Econometrics - Lecture 1
Econometrics – First Steps


Contents
nOrganizational Issues
nSome History of Econometrics
nAn Introduction to Linear Regression
qOLS: An Algebraic Tool
qThe Linear Regression Model
qSmall Sample Properties of the OLS Estimator
nIntroduction to GRETL
n
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Organizational Issues
nCourse schedule
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n Time: 10:00-14:00
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Sept 23, 2016
Class
Date
1
Fr,  Sept 23
2
Fr, Nov 4
3
Fr, Nov 18
4
Fr, Nov 25
5
Fr, Dec 2
6
Fr, Dec 9
Hackl, Econometrics, Lecture 1

Organizational Issues, cont’d
nAims of the course
nUse of econometric tools for analyzing economic data: specification of adequate models,
identification of appropriate econometric methods, estimation of model parameters, interpretation
of results
nIntroduction to commonly used econometric tools and techniques
nUnderstanding of econometric concepts and principles
nUse of GRETL
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Example: Individual Wages
nSample (US National Longitudinal Survey, 1987)
nN = 3294 individuals (1569 females)
nVariable list
qWAGE: wage (in 1980 $) per hour (p.h.)
qMALE: gender (1 if male, 0 otherwise)
qSCHOOL: years of schooling
qEXPER: experience in years
qAGE: age in years
nQuestions of interest
qEffect of gender on wage p.h.: Average wage p.h.: 6,31$ for males, 5,15$ for females
qEffects of education, of experience, of interactions, etc. on wage p.h.
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Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Example: Income and Consumption
Hackl, Econometrics, Lecture 1
PCR: Private Consumption,
   real, in bn. EUROs
PYR: Household's Dispos-
   able Income, real, in bn.
   EUROs
1970:1-2003:4
Basis: 1995
Source: AWM-Database
Sept 23, 2016

Organizational Issues, cont’d
nLiterature
nCourse textbook
nMarno Verbeek, A Guide to Modern Econometrics, 4rd ed., Wiley, 2012
nSuggestions for further reading
nPeter Kennedy, A guide to econometrics. 6th ed., Blackwell, 2008.
nWilliam H. Greene, Econometric Analysis. 7th Ed., Prentice Hall, 2011
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Organizational Issues, cont’d
nPrerequisites
nLinear algebra: linear equations, matrices, vectors (basic operations and properties) nDescriptive
statistics: measures of central tendency, measures of dispersion, measures of association,
frequency tables, histogram, scatter plot, quantile nTheory of probability: probability and its
properties, random variables and distribution functions in one and in several dimensions, moments,
convergence of random variables, limit theorems, law of large numbers nMathematical statistics:
point estimation, confidence interval, hypothesis testing, p-value, significance level
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Organizational Issues, cont’d
nTeaching and learning method
nCourse in six blocks
nClass discussions, written homework (computer exercises, GRETL) submitted by groups of (3-5)
students, presentations of homework by participants
nFinal exam
nAssessment of student work
nFor grading, the written homework, presentation of homework in class and a final written exam will
be of relevance
nWeights: homework 40 %, final written exam 60 %
nPresentation of homework in class: students must be prepared to be called at random
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Contents
nOrganizational Issues
nSome History of Econometrics
nAn Introduction to Linear Regression
qOLS: An Algebraic Tool
qThe Linear Regression Model
qSmall Sample Properties of the OLS Estimator
nIntroduction to GRETL
n
Sept 23, 2016
•
Hackl, Econometrics, Lecture 1

