Econometrics - Lecture 6
GMM-Estimator and Econometric Models


Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
Nov 10, 2017
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From OLS to IV Estimation
nLinear model yi = xi‘β + εi
nOLS estimator: solution of the K normal equations
n 1/N Σi(yi – xi‘b) xi = 0
nCorresponding moment conditions
n E{εi xi} = E{(yi – xi‘β) xi} = 0
nIV estimator given R instrumental variables zi which may overlap with xi: based on the R moment
conditions
n E{εi zi} = E{(yi – xi‘β) zi} = 0
nIV estimator: solution of corresponding sample moment conditions
n
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The IV Estimator
nGeneral case: R moment conditions
nR = K: one unique solution, the IV estimator; identified model
n
nR > K: minimization  of the weighted quadratic form in the sample moments
n
n with a RxR positive definite weighting matrix WN gives the generalized instrumental variable
(GIV) estimator
n
n Optimal weighting matrix: WNopt = [1/N(Z’Z)]-1; corresponds to the most efficient IV estimator
n
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Generalized Method of Moments (GMM) Estimation
nThe model is characterized by R moment conditions and the corresponding equations
n E{f(wi, zi, θ)} = 0
n [cf. E{(yi – xi‘β) zi} = 0]
nf(.): R-vector function
nwi: vector of observable variables, exogenous or endogenous
nzi: vector of instrumental variables
nθ: K-vector of unknown parameters
nSample equivalents gN(θ) of moment conditions should fulfill
n
n
nEstimates     are chosen such that the sample moment conditions are fulfilled
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GMM Estimation
nR ≥ K is a necessary condition for GMM estimation
nR = K: unique solution, the K-vector    , of
n gN(θ) = 0
n if f(.) is nonlinear in θ, numerical solution might be derived
nR > K: in general, no choice     for the K-vector θ will result in gN(  ) = 0 for all R equations;
for a good choice   , gN(  ) ~ 0, i.e., all components of gN(  ) are close to zero
n estimate    is obtained through minimization wrt θ of the quadratic form
n QN(θ) = gN(θ)‘ WN gN(θ)
n    WN: symmetric, positive definite weighting matrix
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The GMM Estimator
nWeighting matrix WN
nDifferent weighting matrices result in different consistent estimators with different covariance
matrices
nOptimal weighting matrix
n WNopt = [E{f(wi, zi, θ) f(wi, zi, θ)’}]-1
n i.e., the inverse of the covariance matrix of the sample moments
nFor R = K : WN = IN with unit matrix IN
nMinimization of QN(θ) = gN(θ)‘ WN gN(θ): For nonlinear f(.)
nNumerical optimization algorithms
nWN depends on θ; iterative optimization
n
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Example: The Linear Model
nModel: yi = xi‘β + εi with E{εi xi} = 0 and V{εi} = σε²
nMoment or orthogonality conditions:
n E{εt xt} = E{(yt - xt‘β)xt} = 0
n f(.) = (yi - xi‘β)xi, θ = β, instrumental variables: xi; moment conditions are exogeneity
conditions for xi
nSample moment conditions:
n 1/N Σi (yi - xi ‘b) xi = 1/N Σi ei xi = gN(b) = 0
nWith WN = IN, QN(β) = [1/N]2 (Σi εi xi)’(Σi εi xi) = [1/N]2 X’εε’X
nOLS and GMM estimators coincide, give the estimator b, but
qOLS: residual sum of squares SN(b) = 1/N Σi ei2 has its minimum
qGMM: QN(b) = [1/N]2 (Σi ei xi)’(Σi ei xi) = 0
n
n
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Linear Model, E{εt xt}  ≠ 0
nModel yi = xi‘β + εi with V{εi} = σε², E{εi xi}  ≠ 0 and R instrumental variables zi
nMoment conditions:
n E{εi zi} = E{(yi - xi‘β)zi} = 0
nSample moment conditions:
n 1/N Σi (yi - xi‘b) zi = gN(b) = 0
nIdentified case (R = K): the single solution is the IV estimator
n bIV = (Z’X)-1 Z’y
nOver-identified case (R > K): GMM estimator from
n minβ QN(β)= minβ gN(β)’WN gN(β)
n
n
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Linear Model: GMM Estimator
nMinimization of QN(β)= minβ gN(β)’WN gN(β) wrt β:
nFor WN = I, the first order conditions are
n
n
n resulting in the estimator
n b = [(X’Z)(Z’X)]-1 (X’Z)Z’y
n b coincides with the IV estimator if R = K
nThe optimal weighting matrix WNopt  = (E{εi2zizi‘}) -1 is estimated by
n
n
n generalizes the covariance matrix of the GIV estimator to White‘s heteroskedasticity-consistent
covariance matrix estimator (HCCME)
n
n
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Example: Labour Demand
nVerbeek’s data set “labour2”: Sample of 569 Belgian companies (data from 1996)
nVariables
qlabour: total employment (number of employees)
qcapital: total fixed assets
qwage: total wage costs per employee (in 1000 EUR)
qoutput: value added (in million EUR)
nLabour demand function
n labour = b1 + b2*output + b3*capital
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Labour Demand Function: OLS Estimation
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In logarithmic transforms: Output from GRETL
Dependent variable : l_LABOUR
Heteroskedastic-robust standard errors, variant HC0,
             coefficient   std. error   t-ratio    p-value
  -------------------------------------------------------------
  const            3,01483 0,0566474          53,22     1,81e-222 ***
  l_ OUTPUT  0,878061 0,0512008   17,15     2,12e-053 ***
  l_CAPITAL   0,003699 0,0429567            0,08610      0,9314
Mean dependent var                       4,488665   S.D. dependent var    1,171166
Sum squared resid 158,8931   S.E. of regression   0,529839
R- squared               0,796052   Adjusted R-squared 0,795331
F(2, 129)               768,7963  P-value (F)               4,5e-162
Log-likelihood          -444,4539 Akaike criterion       894,9078
Schwarz criterion                          907,9395   Hannan-Quinn 899,9928
Nov 10, 2017

