Econometrics 2 - Lecture 5
Multi-equation Models


Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Multiple Dependent Variables
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In general, economic processes involve a multiple set of variables which show a simultaneous and
interrelated development
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, labor, 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 modeling 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 modeled 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 equation 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 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: 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 the equilibrium transaction quantity and price
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 equation 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
nP is not exogenous: Cov{P, ε1} = (σ12 - σ12)/(β2 - α2) ≠ 0
nCovariance matrix of ε = (ε1, ε2)’
n
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 equation 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 Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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The SUR Model
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Seemingly unrelated regression model
Multivariate regression model: the general case, m equations
yt1 = x‘t1β1 + εt1
…
ytm = x‘tmβm + εtm
with V{εti} = σi2 for i = 1,…,m; Cov{εti, εtj} = σij ≠ 0 for i ≠ j , i,j = 1,…,m, and t = 1,…,T,
i.e., contemporaneously correlated error terms
Regressors
nCan be specific for each equation
nMultivariate regression with common regressors
 xti = xt  for i = 1,…,m
e.g., the CAP model
n

Example: Investment Model
nInvestment model based on the Grunfeld data set [Greene, (2003), Chpt.13; Grunfeld & Griliches
(1960)]
n Iit = bi1 + bi2Fit + bi3Cit + εit
n with
qIit: gross investment Iit of firm i
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
nExplanatory variables observed for firm i may affect other firms due to the dependence structure
of the error terms
nEstimation methods
qEach equation separately using OLS
qGeneralized least squares estimation may take dependence structure of the error terms into account
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The SUR Model: Notation
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For m = 2
nWith T-vectors yi, εi, and (TxK)-matrix Xi
yi = Xi βi + εi,  i = 1, 2
and V{εti} = σi2, Cov{εt1, εt2} = σ12 ≠ 0 for t = 1,…,T,
nWith 2T-vectors
n
n
n
n
and

The Kronecker Product
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Definition of the Kronecker product of matrices A (order nxm) and B (order pxq)
The product matrix has the order npxmq
Some rules:                                                 suitable A, B, C, D
                                                            square A, B

Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Parameter Estimation
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For m = 2
OLS estimators for β1 and β2 on basis of yi = Xi βi + εi,  i = 1, 2
nbi = (Xi‘Xi)-1 Xi‘yi, i = 1, 2
nOr
nWith V{bi} = σi2(Xi‘Xi)-1
nIgnores Σ ≠ I, i.e., the contemporaneous correlation of error terms
GLS estimators
n
nwith
Analogous for any number m of equations
n
n
n
n
and

Investment Model
nInvestment models
n Iit = b1 + b2Fit + b3Cit + εit
n for General Motors, Chrysler, and General Electric
n
n
n
n
n
n
n
n
n
n F: market value, C: value of stock
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b1
b2
b3
R2
OLS
GLS
OLS
GLS
OLS
GLS
GM
-149.8
-133.6
0.119
0.115
0.371
0.376
0.92
p-val.
0.175
0.178
0.0002
0.0001
1.5E-8
3.1E-9
Crysler
-6.190
-3.266
0.078
0.073
0.316
0.320
0.91
p-val.
0.653
0.794
0.0011
0.0009
4.0E-9
8.6E-10
GE
-9.96
-11.96
0.027
0.028
0.152
0.152
0.71
p-val.
0.755
0.680
0.106
0.067
1.7E-5
6.1E-6

The General SUR Model
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SUR model with m equations
nThe i-the equation: T-vectors yi, εi, and (TxKi)-matrix Xi
yi = Xi βi + εi,  i = 1, …, m
V{εti} = σi2, Cov{εti, εtj} = σij ≠ 0 for i, j = 1, …, m, t = 1,…,T,
nFull model
n
n
n
n
with

FGLS Estimator
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2-step procedure: estimation of GLS estimators
n
requires knowledge of covariance matrix V of the error terms
1.OLS estimation of each of the m equations; calculation of the m-squared matrix S = E’E/T,
estimator of Σ, using the Txm matrix E = (e1, …, em) of OLS residuals, with elements sij of S
sij = (Σt etietj)/T
for i,j = 1, …, m; suitable degree of freedom correction
2.GLS estimation using instead of V the estimated matrix                  i.e., V with
sij substituted for σij for i, j = 1, …, m
n
n
n
n
and