Empirical Economics Prior to 1930ies
nThe situation in the early 1930ies:
nTheoretical economics aims at “operationally meaningful theorems“; “operational” means purely
logical mathematical deduction
nEconomic theories or laws are seen as deterministic relations; no inference from data as part of
economic analysis
nIgnorance of the stochastic nature of economic concepts
nData: limited availability; time-series on agricultural commodities, foreign trade
nUse of statistical methods for
qmeasuring theoretical coefficients, e.g., demand elasticities
qrepresenting business cycles
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Early Institutions
nApplied demand analysis: US Bureau of Agricultural Economics
nStatistical analysis of business cycles: H.L.Moore (Columbia University): Fourier periodogram;
W.M.Persons et al. (Harvard): business cycle forecasting; US National Bureau of Economic Research
(NBER)
nCowles Commission for Research in Economics
qFounded 1932 by A.Cowles: determinants of stock market prices?
qFormalization of econometrics, development of econometric methodology
qR.Frisch, G.Tintner; European refugees
qJ.Marschak (head 1943-55) recruited people like T.C.Koopmans, T.M.Haavelmo, T.W.Anderson,
L.R.Klein
qInterests shifted to theoretical and mathematical economics after 1950
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Early Actors
nR.Frisch (Oslo Institute of Economic Research): econometric project, 1930-35; T.Haavelmo,
O.Reiersol
nJ.Tinbergen (Dutch Central Bureau of Statistics, Netherlands Economic Institute; League of
Nations, Genova): macro-econometric model of Dutch economy, ~1935; T.C.Koopmans, H.Theil
nAustrian Institute for Trade Cycle Research: O.Morgenstern (head), A.Wald, G.Tintner
nEconometric Society, founded 1930 by R.Frisch et al.
qFacilitates exchange of scholars from Europe and US
qCovers econometrics and mathematical statistics
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n
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Trygve Magnus Haavelmo, Olav Reiersöl

First Steps
nR.Frisch, J.Tinbergen:
qMacro-economic modelling based on time-series, ~ 1935
qAiming at measuring parameters, e.g., demand elasticities
qAware of problems due to quality of data
qNobel Memorial Prize in Economic Sciences jointly in 1969 (“for having developed and applied
dynamic models for the analysis of economic processes”)
nT.Haavelmo
q“The Probability Approach in Econometrics”: PhD thesis (1946)
qEconometrics as a tool for testing economic theories
qNobel Memorial Prize in Economic Sciences in 1989 ("for his clarification of the probability
theory foundations of econometrics and his analyses of simultaneous economic structures”)
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Trygve Haavelmo, 1911-1999

First Steps, cont’d
nCowles Commission
qMethodology for macro-economic modeling based on Haavelmo’s approach
qCowles Commission monographs by G.Tintner, T.C.Koopmans, et al.
Sept 23, 2016
Hackl, Econometrics, Lecture 1

The Haavelmo Revolution
nIntroduction of probabilistic concepts in economics
qObvious deficiencies of traditional approach: Residuals, measurement errors, omitted variables;
stochastic time-series data
qAdvances in probability theory in early 1930ies
qFisher‘s likelihood function approach
nHaavelmo‘s ideas
qCritical view of Tinbergen‘s macro-econometric models
qThorough adoption of probability theory in econometrics
qConversion of deterministic economic models into stochastic structural equations
nHaavelmo‘s “The Probability Approach in Econometrics”
qWhy is the probability approach indispensible?
qModeling procedure based on ML estimation and hypothesis testing
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Haavelmo’s Arguments for the Probabilistic Approach
nEconomic variables in economic theory and econometric models
q“Observational” vs. “theoretical” vs. “true” variables
qModels have to take into account inaccurately measured data and passive observations
nUnrealistic assumption of permanence of economic laws
qSimplifying economic theories
qSelection of economic variables and relations out of the whole system of fundamental laws
qEconomic time-series data
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Cowles Commission Methodology
nAssumptions based to macro-econometric modelling and testing of economic theories
nTime series model
n Yt = aXt + bWt+ u1t
n Xt = gYt + dZt+ u2t
1.Specification of the model equation(s) includes the choice of variables; functional form is
(approximately) linear 2.Time-invariant model equation(s): the model parameters a, …, d are
independent of time t 3.Parameters a, …, d are structurally invariant, i.e., invariant wrt changes
in the variables
4.Causal ordering (exogeneity, endogeneity) of variables is known
5.Statistical tests can falsify but not verify a model
6.
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Classical Econometrics and More
n“Golden age” of econometrics until ~1970
qMulti-equation models for analyses and forecasting
qGrowing computing power
qDevelopment of econometric tools
nSkepticism
qPoor forecasting performance
qDubious results due to
nwrong specifications
nimperfect estimation methods
nTime-series econometrics: non-stationarity of economic time-series
qConsequences of non-stationarity: misleading t-, DW-statistics, R²
qNon-stationarity: needs new models (ARIMA, VAR, VEC); Box & Jenkins (1970: ARIMA-models), Granger
& Newbold (1974, spurious regression), Dickey-Fuller (1979, unit-root tests)
Sept 23, 2016
Model
year
eq‘s
Tinbergen
1936
24
Klein
1950
6
Klein & Goldberger
1955
20
Brookings
1965
160
Brookings Mark II
1972
~200
Hackl, Econometrics, Lecture 1