GMM Estimation in GRETL
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Specification of function and orthogonality conditions for labour demand model
# initializations go here
matrix X = {const , l_OUTPUT, l_CAPITAL}
series e = 0
scalar b1 = 0
scalar b2 = 0
scalar b3 = 0
matrix V = I(3)
gmm e = l_LABOuR - b1*const – b2*l_OUTPUT – b3*l_CAPITAL
  orthog e; X
  weights V
  params b1 b2 b3
end gmm
Nov 10, 2017

Labour Demand Function: GMM Estimation
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In logarithmic transforms: Output from GRETL
Using numerical derivatives
Tolerance = 1,81899e-012
Function evaluations: 44
Evaluations of gradient: 8
Model 8: 1-step GMM, using observations 1-569
e = l_LABOUR - b1*const - b2*l_OUTPUT - b3*l_CAPITAL
              estimate    std. error   t-ratio     p-value
  --------------------------------------------------------------------------
  b1         3,01483      0,0566474    53,22      0,0000    ***
  b2         0,878061     0,0512008    17,15      6,36e-066 ***
  b3         0,00369851   0,0429567     0,08610   0,9314
  GMM criterion: Q = 1,1394e-031 (TQ = 6,48321e-029)
Nov 10, 2017

Labour Demand Functions: Comparison of Estimates
nOLS and GMM estimates coincide
n
n
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OLS
GMM
const
3,015
3,015
0,057
0,057
L_OUTPUT
0,878
0,878
0,051
0,051
l_CAPITAL
0,0037
0,0037
0,0430
0,0430

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GMM Estimator: Properties
nUnder weak regularity conditions, the GMM estimator is
nconsistent (for any WN)
nmost efficient if WN = WNopt = [E{f(wi, zi, θ) f(wi, zi, θ)’}]-1
nasymptotically normal:
n where V = D WNopt D’ with the KxR matrix of derivatives
n
n
nThe covariance matrix V-1 can be estimated by substituting the population parameters θ by sample
equivalents    evaluated at the GMM estimates in D and WNopt
n
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GMM Estimator: Calculation
1.One-step GMM estimator: Choose a positive definite WN, e.g., WN = IN, optimization gives
(consistent, but not efficient)
2.Two-step GMM estimator: use the one-step estimator      to estimate V = D WNopt D‘, repeat
optimization with WN = V-1; this gives
3.Iterated GMM estimator: Repeat step 2 until convergence
nIf R = K, the GMM estimator is the same for any WN, only step 1 is needed; the objective function
QN(θ) is zero at the minimum
nIf R > K, step 2 is needed to achieve efficiency
Nov 10, 2017