GLS and OLS Estimators
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 GLS estimators
n
nMore efficient than OLS estimators
nEfficiency gain increases
qwith growing correlation of error terms
qwith shrinking correlation of regressors
nGLS estimator for βi coincides with OLS estimator bi if
qmatrix Xi of regressors is the same for all equations: Xi = X
qεti is uncorrelated with all εtj, j ≠ i
nFGLS estimates are consistent, asymptotically efficient

Goodness of Fit Measures
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 Definition of an R2-like measure:
n
n
n
with
using the Txm matrix                              of FGLS residuals and analogously the mxm matrix
Syy of sample covariances of the yti
An alternative measure is

SUR Model in GRETL
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Model > simultaneous equation …
nto be specified
qthe simultaneous equations
qoption for estimator: Seemingly Unrelated Regression (sur)
nFGLS estimation

Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Example: Two-equation Model
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Dependent variables Y1, Y2; the model is
Y1 = α1 + α2Y2 + α3X1 + ε1  (equation A)
Y2 = β1 + β2Y1 + β3X2 + ε2  (equation B)
1.Violation of assumption A2 (exogeneity of regressors): a positive ε1
qresults in an increase of Y1 (see equation A)
qConsequence of this (see equation B) is, given a positive β2, an increase of Y2
qi.e., ε1 and Y2 are correlated
2.Biased OLS estimators
qLarge values of Y1 are observed – due to positive values of ε1 – together with large values of Y2
qOverestimated α2

Example: Market Model
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Describes the transactions in the market of a commodity, e.g., hogs
  Qd = α1 + α2P + α3Y + ε1  (demand equation)
Qs = β1 + β2P + β3Z + ε2   (supply equation)
Qd = Qs (equilibrium condition, market clearing assumption)
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 the equilibrium transaction quantity Q and price P
nExogeneity assumption for variables Y, Z
qModel determines simultaneously Q and P, given Y and Z
qEndogenous variables: Q, P

Market Model
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Structural form of the model
  Q = α1 + α2P + α3Y + ε1  (demand equation)
Q = β1 + β2P + β3Z + ε2   (supply equation)
Corresponding reduced form
Q = π11 + π12Y + π13Z + u1
P = π21 + π22Y + π23Z + u2
with π11 = (α1 β2–α2 β1)/(β2–α2), u1 = (β2 ε1–α2 ε2)/(β2–α2), etc.
nGiven values for Y and Z, values for Q and P can be calculated
nParameter estimation:
qEstimates from structural form parameters are biased, inconsistent; see above
qReduced form equations are a SUR model; FGLS estimates are consistent, asymptotically efficient

Simultaneous Equation Models: Estimation
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Issues:
nEstimation problem: What methods can be applied to multi-equation model, what properties will the
estimates have?
nIdentification problem: Given estimates of reduced form parameters, can from them structural
parameters be derived?

Types of Variables
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Endogenous variables
nDetermined by the model
Exogenous variables
nDetermined from outside the model
nTypes of exogenous variables
qStrictly exogenous variables: uncorrelated with past, actual, and future error terms
qPredetermined variables: uncorrelated with actual and future error terms, e.g., lagged endogenous
variables
Complete system of equations: number of equations equals the number of endogenous variables

Structural and Reduced Form
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Structural form: represents relations between endogenous variables and exogenous (and
predetermined) variables according to economic theory
Reduced form: describes the dependence of endogenous variables upon exogenous or predetermined
variables
Coefficients of
nStructural form: Interpretation as structural parameters corresponding to economic theory
nReduced form: Interpretation as impact multiplicator, indicating the effects of changes of the
predetermined variables on endogenous variables

Market Model: Structural Form
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Structural form of the 2-equation model
Qt = α1 + α2Pt + α3Yt + εt1  (demand equation)
Qt = β1 + β2Pt + β3Zt + εt2   (supply equation)
with εt = (εt1, εt2)‘: bivariate white noise with
Structural form in matrix notation:
Ayt = Γzt + εt
with
yt = (Qt, Pt)‘, zt = (1, Yt, Zt)‘

Market Model: Reduced Form
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Reduced form in matrix notation
yt = A-1 Γzt + A-1εt = Πzt + ut
with
and Ω = V{ut} = A-1 Σ(A-1)’
Reduced form equations:
Qt = π11 + π12Yt + π13Zt + ut1
Pt = π21 + π22Yt + π23Zt + ut2

