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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•2

Multiple Dependent Variables
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•3
•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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•4
Economic processes encompass the simultaneous developments as well as interrelations of a set of
dependent variables
nFor modelling economic processes: 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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•5
•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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•6
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’s optimal 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, e.g., inflation rate, 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 i 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: efficient use of information
qError terms for the firms include common factors such as economic climate
qCoefficients may be the same for the firms
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•7

The Hog Market
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•8
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: cost 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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•9
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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•10
•Multivariate regression models
nCapital asset pricing (CAP) model: for all assets, return Ri (or risk premium Ri – Rf) 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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•11
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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•12
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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•13

Hackl, Econometrics 2, Lecture 5
•14
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•15
The VAR(p) Model for the k-Vector
nVAR(p) model for the k-vector Yt: generalization of the AR(p) model
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•16
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 + …
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•17
VAR(p) Model: Extensions
nof the VAR(p) model
n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt
n for the k-vector Yt
nVARMA(p,q) Model: Extension of the VAR(p) model by multiplying εt (from the left) with a matrix
lag polynomial MA(L) of order q
nVARX(p) model with m-vector Xt  of exogenous variables, kxm-matrix Γ
n Yt = Θ1Yt-1 + … + ΘpYt-p + ΓXt + εt
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•18
Reasons for Using a VAR Model
nVAR model represents a set of univariate AR(MA) 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)
April 20, 2018
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
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•19

Hackl, Econometrics 2, Lecture 5
•20
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•21
Simultaneous Equations Model 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 the set of variables by regressors xt: the matrix δt becomes a vector of
deterministic components (intercepts)
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•22
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
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
nCf. with estimation of SUR model
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•23
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•24
Income and Consumption: Estimation of VAR-System
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 with AIC = -14.60
nVAR(2) model: AIC = -14.55
nLR-test of H0: VAR(1) against H1: VAR(2): p-value 0.51
n
April 20, 2018
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

Hackl, Econometrics 2, Lecture 5
•25
Income and Consumption: Other Estimation Methods
nAlternative estimation methods
nOLS equation-wise
nSUR
n
nVAR estimation, SUR
n   estimation, and OLS
n   equation-wise estimation
n   give very similar results
April 20, 2018
δ
Y-1
C-1
adj.R2
OLS
Y
0.001
0.815
0.106
0.82
0.39
11.33
1.30
C
0.003
0.085
0.796
0.79
2.52
1.23
10.16
SUR
Y
0.001
0.815
0.106
0.82
0.39
11.47
1.31
C
0.003
0.085
0.796
0.79
2.55
1.25
10.28

Hackl, Econometrics 2, Lecture 5
•26
VAR Model Estimation in GRETL
nVAR systems
nModel > Time Series > Multivariate > Vector Autoregression
nEstimates the specified VAR system for the chosen lag order; calculates information criteria like
AIC and BIC, F-tests for various zero restrictions for the equations and for the system as a whole
nSUR model
nModel > Simultaneous equations
nAllows for various estimation methods, among them OLS and SUR; estimates the specified equations
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•27
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
April 20, 2018

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 20, 2018
Hackl, Econometrics 2, Lecture 5
•28