GMM and Other Estimation Methods
nGMM estimation generalizes the method of moments estimation
nAllows for a general concept of moment conditions
nMoment conditions are not necessarily linear in the parameters to be estimated
nEncompasses various estimation concepts such as OLS, GLS, IV, GIV, ML
n
n
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moment conditions
OLS
E{(yi – xi’β) xi} = 0
GLS
E{(yi – xi’β) xi /σ2 (xi)} = 0
IV
E{(yi – xi’β) zi} = 0
ML
E{∂/∂β f[εi(β)]} = 0

Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
Nov 10, 2017
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Klein‘s Model 1
n Ct = a1 + a2Pt + a3Pt-1 + a4(Wtp+ Wtg) + et1   (consumption)
n It = b1 + b2Pt + b3Pt-1 + b4Kt-1 + et2   (investments)
n Wtp = g1 + g2Xt + g3Xt-1 + g4t + et3    (private wages and salaries)
n Xt = Ct + It + Gt
n Kt = It + Kt-1
n Pt = Xt – Wtp – Tt
nC (consumption), P (profits), Wp (private wages and salaries), Wg (public wages and salaries), I
(investments), K (capital stock), X (production), G (governmental expenditures without wages and
salaries), T (taxes) and t [time (trend)]
nEndogenous: C, I, Wp, X, P, K; exogeneous: Wg, G, T, t, P-1, K-1, X-1
Nov 10, 2017

EViews: n=100; series y1 = nrnd; y2 = y2(-1)+y1

Early Econometric Models
nKlein‘s Model
nAims:
qto forecast the development of business fluctuations and
qto study the effects of government economic-political policy
nSuccessful forecasts of
qeconomic upturn rather than a depression after World War II
qmild recession at the end of the Korean War
n
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Model
year
eq‘s
Tinbergen
1936
24
Klein
1950
6
Klein & Goldberger
1955
20
Brookings
1965
160
Brookings Mark II
1972
~200

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Econometric Models
nBasis: the multiple linear regression model
nAdaptations of the model
qDynamic models
qSystems of regression models
qTime series models
nFurther  developments
qModels for panel data
qModels for spatial data
qModels for limited dependent variables
q
Nov 10, 2017

Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
Nov 10, 2017
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Dynamic Models: Examples
nDemand model: describes the quantity Q demanded of a product as a function of
its price P and consumers’ income Y
n(a) Current price and current income determine the demand (static model):
n Qt = β1 + β2Pt + β3Yt + et
n(b) Current price and income of the previous period determine the demand (dynamic model):
n Qt = β1 + β2Pt + β3Yt-1 + et
n(c) Current  price and demand of the previous period determine the demand (autoregressive model):
n Qt = β1 + β2Pt + β3Qt-1 + et
Nov 10, 2017

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Dynamic of Processes
nStatic processes: independent variables have a direct effect, the adjustment of
the dependent variable on the realized values of the independent variables is completed within the
current period, the process is assumed to be always in equilibrium
nStatic models may be unsuitable:
n(a) Some activities are determined by the past, such as: energy consumption depends on past
investments into energy-consuming systems and equipment
n(b) Actors of the economic processes often respond with delay, e.g.,  due to the
duration of decision-making and procurement processes
n(c) Expectations: e.g., consumption depends not only on current income but also on income
expectations in future; modelling of income expectation based on past income development
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Elements of Dynamic Models
1.Lag-structures, distributed lags: describe the delayed effects of one
or more regressors on the dependent variable; e.g., the lag-structure of order s or DL(s) model
(DL: distributed lag)
n Yt = a + Ssi=0βiXt-i + et
2.Geometric lag-structure, Koyck’s model: infinite lag-structure with bi = l0li (0 < l< 1)
3.ADL-model: autoregressive model with lag-structure, e.g., the ADL(1,1)-model
n Yt = a + jYt-1 + β0Xt + β1Xt-1 + et
4.Error-correction model
n DYt = - (1-j)(Yt-1 – m0 – m1Xt-1) + β0D Xt + et
n obtained from the ADL(1,1)-model with m0 = a/(1-j) und m1 = (b0+b1)/(1-j)
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The Koyck Transformation
nTransforms the model
n Yt = l0SiliXt-i + et
n into an autoregressive model (vt = et - let-1):
n Yt = lYt-1 + l0Xt + vt
nThe model with infinite lag-structure in X becomes a model
qwith an autoregressive component  lYt-1
qwith a single regressor Xt and
qwith autocorrelated error terms
nEconometric applications
qThe adaptive expectations model
q Example: Investments determined by expected profit Xe:
q Xet+1 = l Xet + (1 - l) Xt  (with 0 < l< 1)
qThe partial adjustment model
n Example: Kpt: planned stock for t; strategy for adapting Kt on Kpt
n Kt – Kt-1 = d(Kpt – Kt-1)
Nov 10, 2017

Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
Nov 10, 2017
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Example: Income and Consumption
nConsumption Ct and disposable income Yt are simultaneously determined by
n Ct = β1 + β2Yt + εt (A)
n Yt = Ct + It  (B)
nThe disposable incomeYt  is determined by the consumption Ct
nEquations (A) and (B) are the structural equations or the structural form of the simultaneous
equation model that describes both Ct and Yt
nThe coefficients β1 and β2 are behavioural parameters
nIn equation (A), Yt is endogenous: The OLS estimates b1 and b2 are biased and not consistent
n
n
n
n
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Multi-equation Models
nEconomic phenomena are usually characterized by the behaviour of more than one dependent variable
nMulti-equation model: the number of equations determines the number of dependent variables which
are described by the model
n
nCharacteristics of multi-equation models:
nTypes of equations
nTypes of variables
nIdentifiability
n
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Klein‘s Model 1
n Ct = a1 + a2Pt + a3Pt-1 + a4(Wtp+ Wtg) + et1   (consumption)
n It = b1 + b2Pt + b3Pt-1 + b4Kt-1 + et2   (investments)
n Wtp = g1 + g2Xt + g3Xt-1 + g4t + et3    (private wages and salaries)
n Xt = Ct + It + Gt
n Kt = It + Kt-1
n Pt = Xt – Wtp – Tt
nC (consumption), P (profits), Wp (private wages and salaries), Wg (public wages and salaries), I
(investments), K (capital stock), X (production), G (governmental expenditures without wages and
salaries), T (taxes) and t [time (trend)]
nEndogenous: C, I, Wp, X, K, P; exogeneous: Wg, G, T, t, P-1, K-1, X-1
Nov 10, 2017

EViews: n=100; series y1 = nrnd; y2 = y2(-1)+y1

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Types of Equations
nBehavioural or structural equations: describe the behaviour of a dependent variable as
a function of explanatory variables
nDefinitional identities: define how a variable is defined as the sum of
other variables, e.g., decomposition of gross domestic product as the sum
of its consumption components
n Example: Klein’s model 1: Xt = Ct + It + Gt
nEquilibrium conditions: assume a certain relationship, which can be interpreted as an equilibrium
n Example: equality of demand (Qd) and supply (Qs) in a market    model: Qtd = Qts
n
nDefinitional identities and equilibrium conditions have no error terms
Nov 10, 2017

EViews: n=100; series u = nrnd; y1 = y1(-1)+u; y2 = 0.1+y2(-1)+u; y3 = 0.2+0.7*y3(-1)+u;

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Types of Variables
nSpecification of a multi-equation model: definition of
nvariables which are explained by the model (endogenous variables)
nother variables which are used in the model
n
nNumber of equations needed in the model: same number as that of the endogenous variables in the
model
n
nExplanatory or exogenous variables: uncorrelated with error terms
nstrictly exogenous variables: uncorrelated with error terms et+i (for any i ≠ 0)
npredetermined variables: uncorrelated with current and future error terms (et+i, i ≥ 0); lagged
explanatory variables
n
nError terms:
nUncorrelated over time
nError terms from different equations and same observation period typically correlated,
contemporaneous correlation
Nov 10, 2017

EViews: n=100; series u = nrnd; y1 = y1(-1)+u; y2 = 0.1+y2(-1)+u; y3 = 0.2+0.7*y3(-1)+u;