Multi-equation Model: General Structural Form
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Model with m endogenous variables (and equations), K regressors
Ayt = Γzt + εt
with m-vectors yt and εt, K-vector zt, (mxm)-matrix A, (mxK)-matrix Γ, and (mxm)-matrix Σ = V{εt}
Structure of the multi-equation model: (A, Γ, Σ)
Structural parameters: Elements of A and Γ
Normalized matrix A: αii = 1 for all i
Complete multi-equation model: A is a square matrix with full rank, i.e., A is invertible
Recursive multi-equation model: A is a lower triangular matrix, i.e., yti can be regressor in
equations j with j > i only

Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Example: A Simple Market Model
nStructural form of the 2-equation model
n Qt = α1 + α2Pt + εt1  (demand equation)
n Qt = β1 + β2Pt + εt2   (supply equation)
nObservations (Qt, Pt), t = 1, …, T,
qA cloud of points in the scatter diagram
qOLS estimation gives slope and intercept: not clear whether these parameters correspond to demand
or supply equation
qDemand and supply equations are “not identified”
nReduced form equations
n Qt = π11 + ut1, Pt = π21 + ut2
n with π11 = (α1β2 – α2β1)/(β2 – α2), π21 = (α1 – β1)/(β2 – α2)
n
n
n
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A Simple Market Model, cont’d
nGiven estimates for π11 and π21, the two equations
n π11 = (α1β2 – α2β1)/(β2 – α2)
n π21 = (α1 – β1)/(β2 – α2)
n have no unique solution for four structural parameters α1, α2, β1, and β2
n“Identifying” the demand or supply equation from the data is linked to a unique solution for the
structural parameters
nBoth the demand and supply equations are not identified or unidentified
n
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A Modified Market Model
n Qt = α1 + α2Pt + α3Yt + εt1  (demand equation)
n Qt = β1 + β2Pt + εt2   (supply equation)
nCoefficients of the reduced form
n π11 = (α1β2 – α2β1)/(β2 – α2), π12 = α3β2/(β2 – α2)
n π21 = (α1 – β1)/(β2 – α2), π22 = α3/(β2 – α2)
n with OLS estimates pij, i, j =1, 2, j=1, 2
nSupply equation: estimates for β1, β2 are uniquely determined
n b2 = p12/p22, b1 = p11 – p21b2
nDemand equation: only two equations for α1, …, α3
n a1 = p11 – p21a2, a3 = p22(b2 – a2)
n No unique solutions
nThe supply equation is identified, the demand equation is unidentified
n
n
n
n
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One more Market Model
n Qt = α1 + α2Pt + α3Yt + εt1  (demand equation)
n Qt = β1 + β2Pt + β3Zt + εt2   (supply equation)
nOLS estimates for reduced form parameters pij, i=1, 2, j=1, ..., 3, give estimates
n a2 = p13/p23, b2 = p12/p22
nThe parameters of the demand equation are uniquely determined:
n a3 = p22(b2 – a2), a1 = p11 – p21a2
nThe supply equation parameters are uniquely determined:
n b3 = – p23(b2 – a2), b1 = p11 – p21b2
nBoth the supply equation and the demand equation are identified
n
n
n
n
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Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Counting the Parameters
nNumber of structural parameters:
qA: mxm non-singular matrix, i.e., m2 parameters
qΓ: mxK matrix, i.e., mK parameters
qΣ: mxm symmetric, positive definite matrix, i.e., m(m+1)/2 parameters
nNumber of reduced form parameters:
qΠ: mxK matrix, i.e., mK parameters
qΩ: mxm symmetric, positive definite matrix , i.e., m(m+1)/2 parameters
nNumber of structural parameters exceeds that of reduced form parameters by m2
nIdentification requires further information such as restrictions for parameters
n
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Identification: Parameter Restrictions
nRestrictions on structural parameters: reduce the number of parameters to be estimated, so that
equations are identified
nNormalization: in each structural equation, one coefficient is a “1”
nExclusion: the omission of a regressor in an equation results in a zero in A or Γ, i.e., reduces
the number of structural parameters
nIdentities, like equations 4 through 6 in Klein’s I model, reduce the number of structural
parameters to be estimated
nLinear – or non-linear – restrictions on structural parameters, restrictions on the elements of Σ
also, reduce the number of structural parameters to be estimated
nCheck of identification
nOrder condition
nRank condition
n
n
n
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Order Condition
nModel with m endogenous variables (and equations), K regressors
n Ayt = Γzt + εt
nEquation j:
qmj: number of explanatory endogenous variables
qmj*: number of excluded endogenous variables (mj* = m – mj – 1)
qKj*: number of excluded exogenous variables (Kj* = K – Kj)
nOrder Condition: Equation j is identified if
n Kj* ≥ mj
n i.e., the number of exogenous variables excluded from equation j is at least as large as the
number of explanatory endogenous variables included in the equation
nThe order condition is a necessary but not sufficient condition for identification
n
n
n
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Market Model
nModel:
n Qt = α1 + α2Pt + α3Yt + εt1  (demand equation)
n Qt = β1 + β2Pt + εt2   (supply equation)
n m = 2 (Q, P), K = 2 (1 for the intercept, Y)
nSupply equation (j = 2):
n m2* = 0, m2 = 1, K2* = 1, K2 = 1
n Order condition is fulfilled: K2* = 1 = m2 = 1; the supply equation is identified
nDemand equation (j = 1):
n m1* = 0, m1 = 1, K1* = 0, K1 = 2
n Order condition is not fulfilled: K1* = 0 < m1 = 1; the demand equation is not identified
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Rank Condition
nModel with m endogenous variables (and equations), K regressors
n Ayt = Γzt + εt
nEquation j:
qA*: obtained by deleting from A the j-th row and all column with a non-zero element in the j-th
row
qΓ*: obtained by deleting from Γ the j-th row and all column with a non-zero element in the j-th
row
nRank Condition: Equation j is identified if
n r(A*| Γ*) ≥ m – 1
n i.e., the rank of the matrix (A*| Γ*) is at least as large as the number of endogenous variables
minus 1
nThe order condition is a sufficient condition for identification of equation j
n
n
n
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IS-LM Model
n Ct = g11 – a14Yt + εt1
n It  = g21 – a23Rt + εt2
n Rt = – a34Yt + g32Mt + εt3
n Yt = Ct + It + Zt
nEquation 1:
qOrder condition: K1* = 2 ≥ m1 = 1
qRank condition:  r(A*| Γ*) = 3 ≥ m – 1 = 3
n
n
n
n
n Both conditions are fulfilled, equation 1 is identified
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C: consumption, I: investments, R: interest rate, Y: production/income, M: money, Z: autonomous
expendi-tures
endo.: C,I,R,Y (m=4); exo.: 1,M,Z (K=3)
n

