Econometrics 2
Complete Course Summary
Endogeneity & Instrumental Variables · GMM · Maximum Likelihood
Binary, Multinomial & Ordered Choice · Truncation, Censoring & Sample Selection
Lecture Notes, Step-by-Step Solution Guides, Worked Exercises & Exam Practice Questions
How to use this document.
Part I lecture notes for Weeks 16 with the key derivations and
results.
Part II quick-reference, step-by-step guides: given this question
type, do this.
Part III fully worked exam-style exercises (with numbers).
Part IV practice questions for midterm and
nal, with an answer
key, plus a formula sheet.
Boxes: blue = de
nitions, yellow = key results, green = procedures,
red = exam tips.
,Econometrics 2 Contents
Contents
I Lecture Notes & Derivations 3
1 Week 1: Endogeneity & Instrumental Variables 3
1.1 Endogeneity and its causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Consequences of endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Instrumental variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 More instruments than regressors (m > k ): 2SLS . . . . . . . . . . . . . . . . . . . . 5
1.5 Testing instruments and endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Week 2: Generalized Method of Moments (GMM) 6
2.1 Method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 GMM set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Asymptotic distribution of GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Testing the moment conditions: Hansen's J -test . . . . . . . . . . . . . . . . . . . . . 7
2.5 Invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 E
ciency of 2SLS relative to GMM with W =I . . . . . . . . . . . . . . . . . . . . 8
2.7 Worked example: GMM for an AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Week 3: Maximum Likelihood 8
3.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Properties of the MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 The linear regression model with normal errors . . . . . . . . . . . . . . . . . . . . . 9
3.4 Poisson regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 The three classical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.6 Identi
cation of the uniform example (tutorial) . . . . . . . . . . . . . . . . . . . . . 10
4 Week 4: Binary Response & Multinomial Data 11
4.1 Random utility maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Binary response models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Multinomial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.4 Inference for ML estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Week 5: Ordered Choice Models 14
5.1 Ordered probit / logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Interpreting utility coe
cients: value of time . . . . . . . . . . . . . . . . . . . . . . 14
6 Week 6: Truncation, Censoring & Sample Selection 16
6.1 Truncated regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Censored regression (Tobit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
,Econometrics 2 Contents
6.3 Sample selection (Tobit type 2 / Heckman) . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Choosing the right model overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.5 Interpreting regression output (log forms) . . . . . . . . . . . . . . . . . . . . . . . . 18
II Quick-Reference Step-by-Step Guides 18
7 Guide: Endogeneity, Instruments & 2SLS 18
8 Guide: GMM 20
9 Guide: Maximum Likelihood & the Three Tests 20
10 Guide: Binary & Multinomial Models 20
11 Guide: Ordered Models 21
12 Guide: Truncated, Censored & Selected Samples 22
13 Guide: Common Derivations Checklist 23
III Worked Exam-Style Exercises 23
14 Endogeneity, IV & GMM 23
15 Maximum Likelihood & Tests 24
16 Binary & Multinomial Choice 25
17 Ordered Models 26
18 Truncation, Censoring & Selection 27
IV Practice Questions, Answer Key & Formula Sheet 28
19 Practice Questions 28
20 Answer Key 32
21 Formula Sheet 35
2
, Econometrics 2 1 Week 1: Endogeneity & Instrumental Variables
Part I
Lecture Notes & Derivations
1 Week 1: Endogeneity & Instrumental Variables
1.1 Endogeneity and its causes
Model: y = Xβ + ε, with y, ε ∈ Rn , X ∈ Rn×k . Regressors are exogenous if E[εi | xi ] = 0; they
are endogenous if xi and εi are correlated, i.e. E[xi εi ] ̸= 0.
Four causes of endogeneity
1. Omitted variables (unobserved heterogeneity): true model y = x′ β + w′ γ + ε with
Cov(x, w) ̸= 0, but we estimate y = x′ β+u so u= w′ γ+ε and Cov(x, u) = Cov(x, w)γ ̸=
0.
2. Measurement error in the regressors: we observe x = x∗ +εx , which creates correlation
between x and the composite error.
3. Simultaneity / reverse causality: y depends on x and x depends on y (e.g. price
and quantity in a market).
4. Sample selection (non-random sample) and misspeci
ed dynamics.
1.2 Consequences of endogeneity
If E[ε | X] ̸= 0:
1. E[y | X] ̸= Xβ : the regression function is not linear in the way assumed.
2. OLS is biased: E[β̂ | X] = β + (X ′ X)−1 X ′ E[ε | X] ̸= β .
3. OLS is inconsistent: by the LLN plim n1 X ′ ε ̸= 0, and by Slutsky
plim β̂ = β + Q−1 1 ′
xx plim n X ε ̸= β, Qxx = plim n1 X ′ X.
Lemma. E[ε | X] = 0 ⇒ Cov(ε, Xj ) = 0. Proof: E[ε] = EX [E[ε|X]] = 0 (LIE), so Cov(Xj , ε) =
E[Xj ε] = EX [Xj E[ε|X]] = 0. The converse is false: zero correlation does not imply mean indepen-
dence.
Omitted-variable bias (one omitted regressor)
True: y = β0 + β1 x1 + β2 x2 + u. Regressing y on x1 only:
Cov(x1 , x2 )
plim β̂1 = β1 + β2 .