Systems of Regression Equations
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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
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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”
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”
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’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 (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 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

Examples of Multi-equation Models
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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
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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

Multi-equation Models: Estimation of Parameters
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Estimation procedures
nMultivariate regression models
qFGLS , GLS, 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
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
Nov 10, 2017
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Types of Trend
nTrend: The expected value of a process {Yt, t = 1, 2, ... } increases or decreases with time
nDeterministic trend: a function f(t) of the time t, describing the evolution of E{Yt} over time
n Yt = f(t) + εt, εt: white noise
n Example: Yt = α + βt + εt describes a linear trend of Y; an increasing trend corresponds to β > 0
nStochastic trend: Yt = δ + Yt-1 + εt or
n ΔYt = Yt – Yt-1 = δ + εt, εt: white noise
qdescribes an irregular or random fluctuation of the differences ΔYt around the expected value δ
qAR(1) – or AR(p) – process with unit root
q“random walk with trend”
n
Nov 10, 2017

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Trends: Random Walk and AR Process
nRandom walk: Yt = Yt-1 + εt; random walk with trend: Yt = 0.1 +Yt-1 + εt; AR(1) process: Yt = 0.2
+ 0.7Yt-1 + εt; εt simulated from N(0,1)
n
Nov 10, 2017
-12
-8
-4
0
4
8
12
16
20
10
20
30
40
50
60
70
80
90
100
random walk