Econometrics …
n… consists of the application of statistical data and techniques to mathematical formulations of
economic theory. It serves to test the hypotheses of economic theory and to estimate the implied
interrelationships. (Tinbergen, 1952)
n… is the interaction of economic theory, observed data and statistical methods. It is the
interaction of these three that makes econometrics interesting, challenging, and, perhaps,
difficult. (Verbeek, 2008)
n… is a methodological science with the elements
qeconomic theory
qmathematical language
qstatistical methods
qcomputer science
qaiming to give empirical content to economic relations. (Pesaran, 1987)
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Our Course
n1. Introduction to linear regression (Verbeek, Ch. 2): the linear regression model, OLS method,
properties of OLS estimators
n2. Introduction to linear regression (Verbeek, Ch. 2): goodness of fit, hypotheses testing,
multicollinearity
n3. Interpreting and comparing regression models (MV, Ch. 3): interpretation of the fitted model,
selection of regressors, testing the functional form
n4. Heteroskedascity and autocorrelation (Verbeek, Ch. 4): causes and consequences, testing,
alternatives for inference
n5. Endogeneity, instrumental variables and GMM (Verbeek, Ch. 5): the IV estimator, the generalized
instrumental variables estimator, the generalized method of moments (GMM)
n6. The practice of econometric modelling
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Econometrics 2: An Advanced Course
nUnivariate and multivariate time series models:  ARMA-, ARCH-, GARCH-models, VAR-, VEC-models
nModels for panel data
nModels with limited dependent variables: binary choice, count data
n
Sept 23, 2016
Hackl, Econometrics, Lecture 1

(G)ARCH: (generalized) autoregressive conditional heteroskedasticity

Contents
nOrganizational Issues
nSome History of Econometrics
nAn Introduction to Linear Regression
qOLS: An Algebraic Tool
qThe Linear Regression Model
qSmall Sample Properties of the OLS Estimator
nIntroduction to GRETL
n
Sept 23, 2016
•
Hackl, Econometrics, Lecture 1

Example: Individual Wages
nSample (US National Longitudinal Survey, 1987)
nN = 3294 individuals (1569 females)
nVariable list
qWAGE: wage (in 1980 $) per hour (p.h.)
qMALE: gender (1 if male, 0 otherwise)
qSCHOOL: years of schooling
qEXPER: experience in years
qAGE: age in years
nPossible questions
qEffect of gender on wage p.h.: Average wage p.h.: 6,31$ for males, 5,15$ for females
qEffects of education, of experience, of interactions, etc. on wage p.h.
n
n
n
n
n
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Individual Wages, cont’d
qWage per hour vs. Years of schooling
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Linear Regression
Sept 23, 2016
•Y: explained variable
•X: explanatory or regressor variable
•The linear regression model describes the data-generating
•    process of Y under the condition X
•   simple linear regression model
•b2: coefficient of X
•b1 : intercept
•
•multiple linear regression model
Hackl, Econometrics, Lecture 1

Fitting a Model to Data
nChoice of values b1, b2 for model parameters b1, b2 of Y = b1 + b2 X,
ngiven the observations (yi, xi), i = 1,…,N
n
nPrinciple of (Ordinary) Least Squares or OLS:
n bi = arg minb1, b2 S(b1, b2), i =1,2
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nObjective function: sum of the squared deviations
n S(b1, b2) = Si [yi - (b1 + b2xi)]2 = Si ei2
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nDeviation between observation and fitted value: ei = yi - (b1 + b2xi)
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Observations and Fitted Regression Line
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nSimple linear regression: Fitted line and observation points (Verbeek, Figure 2.1)
Sept 23, 2016
Hackl, Econometrics, Lecture 1

OLS Estimators
nOLS estimators b1 und b2 result in
Sept 23, 2016
•
•with mean values     and
•and second moments
•
•Equating the partial derivatives of S(b1, b2) to zero: normal equations
Hackl, Econometrics, Lecture 1

