of the matrices
α, β, and Π = αβ‘
nModel > Time Series > VECM
nEstimates the specified VEC model for a given cointegration rank: (1) cointegrating vectors and
standard errors, (2) adjustment vectors, (3) coefficients and various criteria for each of the
equations of the VEC model
n
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Hackl, Econometrics 2, Lecture 5
58
Your Homework
1.Read section 9.6 of Verbeek’s book. Perform the steps 1 – 6 for estimating a VEC model for
Verbeek’s dataset “money”. Is the choice p = 2 appropriate? Compare the VEC(2) models for r = 1 and
2.
2.Derive the VEC form of the VAR(2) model
n Yt = δ + Θ1Yt-1 + Θ2Yt-2 + εt
n assuming a k-vector Yt and appropriate orders of the other vectors and matrices.
n
n
April 9, 2013