Hackl, Econometrics 2, Lecture 5
•29
AR(1) Process: 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 an MA(∞) process: Yt = Θ(L)-1εt
nis non-stationary, if
n z = 1, i.e., θ = 1
n i.e.,Yt ~ I(1), Yt has a stochastic trend
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•30
VAR(1) Model, Non-stationarity, and Cointegration
nVAR(1) model for the k-vector Yt = (Y1t, ..., Ykt)'
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 and ΔY1t = δ1 + ε1t, ΔY2t = δ2 + ε2t
qThe more interesting case: at least one cointegrating relation; number of cointegrating relations
equals the rank r{Θ(1)} of matrix Θ(1)
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•31
Example: A VAR(1) Model
nVAR(1) model Yt = δ + Θ1Yt-1 + εt for k-vector Y
n ΔYt = – Θ(1)Yt-1 + δ + εt
n with (kxk) matrix Θ(L) = I – Θ1L and Θ(1) = Ik - Θ1
n r = r{Θ(1)}: rank of Θ(1), 0 ≤ r ≤ k
1.r = 0: implies 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'Yt (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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•32
Cointegrating Space
nYt: k-vector with Yt ~ 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 from ΔYt = - Θ(1)Yt-1 + δ + εt = - γ β'Yt-1 + δ + εt
nThe kxr matrix β = (β1, …, βr) of vectors βj, j = 1, …, r, that state the cointegrating relations
βj'Yt ~ I(0), j = 1, …, r
nCointegrating rank: the rank of matrix β: r{β} = r
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•33
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.
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•34
Granger‘s Representation Theorem for VAR(p) Models
nVAR(p) model for the k-vector Yt with Yt ~ I(1)
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•35
Vector Error-Correction Form
nVAR(p) model for the k-vector Yt with Yt ~ I(1)
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 errors” Zt-1 = β'Yt-1
nEquation (B) is called the vector error-correction (VEC) form of the VAR(p) model
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•36
Example: Bivariate VAR(1) Model
nVAR(1) model for the 2-vector Yt = (Y1t, Y2t)'
n Yt = ΘYt-1 + εt; and ΔYt = (I2 - Θ)Yt-1 + εt = 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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•37
Example: Bivariate VAR Model, cont’d
nThe equilibrium error
n Zt = (Θ11 – 1)Y1t + Θ12Y2t
n      is stationary:
n ΔZt = (Θ11 – 1, Θ12) ΔYt
n              = (Θ11 – 1, Θ12)[1,Θ21/(Θ11 – 1)]’ Zt-1 + (Θ11 – 1, Θ12) εt
n              = (Θ11 – 1 + Θ22 – 1) Zt-1 + (Θ11 – 1, Θ12) εt
n      or
n Zt = (Θ11 + Θ22 – 1)Zt-1 + vt
n      with vt = (Θ11 – 1) ε1t + Θ12 ε2t; i.e., Zt is I(0)
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•38
Deterministic Components
nVEC(p) model for the k-vector Yt
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt  (B)
nExpectation gives
n (Ik – Γ1 – … – Γp-1)E{ΔYt} = Γ E{ΔYt} = δ + γ E{β'Yt-1}
nThe deterministic component (intercept) δ:
1.No deterministic trend in any component of Yt, i.e., E{ΔYt} = 0: given that Γ = Ik – Γ1 – … –
Γp-1 has full rank:
qΓ E{ΔYt} = δ + γE{β'Yt-1} = 0 with equilibrium error β'Yt-1 = Zt-1
qE{Zt-1} corresponds to the intercepts of the cointegrating relations; with r-dimensional vector
E{Zt-1} = α (and hence δ = - γ E{Zt-1} = - γα)
q ΔYt = Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γ(- α + β'Yt-1) + εt  (C)
qIntercepts only in the cointegrating relations
q„Restricted constant“ case
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•39
Deterministic Component, cont’d
2.Variables with deterministic trend: 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 path with growth rate E{ΔYt} = Γ-1λ
qDeterministic trends are assumed to cancel out in the long run: 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
q„Unrestricted constant“ case
3.„No constant“ case: λ = α = 0
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•40
Choice of Constants
nChoice between the three cases: visual inspection, economic reasoning
nExample 1: Income and consumption
nBoth processes are I(1)
nBoth appear to follow a deterministic linear trend
nEquilibrium relation may show an intercept
nUnrestricted constant case
nExample 2: Interest rates
nGenerally not trended
nDifference between two rates might be stationary around a non-zero mean due to, e.g.,
rate-specific risk premia
nRestricted constant case
n
n
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•41
The Five Cases
nBased on empirical observation and economic reasoning, model specification has to choose 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
April 20, 2018

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 20, 2018
Hackl, Econometrics 2, Lecture 5
•42

