Econometrics
c Michael Creel
Version 0.80, February, 2006
D EPT. OF E CONOMICS AND E CONOMIC H ISTORY, U NIVERSITAT AUTÒNOMA DE BARCELONA ,
MICHAEL . CREEL @ UAB . ES , H T T P :// P A R E T O . U A B . E S / M C R E E L
,
, Contents
List of Figures 14
List of Tables 17
Chapter 1. About this document 18
1.1. License 19
1.2. Obtaining the materials 19
1.3. An easy way to use LYX and Octave today 20
1.4. Known Bugs 22
Chapter 2. Introduction: Economic and econometric models 23
Chapter 3. Ordinary Least Squares 26
3.1. The Linear Model 26
3.2. Estimation by least squares 27
3.3. Geometric interpretation of least squares estimation 30
3.3.1. In X ,Y Space 30
3.3.2. In Observation Space 30
3.3.3. Projection Matrices 32
3.4. Influential observations and outliers 33
3.5. Goodness of fit 35
3.6. The classical linear regression model 38
3.7. Small sample statistical properties of the least squares estimator 40
3.7.1. Unbiasedness 40
3
, CONTENTS 4
3.7.2. Normality 41
3.7.3. The variance of the OLS estimator and the Gauss-Markov theorem 42
3.8. Example: The Nerlove model 47
3.8.1. Theoretical background 47
3.8.2. Cobb-Douglas functional form 48
3.8.3. The Nerlove data and OLS 49
Exercises 53
Chapter 4. Maximum likelihood estimation 54
4.1. The likelihood function 54
4.1.1. Example: Bernoulli trial 56
4.2. Consistency of MLE 58
4.3. The score function 60
4.4. Asymptotic normality of MLE 62
4.6. The information matrix equality 66
4.7. The Cramér-Rao lower bound 68
Exercises 71
Chapter 5. Asymptotic properties of the least squares estimator 73
5.1. Consistency 73
5.2. Asymptotic normality 74
5.3. Asymptotic efficiency 75
Chapter 6. Restrictions and hypothesis tests 77
6.1. Exact linear restrictions 77
6.1.1. Imposition 78
6.1.2. Properties of the restricted estimator 82
6.2. Testing 83
c Michael Creel
Version 0.80, February, 2006
D EPT. OF E CONOMICS AND E CONOMIC H ISTORY, U NIVERSITAT AUTÒNOMA DE BARCELONA ,
MICHAEL . CREEL @ UAB . ES , H T T P :// P A R E T O . U A B . E S / M C R E E L
,
, Contents
List of Figures 14
List of Tables 17
Chapter 1. About this document 18
1.1. License 19
1.2. Obtaining the materials 19
1.3. An easy way to use LYX and Octave today 20
1.4. Known Bugs 22
Chapter 2. Introduction: Economic and econometric models 23
Chapter 3. Ordinary Least Squares 26
3.1. The Linear Model 26
3.2. Estimation by least squares 27
3.3. Geometric interpretation of least squares estimation 30
3.3.1. In X ,Y Space 30
3.3.2. In Observation Space 30
3.3.3. Projection Matrices 32
3.4. Influential observations and outliers 33
3.5. Goodness of fit 35
3.6. The classical linear regression model 38
3.7. Small sample statistical properties of the least squares estimator 40
3.7.1. Unbiasedness 40
3
, CONTENTS 4
3.7.2. Normality 41
3.7.3. The variance of the OLS estimator and the Gauss-Markov theorem 42
3.8. Example: The Nerlove model 47
3.8.1. Theoretical background 47
3.8.2. Cobb-Douglas functional form 48
3.8.3. The Nerlove data and OLS 49
Exercises 53
Chapter 4. Maximum likelihood estimation 54
4.1. The likelihood function 54
4.1.1. Example: Bernoulli trial 56
4.2. Consistency of MLE 58
4.3. The score function 60
4.4. Asymptotic normality of MLE 62
4.6. The information matrix equality 66
4.7. The Cramér-Rao lower bound 68
Exercises 71
Chapter 5. Asymptotic properties of the least squares estimator 73
5.1. Consistency 73
5.2. Asymptotic normality 74
5.3. Asymptotic efficiency 75
Chapter 6. Restrictions and hypothesis tests 77
6.1. Exact linear restrictions 77
6.1.1. Imposition 78
6.1.2. Properties of the restricted estimator 82
6.2. Testing 83