Individual Wages, cont’d
nSample (US National Longitudinal Survey, 1987): wage per hour, gender, experience, years of
schooling; N = 3294 individuals (1569 females)
nAverage wage p.h.: 6,31$ for males, 5,15$ for females
nModel:
n wagei = β1 + β2 malei + εi
nmaleI: male dummy, has value 1 if individual is male, otherwise value 0
nOLS estimation gives
n wagei = 5,15 + 1,17*malei
nCompare with averages!
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Individual Wages, cont’d
nOLS estimated wage equation (Table 2.1, Verbeek)
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n wagei = 5,15 + 1,17*malei
n estimated wage p.h for males: 6,313
n        for females: 5,150
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

OLS Estimators: General Case
nModel for Y contains K-1 explanatory variables
n Y = b1 + b2X2 + … + bKXK = x’b
nwith x = (1, X2, …, XK)’ and b = (b1, b2, …, bK)’
nObservations: (yi, xi’) = (yi, (1, xi2, …, xiK)), i = 1, …, N
nOLS estimates b = (b1, b2, …, bK)’ are obtained by minimizing the objective function wrt the bk’s
n
n
nthis results in
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n
n
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

OLS Estimators: General Case, cont’d
nor
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n
nthe so-called normal equations, a system of K linear equations for the components of b
nGiven that the symmetric KxK-matrix                  has full rank K and is hence invertible, the
OLS estimators are
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Best Linear Approximation
nGiven the observations: (yi, xi’) = (yi, (1, xi2, …, xiK)), i = 1, …, N
nFor yi, the linear combination or the fitted value
n
n
nis the best linear combination for Y from X2, …, XK and a constant (the intercept)
n
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Some Matrix Notation
nN observations
n (y1,x1), … , (yN,xN)
n
nModel: yi = b1 + b2xi + εi, i  = 1, …,N, or
n y = Xb + ε
nwith
n
n
n
n
Sept 23, 2016
Hackl, Econometrics, Lecture 1

OLS Estimators in Matrix Notation
nMinimizing
n S(b) = (y  - Xb)’ (y  - Xb) = y’y – 2y’Xb + b’ X’Xb
n with respect to b gives the normal equations
n
n
n resulting from differentiating S(b) with respect to b and setting the first derivative to zero
nThe vector of OLS solution or OLS estimators for b is
n b = (X’X)-1X’y
nThe best linear combinations or predicted values for Y given X or projections of y into the space
of X are obtained as
n ŷ = Xb = X(X’X)-1X’y = Pxy
n the NxN-matrix Px is called the projection matrix or hat matrix
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Residuals in Matrix Notation
nThe vector y can be written as y  = Xb + e  = ŷ + e with residuals
n e  = y – Xb or ei = yi – xi‘b, i = 1, …, N
nFrom the normal equations follows
n -2(X‘y – X‘Xb) = -2 X‘e = 0
n i.e., each column of X is orthogonal to e
nWith
n e = y – Xb = y – Pxy = (I – Px)y = Mxy
n the residual generating matrix Mx is defined as
n Mx = I – X(X’X)-1X’ = I – Px
n Mx projects y into the orthogonal complement of the space of X
nProperties of Px and Mx: symmetry (P’x = Px, M’x = Mx) idempotence (PxPx = Px, MxMx = Mx), and
orthogonality (PxMx = 0)
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Properties of Residuals
nResiduals: ei = yi – xi‘b, i = 1, …, N
nMinimum value of objective function
n S(b) = e’e = Si ei2
nFrom the orthogonality of e = (e1, …, eN)‘ to each xi = (x1i, …, xNi)‘, i = 1,…, K, i.e., e‘xi =
0, follows that
n Si ei = 0
n i.e., average residual is zero, if the model has an intercept
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Contents
nOrganizational Issues
nSome History of Econometrics
nAn Introduction to Linear Regression
qOLS: An Algebraic Tool
qThe Linear Regression Model
qSmall Sample Properties of the OLS Estimator
nIntroduction to GRETL
n
Sept 23, 2016
•
Hackl, Econometrics, Lecture 1