Hackl, Econometrics 2, Lecture 5
•43
Treatment of VEC Models
nThe following steps
1.Test of the k variables in Yt for stationarity
2.Determination of the number p of lags
3.Specification of
qdeterministic trends of the variables in Yt
qintercept in the cointegrating relations
4.Determination of the number r of cointegrating relations
5.Estimation of the coefficients β of the cointegrating relations and the adjustment coefficients γ
6.Estimation of the VEC model
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•44
Choice of the Cointegrating Rank
nThe k-vector Yt obeys Yt ~ I(1)
nYt follows the process
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt
nEstimation procedure needs as input the cointegrating rank r , i.e., the rank r = r{γβ‘}
nTesting for cointegration
nEngle-Granger approach
nJohansen‘s R3 method
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•45
The Engle-Granger Approach
nTwo non-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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•46
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, i.e., which variable on left hand side
nTest may be inappropriate due to wrong specification of cointegrating relation
nPower of the test may suffer from inefficient use of information (dynamic interactions not taken
into account)
nTest gives no information about the rank r
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•47
Johansen‘s R3 Method
nReduced rank regression (R3) method, also called Johansen's test: a method for specifying the
cointegrating rank r
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
qHas the same rank as the kxk long run matrix γβ' = �
qEigenvalues λi fulfil 0 ≤ λi < 1 for all i
qIf r{γβ'} = r, the k-r smallest eigenvalues obey
q log(1 – λj) = λj  = 0,  j = r+1, …, k
nJohansen’s iterative test procedures, based on estimates Îj of λj
qTrace test
qMaximum eigenvalue test or max test
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•48
Max Test
nLR test, based on the assumption of normally distributed errors
nCounts the number of non-zero eigenvalues
nFor r0 = 0, 1, 2, …, the null-hypothesis H0: λr0 = 0 is tested; stops when H0 is not rejected for
the first time, number of cointegrating relations is the number of rejections
nFor r0 = 0, 1, …:
qTest of H0: r ≤ r0 against H1: r = r0+1
qTest statistic
n λmax(r0) = - T log(1 - Îr0+1)
qStops when H0 is not rejected for the first time
qCritical values from simulations
nRejection of H0: r = 0 in favour of H1: r = 1: Test of no cointegrating relation
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•49
Trace Test
nLR tests, based on the assumption of normally distributed errors
nFor r0 = 1, 2, …, the null-hypothesis is tested that the sum of the eigenvalues λj, j≥r0, is zero;
stops when H0 is not rejected for the first time, number of cointegrating relations is the number
of rejections
nFor r0 = 0, 1, …:
qTest of H0: r ≤ r0 against H1: r > r0  (r0 < r ≤ k)
n λtrace(r0) = - T Σkj=r0+1log(1- Îj)
qTests whether the k-r0 smallest λj are zero
qH0 is rejected for large values of λtrace(r0)
qStops when H0 is not rejected for the first time
qCritical values from simulations
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•50
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 (“restricted constant”)
qCase 2: k unrestricted intercepts in the VAR model, i.e., k - r deterministic trends, r intercepts
in cointegrating relations (“unrestricted constant”)
nDepend on k – r0
nSmall sample correction, e.g., factor (T-pk)/T  for the test statistic: avoids too large values of
r
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•51
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
n LNXt = LNPt (A)
nRelative PPP: equality fulfilled only in the long run
n LNXt = α + β LNPt (B)
n with LNPt = LNITt – LNFRt, i.e., the log of the price index ratio France/Italy
nGeneralization:
n LNXt = α + β1 LNITt – β2 LNFRt  (C)
April 20, 2018

PPP: Cointegrating Rank r
nAs discussed by Verbeek: Johansen test for k = 3 variables, based on a VEC(3) model; cf. equation
(C)
n
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, two cointegrating relations are identified among
the variables LNIT, LNFR, and LNX
April 20, 2018
Hackl, Econometrics 2, Lecture 5
•52
r0
eigen-value
H0
H1
λtr(r0)
p-value
H1
λmax(r0)
p-value
0
0.301
r = 0
r ≥ 1
93.9
0.0000
r = 1
65.5
0.0000
1
0.113
r  ≤ 1
r ≥ 2
28.4
0.0023
r = 2
22.0
0.0035
2
0.034
r  ≤ 2
r = 3
6.4
0.169
r = 3
6.4
0.1690