Identification Checking: The Practice
1.A multi-equation model is identified if each equation is identified
2.Most equations which fulfill the order condition also fulfill the rank condition
3.Identification checking for small models is usually easy; equations of large models usually are
identified (large models contain large numbers of predetermined variables)
4.Addition of an equation to an identified model: the resulting model is identified if the new
equation contains at least one additional variable
5.
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Identification: More Notation
nEquation j is
1.Exactly identified: Kj* = mj and rank condition is met
2.Overidentified: Kj* > mj and rank condition is met
3.Underidentified: Kj* < mj or rank condition fails
n
n
n
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Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Simultaneous Equation Models: Estimation Methods
1.Single equation methods, also limited information methods
qIndirect least squares estimation (ILS)
qTwo stage least squares estimation (2SLS or TSLS)
qLimited information ML estimation (LIML)
2.(Complete) system methods, also full information methods
qThree stage least squares estimation (3SLS )
qFull information ML estimation (FIML)
n
n
n
n
n
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The Modified Market Model
nEstimator for β2 from
n Qt = α2Pt + α3Yt + εt1  (demand equation)
n Qt = β2Pt + εt2   (supply equation)
n with contemporaneously correlated error terms
n T-vectors p and q;
nOLS estimate for β2 from the supply equation: b2 = (p‘p)-1p‘q; is biased
nIV estimate for β2 with instrumental variable Y: b2IV = (y‘p)-1y‘q; is consistent
nILS estimate: b2ILS = p2/p1 = (y‘p)-1y‘q
n with OLS estimates p1 and p2 for π1 and π2 from the reduced form
n P = π1Y + u1
n Q = π2Y + u2
n
n
n
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Modified Market Model, cont’d
4.2SLS estimate for β2 of the supply equation
qStep 1: Regression of the explanatory variable P on the instrumental variable Y, calculation of
fitted values
q     = [(y‘y)-1y‘p] y
qStep 2: OLS estimation of β2 from Qt = β2     + vt
q
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OLS Estimation
nOLS estimators of structural parameters: in general
qbiased
qnot consistent
nBut often a feasible alternative
qOLS estimator is efficient, i.e., has minimal variance; may be a good estimator in spite of
unbiasedness
qTends to be robust against not fulfilled assumptions
qMay be advantageous for small or moderate sample sizes; not depending upon asymptotics
nOLS estimators for parameters of recursive simultaneous equation models: asymptotically unbiased
nOLS technique: important procedure in all estimation methods for simultaneous equation models
n
n
n
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Indirect Least Squares (ILS) Estimation
nModel with m endogenous variables (and equations), K regressors
nStructural form
n Ayt = Γzt + εt, V{εt} = Σ
nReduced form
n yt = A-1 Γzt + A-1εt = Πzt + ut , V{ut} = Ω
nEstimation of the structural parameters of equation j:
qStep 1: OLS estimation of reduced form parameters �
qStep 2: Calculation of estimates for structural parameters, solving AΠ = Γ for the structural
parameters of equation j
nEstimation of structural parameters of equation j: equation j needs to be identified
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Two Stage Least Squares (2SLS) Estimation
nEstimation of structural parameters of equation j
n yj = Xjβj + εj = Yjαi + Zjγj + εj
n with Xj: [Tx(mj-1+Kj)]-matrix of explanatory variables, Yj: [Tx(mj-1)]-matrix of explanatory
endogenous variables, Zj: (TxKj)-matrix of exogenous variables
n2SLS (or TSLS) estimation in two steps:
qStep 1: OLS estimation of reduced form parameters Π, calculation of predictions Ŷj
qStep 2: OLS estimation of structural parameters βj, using
q yj =      βj + vj
n       with       = (Ŷj Zj)
n2SLS (or TSLS) estimation of structural parameters of equation j requires the equation to be
identified
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Example: Hog Market
nUS hog market 1922-1941 (Merill & Fox, 1971): P: retail price for hog (US cents p.lb.), Q:
hog-consumption p.c., Y: income p.c. (USD), Z: exogenous production factor
n
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Hog Market: The Model
nModel with endogenous variables Q, P, exogenous variables Y, Z
n Qt = α1 + α2Pt + α3Yt + εt1  (demand equation)
n Qt = β1 + β2Pt + β3Zt + εt2   (supply equation)
n both equations are exactly identified
nCoefficients of demand and supply equation estimated by three single equation methods
nSeparate OLS estimation of both equations
nILS estimation
n2SLS estimation
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Example: Hog Market
nComparison of three single equation estimation methods
nStrong coincidence of ILS and 2SLS estimates
nOLS estimates of demand equation deviates substantially from ILS and 2SLS estimates
n
n
n
n
n
n
n
April 29, 2011
Demand
Supply
const
P
Y
const
P
Z
OLS
coeff
56.962
-1.410
0.080
15.355
-0.030
0.744
p-val
8.7e-9
0.002
0.003
0.004
0.701
1.4e-10
2SLS
coeff
60.885
-3.088
0.149
8.318
0.177
0.770
p-val
4.2e-9
0.001
0.000
0.241
0.222
2.9e-24
ILS
coeff
60.885
-3.088
0.149
8.318
0.177
0.770