random walk with trend



AR(1) process, δ=0.2, θ=0.7

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Example: Private Consumption
nPrivate consumption, AWM database; level values (PCR) and first differences (PCR_D); random walk?
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nMean of PCD_D: 3740
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How to Model Trends?
nSpecification of a
ndeterministic trend, e.g., Yt = α + βt + εt: risk of spurious regression, wrong decisions
nstochastic trend: analysis of differences ΔYt  if a random walk, i.e., a unit root, is suspected
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Spurious Regression: An Illustration
nIndependent random walks: Yt = Yt-1 + εyt, Xt = Xt-1 + εxt
n εyt, εxt: independent white noises with variances σy² = 2, σx² = 1
nFitting the model
n Yt = α + βXt + εt
n gives
n Ŷt = - 8.18 + 0.68Xt
nt-statistic for X: t = 17.1
n p-value = 1.2 E-40
nR2 = 0.50,  DW = 0.11
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Models in Non-stationary Time Series
nGiven that  Xt ~ I(1), Yt ~ I(1) and the model is
n Yt = α + βXt + εt
n it follows in general that εt ~ I(1), i.e., the error terms are non- stationary
nConsequences for OLS estimation of α and β
n(Asymptotic) distributions of t- and F -statistics are not the t- and F -distribution
nt-statistic, R2 indicate explanatory potential
nHighly autocorrelated residuals, DW statistic converges for growing length of time series to zero
nNonsense or spurious regression (Granger & Newbold, 1974)
nNon-stationary time series are trended; non-stationarity causes an apparent relationship
n
n
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Avoiding Spurious Regression
nIdentification of non-stationarity: unit-root tests
nModels for non-stationary variables
qElimination of stochastic trends: specifying the model for differences
qInclusion of lagged variables may result in stationary error terms
qExplained and explanatory variables may have a common stochastic trend, are cointegrated:
equilibrium relation, error-correction models
n
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Unit Root Tests
nAR(1) process Yt = δ + θYt-1 + εt with white noise εt
nDickey-Fuller or DF test (Dickey & Fuller, 1979)
n Test of H0: θ = 1 against H1: θ < 1
nKPSS test (Kwiatkowski, Phillips, Schmidt & Shin, 1992)
n Test of H0: θ < 1 against H1: θ = 1
nAugmented Dickey-Fuller or ADF test
n extension of DF test
nVarious modifications like Phillips-Perron test, Dickey-Fuller GLS test, etc.
n
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The Error-correction Model
nADL(1,1) model with Yt ~ I(1), Xt ~ I(1)
n Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt
nCommon trend implies an equilibrium relation, i.e.,
n Yt-1 – βXt-1 ~ I(0)
n error-correction form of the ADL(1,1) model
q ΔYt = φ0ΔXt – (1 – θ)(Yt-1 – α – βXt-1) + εt
nError-correction model describes
nthe short-run behaviour
nconsistently with the long-run equilibrium Yt = α + βXt
n
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Cointegration
nNon-stationary variables Xt ~ I(1), Yt ~ I(1)
n Yt = α + βXt + εt
nXt and Yt are cointegrated: εt ~ I(0)
nXt and Yt are not cointegrated: εt ~ I(1)
nTests for cointegration:
nIf β is known, unit root test based on differences Yt - βXt
nTest procedures
qUnit root test (DF or ADF) based on residuals et
qCointegrating regression Durbin-Watson (CRDW) test: DW statistic
qJohansen technique: extends the cointegration technique to the multivariate case
n
n
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Vector Error-Correction Model
nYt: k-vector, each component I(1)
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) form of the VAR(p) model
n
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Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
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Example
nTo be explained whether a household owns a car: explanatory power have
nincome
nhousehold size
netc.
nRegression for describing car-ownership is not suitable!
nOwning a car has two manifestations: yes/no
nIndicator for owning a car is a binary variable
nModels are needed that allow describing a binary dependent variable or a, more generally, limited
dependent variable
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Cases of Limited Dependent Variable
nTypical situations: functions of explanatory variables are used to describe or explain
nDichotomous dependent variable, e.g., ownership of a car (yes/no), employment status
(employed/unemployed), etc.
nOrdered response, e.g., qualitative assessment (good/average/bad), working status
(full-time/part-time/not working), etc.
nMultinomial response, e.g., trading destinations (Europe/Asia/Africa), transportation means
(train/bus/car), etc.
nCount data, e.g., number of orders a company receives in a week, number of patents granted to a
company in a year
nCensored data, e.g., expenditures for durable goods, duration of study with drop outs
n
n
n
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Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
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Panel Data
nPopulation of interest: individuals, households, companies, countries
nTypes of observations
nCross-sectional data: Observations of all units of a population, or of a (representative) subset,
at one specific point in time; e.g., wages in 2015
nTime series data: Series of observations on units of the population over a period of time; e.g.,
wages of a worker in 2009 through 2015
nPanel data: Repeated observations of (the same) population units collected over a number of
periods; data set with both a cross-sectional and a time series aspect; multi-dimensional data
nCross-sectional and time series data are one-dimensional, special cases of panel data
nPooling independent cross-sections: (only) similar to panel data
n
n
n
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Panel Data: Three Types
nTypically data at micro-economic level (individuals, households, firms), but also at
macro-economic level (e.g., countries)
nNotation:
nN: Number of cross-sectional units
nT: Number of time periods
nTypes of panel data:
nLarge T, small N: “long and narrow”
nSmall T, large N: “short and wide”
nLarge T, large N: “long and wide”
n
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Some Examples
nVerbeek’s data set “males”: Wages  and related variables
nshort and wide panel (N = 545, T = 8)
nrich in information (~40 variables)
nGrunfeld investment data: Investments in plant and equipment by
nN = 10 firms
nfor each of T = 20 yearly observations for 1935-1954
nPenn World Table: Purchasing power parity and national income accounts for
nN = 189 countries/territories
nfor some or all of the years 1950-2009 (T ≤ 60)
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Example: Individual Wages
nVerbeek’s data set “males”
nSample of
q545 full-time working males, end of schooling in 1980
qfrom each person: yearly data collection from 1980 till 1987
nVariables
qwage: log of hourly wage (in USD)
qschool: years of schooling
qexper: age – 6 – school
qdummies for union membership, married,  black, Hispanic, public sector
qothers
n
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Use of Panel Data
nEconometric models for describing the behaviour of cross-sectional units over time
nPanel data models
nAllow controlling individual differences, comparing behaviour, analysing dynamic adjustment,
measuring effects of policy changes
nMore realistic models than cross-sectional and time-series models
nAllow more detailed or sophisticated research questions
nMethodological implications
nDependence of sample units in time-dimension
nSome variables might be time-constant (e.g., variable school in “males”, population size in the
Penn World Table dataset)
nMissing values
n
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Examples for Fixed- and Random-effects
nGrunfeld investment data: Investment model
n Iit = αi + βi1Fit + βi2Cit + uit
qwith Fit: market value, Cit: value of stock of plant and equipment, both of firm i at the end of
year t-1
qN = 10 firms, T = 20 yearly observations
qFixed effects αi allow for firm-specific, time-constant factors
nWage equation
n wageit = β1 + β2 experit + β3 exper2it + β4 schoolit + β5 unionit
n + β6 marit + β7 blackit + β8 ruralit + αi + uit
n with composite error εit = αi + uit
qαi: unit-specific parameter for each of 545 units
qTime-constant factors αi: stochastic variables with identical distribution
qRegressors are uncorrelated with uit
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Models for Panel Data
nModel for y, based on panel data from N cross-sectional units and T periods
n yit = β0 + xit’β1 + εit
n i = 1, ..., N: sample unit
n t = 1, ..., T: time period of sample
n xit and β1: K-vectors
nβ0 and β1: represent intercept and K regression coefficients; are assumed to be identical for all
units and all time periods
nεit: represents unobserved factors that may affect yit
qAssumption that εit are uncorrelated over time not realistic; refer to the same unit or individual
qStandard errors of OLS estimates misleading, OLS estimation not efficient (does not exploit
dependence structure over time)
n
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Fixed Effects Model
nThe general model
n yit = β0 + xit’β1 + εit
nSpecification for the error terms: two components
n εit = αi + uit
qαi fixed, unit-specific, time-constant factors, also called unobserved (individual) heterogeneity;
may be correlated with xit
quit ~ IID(0, σu2); homoskedastic, uncorrelated over time; represents unobserved factors that
change over time, also called idiosyncratic or time-varying error
qεit : also called composite error
nFixed effects (FE) model
n yit = Σj αi dij + xit’β1 + uit
n dij: dummy variable for unit i: dij = 1 if i = j, otherwise dij = 0
nOverall intercept β0 omitted; unit-specific intercepts αi
n
n
n
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Fixed Effects Estimator
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Within transformation”: transforms yit into time-demeaned ÿit by subtracting the average ӯi = (Σt
yit )/T: ÿit = yit - ӯi; analogously ẍit and üit, for all i and t
bFE = (ΣiΣt ẍit ẍit’)-1 ΣiΣt ẍit ÿit
nUnbiased if all xit are independent of all uit
nConsistent (for N → ∞) if xit are strictly exogenous, i.e., E{xit uis} = 0 for all s, t
nAsymptotically normally distributed
nCovariance matrix
V{bFE} = σu2(ΣiΣt ẍit ẍit’)-1

