Tilburg University
Econometrics
(35B206-B-6)
Summary Examination 2022
Author:
Jelte Hoevenaars
June 9, 2022
,Contents
1 Linear Regression Model and OLS 2
1.1 Linear Regression Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Estimation by the Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Properties of βb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Geometry of LS and Partial Regression 5
2.1 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Properties of σ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Partitioned Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The Projection Matrix Mi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Coefficient of Determination or R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Confidence Intervals and Hypotheses 10
3.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Vector Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Testing Hypotheses About a Single Coefficient . . . . . . . . . . . . . . . . . . . . 12
3.4 Testing Linear Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Large Sample Theory 14
4.1 Limits and Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Large Sample Properties and Asymptotic Confidence Sets 17
5.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Variance-Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Asymptotic Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Heteroskedasticity and Generalized LS 20
6.1 Generalized Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 Large Sample Properties of GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.3 Feasible GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Endogeneity and Instrumental Variables Estimation 24
7.1 Omitted Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.2 Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.3 Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.4 Instrumental Variables Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A Overview Linear Estimators 28
1
, 1 | Linear Regression Model and OLS
A common question in econometrics is to study the effect of one group of variables X, usually
called the regressors, on another Y, the dependent variable.
In cross-sections1 , the relationship between the regressors and the dependent variable is modelled
through the conditional expectation E[Yi | Xi ]. The deviation of Yi from its conditional expectation
is called the error or residual:
εi = Yi − E[Yi | Xi ]. (1.1)
In the parametric framework, it is assumed that the conditional expectation function depends on a
number of unknown constants or parameters, and that the functional form of E[Yi | Xi ] is known.
In the linear regression model, it is assumed that E[Yi | Xi ] is linear in the parameters:
E[Yi | Xi ] = β1 Xi1 + β2 Xi2 + ... + βk Xik
(1.2)
= Xi⊤ β.
Here, β is a vector of unknown constants. Thus, the error in (1.1) is not observable, as the
conditional expectation is unknown. With econometric theory we aim to estimate these unknowns.
1.1 Linear Regression Model Assumptions
Let us formally define the linear regression model. In the classical regression model, it is assumed
that the variance of the errors is independent of the regressors and the same for all observations.
Var(εi | Xi ) = σ 2
For some constant σ 2 > 0. This property is called homoscedasticity. The following are the four
classical regression assumptions.
(A1) y = Xβ + ε
(A2) E[ε | X] = 0
(A3) Var(ε | X) = σ 2 In
(A4) rank(X) = k
1 A cross-sectional data set consists of a sample of individuals, households, firms, cities, counties, or a variety
of other units, taken at a given point in time.
2
Econometrics
(35B206-B-6)
Summary Examination 2022
Author:
Jelte Hoevenaars
June 9, 2022
,Contents
1 Linear Regression Model and OLS 2
1.1 Linear Regression Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Estimation by the Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Properties of βb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Geometry of LS and Partial Regression 5
2.1 Projection Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Properties of σ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Partitioned Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 The Projection Matrix Mi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Coefficient of Determination or R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Confidence Intervals and Hypotheses 10
3.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Vector Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Testing Hypotheses About a Single Coefficient . . . . . . . . . . . . . . . . . . . . 12
3.4 Testing Linear Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Large Sample Theory 14
4.1 Limits and Convergence Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Convergence in Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Large Sample Properties and Asymptotic Confidence Sets 17
5.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Asymptotic Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Variance-Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Asymptotic Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6 Heteroskedasticity and Generalized LS 20
6.1 Generalized Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6.2 Large Sample Properties of GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.3 Feasible GLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Endogeneity and Instrumental Variables Estimation 24
7.1 Omitted Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.2 Measurement Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.3 Simultaneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7.4 Instrumental Variables Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
A Overview Linear Estimators 28
1
, 1 | Linear Regression Model and OLS
A common question in econometrics is to study the effect of one group of variables X, usually
called the regressors, on another Y, the dependent variable.
In cross-sections1 , the relationship between the regressors and the dependent variable is modelled
through the conditional expectation E[Yi | Xi ]. The deviation of Yi from its conditional expectation
is called the error or residual:
εi = Yi − E[Yi | Xi ]. (1.1)
In the parametric framework, it is assumed that the conditional expectation function depends on a
number of unknown constants or parameters, and that the functional form of E[Yi | Xi ] is known.
In the linear regression model, it is assumed that E[Yi | Xi ] is linear in the parameters:
E[Yi | Xi ] = β1 Xi1 + β2 Xi2 + ... + βk Xik
(1.2)
= Xi⊤ β.
Here, β is a vector of unknown constants. Thus, the error in (1.1) is not observable, as the
conditional expectation is unknown. With econometric theory we aim to estimate these unknowns.
1.1 Linear Regression Model Assumptions
Let us formally define the linear regression model. In the classical regression model, it is assumed
that the variance of the errors is independent of the regressors and the same for all observations.
Var(εi | Xi ) = σ 2
For some constant σ 2 > 0. This property is called homoscedasticity. The following are the four
classical regression assumptions.
(A1) y = Xβ + ε
(A2) E[ε | X] = 0
(A3) Var(ε | X) = σ 2 In
(A4) rank(X) = k
1 A cross-sectional data set consists of a sample of individuals, households, firms, cities, counties, or a variety
of other units, taken at a given point in time.
2