US Wages
nUS wages are gender-specific
nThe relation
n wagei = β1 + β2 malei + εi
n with maleI: male dummy (equals 1 for males, otherwise 0)
ndescribes the wage of individual i as a function of its gender
nis assumed to be true for all US citizens
nGiven sample data (wagei, malei, i = 1,…N), OLS estimation of β1 and β2 may result in
n wagei = 5,15 + 1,17*malei
nThis is not (only) a description of the sample!
nBut reflects a general relationship
n
n
n
n
n
n
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Income and Consumption
Hackl,  Econometrics, Lecture 3
41
PCR: private consumption, real
PYR: household's disposable income, real,
AWM-Database, 1970:1-2003:4, Basis: 1995
Consumption function
        PCRt = β1 + β2 PYRt + εt
§  describes consumption in the
       Euro-zone
Oct 30, 2015

Economic Models
nDescribe economic relationships (not just a set of observations), have an economic interpretation
nLinear regression model:
n yi = b1 + b2xi2 + … + bKxiK + ei = xi’b + εi
nVariables Y, X2, …, XK: observable
nError term εi (disturbance term) contains all influences that are not included explicitly in the
model; not observable; assumption E{εi | xi} = 0 gives
q E{yi | xi} = xi‘β
qthe model describes the expected value of y given x
nSample (yi, xi2, …, xiK, i = 1, …, N) from a well-defined population
nUnknown coefficients b1, …, bK: population parameters
n
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Sampling in the Economic Context
nThe regression model yi = xi’b + εi, i = 1, …, N; or y = Xb + ε
n describes one realization out of all possible samples of size N from the population
nA) Sampling process with fixed, i.e., non-stochastic xi’s
nNew sample: new error terms εi, i = 1, …, N, and, hence, new yi’s
nJoint distribution of εi‘s determines properties of b etc.
nA laboratory setting, does not apply to the economic context
nB) Sampling process with samples of (xi, yi) or (xi, ei)
nNew sample: new error terms εi and new xi, i = 1, …, N
nRandom sampling of (xi, ei), i = 1, …, N: joint distribution of (xi, ei)‘s determines properties
of b etc.
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Sampling in the Economic Context, cont’d
nThe sampling with fixed, non-stochastic xi’s is not realistic for economic data
nSampling process with samples of (xi, yi) is appropriate for modeling cross-sectional data
qExample: household surveys, e.g., US National Longitudinal Survey, EU-SILC
nSampling process with samples of (xi, yi) from time-series data: sample is seen as one out of all
possible realizations of the underlying data-generating process
qExample: time series PYR and PCR of the AWM-Database
q
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Assumptions of the Linear Regression Model
nThe linear regression model yi = xi’b + εi needs assumptions to allow interpretation
nAssumption for εi‘s: E{εi | xi } = 0; exogeneity of variables X
qX contains no information on the error term ε
qE{εi | xi } = 0 implies that εi and xi are uncorrelated
nThis implies
n E{yi | xi } = xi'b
n i.e., the regression line describes the conditional expectation of yi given xi
nCoefficient bk measures the change of the expected value of Y if Xk changes by one unit and all
other Xj values, j ǂ k, remain the same (ceteris paribus condition)
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Regression Coefficients
nLinear regression model:
n yi = b1 + b2xi2 + … + bKxiK + ei = xi’b + εi
nCoefficient bk measures the change of the expected value of Y if Xk changes by one unit and all
other Xj values, j ǂ k, remain the same (ceteris paribus condition); marginal effect of changing
Xk on Y
n
n
nExample
nWage equation: wagei = β1 + β2 malei + β3 schooli + β4 experi + εi
n β3 measures the impact of one additional year at school upon a person’s wage, keeping gender and
years of experience fixed
Sept 23, 2016
•
Hackl, Econometrics, Lecture 1

Estimation of β
nGiven a sample (xi, yi), i = 1, …, N, the OLS estimators for b
n b = (X’X)-1X’y
n can be used as an approximation for b
nThe vector b is a vector of numbers, the estimates
nThe vector b is the realization of a vector of random variables
nThe sampling concept and assumptions on εi‘s determine the quality, i.e., the statistical
properties, of b
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Contents
nOrganizational Issues
nSome History of Econometrics
nAn Introduction to Linear Regression
qOLS: An Algebraic Tool
qThe Linear Regression Model
qSmall Sample Properties of the OLS Estimator
nIntroduction to GRETL
n
Sept 23, 2016
•
Hackl, Econometrics, Lecture 1