Hackl, Econometrics 2, Lecture 5
•53
Identification of Cointegrating Vectors
nAfter determining the number r, identification of the cointegrating vectors of
n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt
n requires finding (kxr)-matrices γ and β with Π = γβ'
qβ: matrix of cointegrating vectors
qγ: matrix of adjustment coefficients
nIdentification problem: linear combinations of cointegrating vectors are also cointegrating
vectors
nUnique solutions for γ and β require restrictions
nMinimum number of restrictions which guarantee identification is r2
nNormalization
qPhillips normalization
qManual normalization
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•54
Phillips Normalization
nCointegrating vectors
n β' = (β1', β2')
n β1: (rxr)-matrix with rank r, β2: [(k-r)xr]-matrix
nNormalization consists in transforming the (kxr)-matrix 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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•55
Example: Money Demand
nVerbeek’s data set “money”: US data 1:54 – 4:1994 (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
nAll variables are I(1)
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•56
Money Demand: Cointegrating Relations
nIntuitive choice of long-run behaviour relations
nMoney demand
n mt = α1 + β14 yt + β15 tbrt + ε1t
n Expected: β14 ≈ 1, β15 < 0
nFisher equation (stationary real interest rate)
n inflt = α2 + β25 tbrt + ε2t
n Expected: β25 ≈ 1
nStationary risk premium
n cprt = α3 + β35 tbrt + ε3t
n Stationarity of difference between cpr and tbr; expected: β35 ≈ 1
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•57
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 20, 2018
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)

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 20, 2018
Hackl, Econometrics 2, Lecture 5
•58

Hackl, Econometrics 2, Lecture 5
•59
Estimation of VEC Models
nEstimation procedure consists of the following steps
1.Test of the k variables in Yt for stationarity: ADF test; VEC models need I(1) variables
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 relation
4.Cointegration test: 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 coefficients
γ; normalization
6.Estimation of the VEC model
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•60
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
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•61
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, results in
n C ~ I(1), Y ~ I(1)
2.Lag order with minimal AIC: p = 4
3.Restricted constant: C and Y without deterministic trend, cointegrating relation with intercept
4.Johansen test for cointegration:
n r = 1 (p < 0.05)
5.The cointegrating relationship is
n C = 8.55 – 1.61Y
n with t(Y) = 18.2
n
n
n
n
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•62
Income and Consumption: VEC(1) Model, cont’d
6.VEC(1) model (same specification, p=4, r=1) with Z = (Y, C)'
n DZt = - γ(β'Zt-1 + δ) + ΓDZt-1 + εt
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 20, 2018
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
•63
VEC Models in GRETL
nModel > Time Series > Multivariate > VAR lag selection
nCalculates information criteria like AIC and BIC for VARs of order 1 to the chosen maximum order
of the VAR; helps to choose the order p
nModel > Time Series > Multivariate > Cointegration test (Johansen), Model > … > Cointegration test
(Engle-Granger)
nCalculate eigenvalues, test statistics for the trace and max tests, and estimates of the matrices
γ, β, and Π = γβ‘; helps to choose r
nModel > Time Series > Multivariate > VECM
nEstimates the specified VEC model for given p and r: (1) cointegrating vectors and standard
errors, (2) adjustment vectors, (3) coefficients and various criteria for each of the equations of
the VEC model
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•64
Your Homework
1.Verbeek’s data set “money”: US data 1:54 – 4:1994 (T=164) with m: log of real M1 money stock,
infl: quarterly inflation rate (change in log prices, % per year), cpr: commercial paper rate (%
per year), y: log real GDP (billions of 1987 dollars), and tbr: treasury bill rate. Answer the
following questions for the three equations for m with regressors y and tbr, infl with regressor
tbr, and cpr with regressor tbr.
a.What order of integration apply to the five variables?
b.Which indications (i) for spurious regressions and (ii) for cointegrating relationships do you
see from analyses of the three equations?
c.For a VAR model for the vector Y = (m, infl, cpr, y, tbr)’, determine the number p of lags in the
cointegration test.
d.Estimate an VAR(1) model for the vector Y = (m, infl, cpr, y, tbr)’.
e.Estimate an VEC model for the vector Y = (m, infl, cpr, y, tbr)’ with p = 2 and (i) r = 1 and
(ii) r = 2. Compare the AICs for the two VEC models and the VAR model; compare the equation for d_m
in the two VEC models.
n
April 20, 2018

Hackl, Econometrics 2, Lecture 5
•65
Your Homework
2.For 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, derive the VEC
form DYt = δ + Γ1 DYt-1 + ΠYt-1 + εt; indicate Γ1 and Π as functions of the parameters Θ1 and Θ2.
April 20, 2018