2SLS Estimator: Properties
n2SLS (or TSLS) estimation of structural parameters of equation j requires the equation to be
identified
nOrder condition Kj* ≥ mj: number of excluded exogenous variables (Kj*) is at least the number of
explanatory endogenous variables (mj)
ni.e., the number of potential instrumental variables is at least the number of variables to be
substituted by predictions
nProperties: 2SLS estimators are
nConsistent
nAsymptotically normally distributed
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LIML Estimator
nLimited information ML (LIML) estimation:
nMaximization of the likelihood function derived from the system  of
qone structural equation
qreduced form equations for the remaining endogenous variables
qAssumes normally distributed error terms
nApplication:
nAsymptotic distribution of LIML estimator equivalent to that of the 2SLS estimator
nWrt computational effort, 2SLS estimation is much easier to use
nIn practical applications, LIML estimation is hardly used
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Contents
nSystems of Regression Equations
nMultivariate Regression Models
nSUR Model: Estimation
nSimultaneous Equation Models
nSimultaneous Equation Models: Identification
nIdentification: Criteria
nSingle Equation Estimation Methods
nSystem Estimation Methods
n
n
n
n
n
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Why System Estimation Methods?
nSingle equation (limited information) estimation methods ignore the contemporaneous correlation of
error terms
nSystem (full information, complete system) estimation methods take contemporaneous correlation of
error terms into account
nEstimation of equation parameters is more efficient: estimation of coefficients of equation j
makes use information contained in other equations
nEstimation methods
q3SLS estimation
qIterative 3SLS estimation
qFull information ML (FIML) estimation
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3SLS Estimator
nThe m equations of the full model in matrix notation
n
n
n
n
n
n with
n
n3SLS estimation: FGLS estimation based on
n2SLS residuals for each of the m equations
nestimate for Σ obtained from these residuals
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3SLS Estimator: Three Steps
nThe steps of the 3SLS estimation are
1.Based on the reduced form equations, calculation of predicted values for all explanatory
endogenous variables; cf. the first stage of 2SLS estimation
2.For the j-th equation, j = 1, …, m,
qCalculation of 2SLS estimators bj and
q2SLS residuals ej = yj – Xjbj
n Estimation of elements σij = Cov{εti, εtj} of Σ: sij = (ej‘ej)/T
3.Calculation of the 3SLS estimator
n
n with the projection matrix Pz = Z(Z‘Z)-1Z‘, S the estimated matrix Σ
n
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3SLS Estimator: Properties
n3SLS estimation requires all equations of the system to be identified
nProperties: 3SLS estimators are
nConsistent
nAsymptotically normally distributed
n3SLS estimates coincide with 2SLS estimates if
nAll equations are exactly identified
nThe error terms are contemporaneously uncorrelated, i.e., Σ is diagonal
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Example: Hog Market
nComparison of 2SLS and 3SLS estimation
nStrong coincidence of 3SLS and 2SLS estimates
nSmaller p-values of most 3SLS estimate indicate higher efficiency
n
n
n
n
n
n
n
April 29, 2011
Demand
Supply
const
P
Y
const
P
Z
3SLS
coeff
60.885
-3.088
0.149
8.318
0.177
0.770
p-val
1.8e-10
0.002
6.9e-5
0.204
0.186
2.9e-28
2SLS
coeff
60.885
-3.088
0.149
8.318
0.177
0.770
p-val
4.2e-9
0.001
0.000
0.241
0.222
2.9e-24