Fitting Economic Models to Data
nObservations allow
nto estimate parameters
nto assess how well the data-generating process is represented by the model, i.e., how well the
model coincides with reality
nto improve the model if necessary
nFitting a linear regression model to data provides
nparameter estimates b = (b1, …, bK)’ for coefficients b  = (b1, …, bK)’
nstandard errors se(bk) of the estimates bk, k=1,…,K
nt-statistics, F-statistic, R2, Durbin Watson test-statistic, etc.
Sept 23, 2016
Hackl, Econometrics, Lecture 1

OLS Estimator and OLS Estimates b
nOLS estimates b are a realization of the OLS estimator
nThe OLS estimator is a random variable
nObservations are a random sample from the population
nObservations are generated by some random sampling process
nDistribution of the OLS estimator
nActual distribution not known
nDistribution determined by assumptions on
qmodel specification
qthe error term εi and regressor variables xi
nQuality criteria (bias, accuracy, efficiency) of OLS estimates are determined by the properties of
this distribution
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Gauss-Markov Assumptions
A1
E{εi} = 0 for all i
A2
all εi are independent of all xi (exogeneous xi)
A3
V{ei} = s2 for all i (homoskedasticity)
A4
Cov{εi, εj} = 0 for all i and j with i ≠ j (no autocorrelation)
Sept 23, 2016
Observation yi is a linear function
yi = xi'b + εi
of observations xik of the regressor variables Xk, k = 1, …, K, and the error term εi
for i = 1, …, N; xi' = (xi1, …, xiK); X = (xik)
n
n
n
n
n
In matrix notation: E{ε} = 0, V{ε} = s2 IN
Hackl, Econometrics, Lecture 1

Systematic Part of the Model
nThe systematic part E{yi | xi } of the model yi = xi'b  + εi, given observations xi, is derived
under the Gauss-Markov assumptions as follows:
n(A2) implies E{ε | X} = E{ε} = 0 and V{ε | X} = V{ε} = s2 IN
nObservations xi, i = 1, …, N, do not affect the properties of ε
nThe systematic part
n E{yi | xi } = xi'b
n can be interpreted as the conditional expectation of yi, given observations xi
n
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Is the OLS Estimator a Good Estimator?
n
nUnder the Gauss-Markov assumptions, the OLS estimator has favourable properties; see below
nGauss-Markov assumptions are very strong but not always satisfied
nRelaxations of the Gauss-Markov assumptions and consequences of such relaxations are important
topics in econometrics
Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Properties of OLS Estimators
n1. The OLS estimator b is unbiased: E{b | X} = E{b} = β
n    Needs assumptions (A1) and (A2)
n
n2. The variance of the OLS estimator b is given by
n V{b | X} = V{b} = σ2(Σi xi xi’)-1 = σ2(X‘ X)-1
n    Needs assumptions (A1), (A2), (A3) and (A4)
n
n3. Gauss-Markov Theorem: The OLS estimator b is a BLUE (best linear unbiased estimator) for β
n    Needs assumptions (A1), (A2), (A3), and (A4) and requires linearity in parameters
n
n
n
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Sept 23, 2016
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Hackl, Econometrics, Lecture 1

The Gauss-Markov Theorem
nOLS estimator b is BLUE (best linear unbiased estimator) for β
nLinear estimator: b* = Ay with any full-rank KxN matrix A
nb*  is an unbiased estimator: E{b*} = E{Ay} = β
nb is BLUE: V{b*} – V{b} is positive semi-definite, i.e., the variance of any linear combination
d’b* is not smaller than that of d’b
n V{d’b*} ≥ V{d’b}
n e.g., V{bk*} ≥ V{bk} for any k
nThe OLS estimator is most accurate among the linear unbiased estimators
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Standard Errors of OLS Estimators
nVariance (covariance matrix) of the OLS estimators:
n V{b} = σ2(X’ X)-1 = σ2(Σi xi xi’)-1
nStandard error of OLS estimate bk: The square root of the kth diagonal element of V{b}
nEstimator V{b} is proportional to the variance σ2 of the error terms
nEstimator for σ2: sampling variance s2 of the residuals ei
n s2 = (N – K)-1 Σi ei2
n Under assumptions (A1)-(A4), s2 is unbiased for σ2
n     Attention: the estimator (N – 1)-1 Σi ei2 is biased
nEstimated variance (covariance matrix) of b:
n Ṽ{b} = s2(X’ X)-1 = s2(Σi xi xi’)-1
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Standard Errors of OLS Estimators, cont’d
nVariance (covariance matrix) of the OLS estimators:
n V{b} = σ2(X’ X)-1 = σ2(Σi xi xi’)-1
nStandard error of OLS estimate bk: The square root of the kth diagonal element of V{b}
n σ√ckk
n with ckk the k-th diagonal element of  (X’ X)-1
nEstimated variance (covariance matrix) of b:
n Ṽ{b} = s2(X’ X)-1 = s2(Σi xi xi’)-1
nEstimated standard error of bk:
n se(bk) = s√ckk
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Two Examples
n1. Simple regression yi = a + b xi + et
n The variance for the OLS estimator of b is
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n b is the more accurate, the larger N and sx² and the smaller s²
n2. Regression with two regressors:
n yi = b1 + b2 xi2 + b3 xi3 + et
n The variance for the OLS estimator of b2 is
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n b2 is most accurate if X2 and X3 are uncorrelated
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Sept 23, 2016
Hackl, Econometrics, Lecture 1