Random Effects Model
nStarting point is again the model
n yit = β0 + xit’β1 + εit
n with composite error εit = αi + uit
nSpecification for the error terms:
quit ~ IID(0, σu2); homoskedastic, uncorrelated over time
qαi ~ IID(0, σa2); represents all unit-specific, time-constant factors; correlation of error terms
over time only via the αi
qαi and uit are assumed to be mutually independent and independent of xjs for all j and s
nRandom effects (RE) model
n yit = β0 + xit’β1 + αi + uit
nUnbiased and consistent (N → ∞) estimation of β0 and β1
nEfficient estimation of β0 and β1: takes error covariance structure into account; GLS estimation
n
n
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Contents
nGMM Estimation
nEconometric Models
nDynamic Models
nMulti-equation Models
nTime Series Models
nModels for Limited Dependent Variables
nPanel Data Models
nEconometrics II
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Econometrics II
1.ML Estimation and Specification Tests (MV, Ch.6)
2.Models with Limited Dependent Variables (MV, Ch.7)
3.Univariate time series models (MV, Ch.8)
4.Multivariate time series models, part 1 (MV, Ch.9)
5.Multivariate time series models, part 2 (MV, Ch.9)
6.Models Based on Panel Data (MV, Ch.10)
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Univariate Time Series Models
nTime Series
nStochastic Processes
nStationary Processes
nThe ARMA Process
nDeterministic and Stochastic Trends
nModels with Trend
nUnit Root Tests
n
n
n
n
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Multivariate Time Series Models
nDynamic Models
nLag Structures, ADL Models
nModels with Non-stationary Variables
nCointegration, Tests for Cointegration
nError-correction Model
nSystems of Equations
nVAR Models
nSimultaneous Equations and VAR Models
nVAR Models and Cointegration
nVEC Models
n
n
n
n
n
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