More System Estimators
nIterative 3SLS estimator
n3SLS estimates b3SLS of structural parameters (or A3SLS and Γ3SLS) give
qrevised reduced form parameters (A-1Γ = Π) and
qpredictions of the explanatory endogenous variables;
nIterative 3SLS estimator: starting with an initial 3SLS estimator, the following iterations are
repeatedly executed until convergence is reached
qOuter iteration: step 1 of 3SLS estimation resulting in improved predictions of the predetermined
variables,
qInner iteration: step 2 of 3SLS estimation, resulting in improved 2SLS residuals and estimate S
for Σ, and step 3, resulting in improved 3SLS estimators b3SLS
qThe inner iteration can be repeated using 3SLS residuals for estimate S
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More System Estimators, cont’d
nFull information ML (FIML) estimator:
nAssumes normally distributed error terms
nMaximizes likelihood function with respect to structural parameters
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Hackl, Econometrics 2, Lecture 4
70
Simultaneous Equation Models in GRETL
nModel > Simultaneous Equations …
nchoice of estimator
qSUR
q2SLS
qLIML
q3SLS
qFIML
nSpecification of equations, instrumental variables, endogenous  variables, and identities
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Your Homework
nKlein’s model I consists of the following equations (see the GRETL data file “klein”):
§Ct = α1 + α2Pt + α3Pt-1 + α4Wt + εt1 (consumption)
§It = β1 + β2Pt + β3Pt-1 + β4Kt-1 + εt2   (investment)
§Wtp = γ1 + γ2Xt + γ3Xt-1 + γ4t + εt3 (wages)
§Wt =Wtp+ Wtg
§Xt = Ct + It + Gt
§Kt = It + Kt-1
§Pt = Xt – Wtp – Tt
n Endogenous variables are: C I Wp X W K P
1.Which of the equations are identified? Use (a) order and (b) rank conditions in answering the
question.
2.Estimate the structural parameters using (a) OLS, (b) SUR, (c) 2SLS, (d) 3SLS, and (e) FIML;
compare the results and explain pros and cons of the methods.
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Your Homework, cont’d
3.The goodness of fit measure
4.
4.
4.
n makes use of
n
n with the Txm matrix                              of FGLS residuals and analogously the mxm matrix
Syy of sample covariances of the yti. Show for T = m = 2 that the two numerators in the definition
of RI2, Sg(.) and tr(.), coincide.
n
April 29, 2011