Normality of Error Terms
nFor the purpose of statistical inference, a distributional assumption for the εi‘s is needed
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nTogether with assumptions (A1), (A3), and (A4), (A5) implies
n εi ~ NID(0,σ2) for all i
ni.e., all εi are
nindependent drawings
nfrom the normal distribution
nwith mean 0
nand variance σ2
nError terms are “normally and independently distributed” (NID)
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Sept 23, 2016
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A5
εi  normally distributed for all i
Hackl, Econometrics, Lecture 1

Properties of OLS Estimators
n1. The OLS estimator b is unbiased: E{b} = β
n2. The variance of the OLS estimator is given by
n V{b} = σ2(X’X)-1
n3. The OLS estimator b is a BLUE (best linear unbiased estimator) for β
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n4. The OLS estimator b is normally distributed with mean β and covariance matrix V{b} = σ2(X‘X)-1
n b ~ N(β, σ2(X’X)-1), bk ~ N(βk, σ2ckk)
n with ckk: (k,k)-element of (X’X)-1
n Needs assumptions (A1) - (A5)
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Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Example: Individual Wages
n wagei = β1 + β2 malei + εi
nWhat do the assumptions mean?
n(A1): β1 + β2 malei contains the entire systematic part of the model; no regressors besides gender
relevant?
n(A2): xi uncorrelated with εi for all i: knowledge of a person’s gender provides no information
about further variables which affect the person’s wage; is this realistic?
n(A3) V{εi} = σ2 for all i: variance of error terms (and of wages) is the same for males and
females; is this realistic?
n(A4) Cov{εi,,εj} = 0, i ≠ j: implied by random sampling
n(A5) Normality of εi: is this realistic? (Would allow, e.g., for negative wages)
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Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Individual Wages, cont’d
nOLS estimated wage equation (Table 2.1, Verbeek)
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n b1 = 5,1479, se(b1) = 0,0812: mean wage p.h. for females: 5,15$, with std.error of 0,08$
n b2 = 1,166, se(b2) = 0,112
n 95% confidence interval for β1: 4,988 £ β1 £ 5,306
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Sept 23, 2016
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Hackl, Econometrics, Lecture 1

Your Homework
1.Verbeek’s data set “wages1” contains for a sample of 3294 individuals the wage p.h. (wage) and
other variables. Using GRETL, (a) cross-tabulate the variable school (years of schooling) over male
for individuals with school at least 8 years; compare (b) the mean values of the males and females;
draw for the whole population (c) scatter plots of wage over school and exper; and (d) a factorized
box plot of wage over school. Discuss the results.
2.For Verbeek’s data set “wages1”, calculate, using GRETL, the mean years of schooling (a) of the
whole sample, (b) of males and females, and (c) the standard deviation of the years of schooling of
males and of females.
Sept 23, 2016
Hackl, Econometrics, Lecture 1

Your Homework, cont’d
3.For the sample (yi, xi), i = 1,...,N, and the linear regression (yi = b1 + b2xi + ei): write out
the matrices X’X and X’y as well as the OLS estimator b = (X’X)-1X’y.
4.For the simple regression yi = a + b xi + ei, i =1,...,N, show that the variance of the OLS
estimate for b is σ2/(Nsx2), where σ2 is the error term variance, sx2 the variance of the xi‘s.
Sept 23, 2016
Hackl, Econometrics, Lecture 1