Var(x1 )
Sign of the bias
Bias = β2 × sign Cov(x1 , x2 ). Example (treatment intensity vs. survival, omitted health):
sicker patients get more intensive treatment, so Cov(Intensity, Health) < 0, and health raises
survival (β2 > 0) ⇒ bias <0⇒ OLS underestimates the treatment eect.
3
Complete Course Summary
Endogeneity & Instrumental Variables · GMM · Maximum Likelihood
Binary, Multinomial & Ordered Choice · Truncation, Censoring & Sample Selection
Lecture Notes, Step-by-Step Solution Guides, Worked Exercises & Exam Practice Questions
How to use this document.
Part I lecture notes for Weeks 16 with the key derivations and
results.
Part II quick-reference, step-by-step guides: given this question
type, do this.
Part III fully worked exam-style exercises (with numbers).
Part IV practice questions for midterm and
nal, with an answer
key, plus a formula sheet.
Boxes: blue = de
nitions, yellow = key results, green = procedures,
red = exam tips.
,Econometrics 2 Contents
Contents
I Lecture Notes & Derivations 3
1 Week 1: Endogeneity & Instrumental Variables 3
1.1 Endogeneity and its causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Consequences of endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Instrumental variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 More instruments than regressors (m > k ): 2SLS . . . . . . . . . . . . . . . . . . . . 5
1.5 Testing instruments and endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Week 2: Generalized Method of Moments (GMM) 6
2.1 Method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 GMM set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Asymptotic distribution of GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Testing the moment conditions: Hansen's J -test . . . . . . . . . . . . . . . . . . . . . 7
2.5 Invariance properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 E
ciency of 2SLS relative to GMM with W =I . . . . . . . . . . . . . . . . . . . . 8
2.7 Worked example: GMM for an AR(1) . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Week 3: Maximum Likelihood 8
3.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Properties of the MLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 The linear regression model with normal errors . . . . . . . . . . . . . . . . . . . . . 9
3.4 Poisson regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.5 The three classical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.6 Identi
cation of the uniform example (tutorial) . . . . . . . . . . . . . . . . . . . . . 10
4 Week 4: Binary Response & Multinomial Data 11
4.1 Random utility maximisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Binary response models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Multinomial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.4 Inference for ML estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5 Week 5: Ordered Choice Models 14
5.1 Ordered probit / logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Interpreting utility coe
cients: value of time . . . . . . . . . . . . . . . . . . . . . . 14
6 Week 6: Truncation, Censoring & Sample Selection 16
6.1 Truncated regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Censored regression (Tobit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
,Econometrics 2 Contents
6.3 Sample selection (Tobit type 2 / Heckman) . . . . . . . . . . . . . . . . . . . . . . . 17
6.4 Choosing the right model overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6.5 Interpreting regression output (log forms) . . . . . . . . . . . . . . . . . . . . . . . . 18
II Quick-Reference Step-by-Step Guides 18
7 Guide: Endogeneity, Instruments & 2SLS 18
8 Guide: GMM 20
9 Guide: Maximum Likelihood & the Three Tests 20
10 Guide: Binary & Multinomial Models 20
11 Guide: Ordered Models 21
12 Guide: Truncated, Censored & Selected Samples 22
13 Guide: Common Derivations Checklist 23
III Worked Exam-Style Exercises 23
14 Endogeneity, IV & GMM 23
15 Maximum Likelihood & Tests 24
16 Binary & Multinomial Choice 25
17 Ordered Models 26
18 Truncation, Censoring & Selection 27
IV Practice Questions, Answer Key & Formula Sheet 28
19 Practice Questions 28
20 Answer Key 32
21 Formula Sheet 35
2
, Econometrics 2 1 Week 1: Endogeneity & Instrumental Variables
Part I
Lecture Notes & Derivations
1 Week 1: Endogeneity & Instrumental Variables
1.1 Endogeneity and its causes
Model: y = Xβ + ε, with y, ε ∈ Rn , X ∈ Rn×k . Regressors are exogenous if E[εi | xi ] = 0; they
are endogenous if xi and εi are correlated, i.e. E[xi εi ] ̸= 0.
Four causes of endogeneity
1. Omitted variables (unobserved heterogeneity): true model y = x′ β + w′ γ + ε with
Cov(x, w) ̸= 0, but we estimate y = x′ β+u so u= w′ γ+ε and Cov(x, u) = Cov(x, w)γ ̸=
0.
2. Measurement error in the regressors: we observe x = x∗ +εx , which creates correlation
between x and the composite error.
3. Simultaneity / reverse causality: y depends on x and x depends on y (e.g. price
and quantity in a market).
4. Sample selection (non-random sample) and misspeci
ed dynamics.
1.2 Consequences of endogeneity
If E[ε | X] ̸= 0:
1. E[y | X] ̸= Xβ : the regression function is not linear in the way assumed.
2. OLS is biased: E[β̂ | X] = β + (X ′ X)−1 X ′ E[ε | X] ̸= β .
3. OLS is inconsistent: by the LLN plim n1 X ′ ε ̸= 0, and by Slutsky
plim β̂ = β + Q−1 1 ′
xx plim n X ε ̸= β, Qxx = plim n1 X ′ X.
Lemma. E[ε | X] = 0 ⇒ Cov(ε, Xj ) = 0. Proof: E[ε] = EX [E[ε|X]] = 0 (LIE), so Cov(Xj , ε) =
E[Xj ε] = EX [Xj E[ε|X]] = 0. The converse is false: zero correlation does not imply mean indepen-
dence.
Omitted-variable bias (one omitted regressor)
True: y = β0 + β1 x1 + β2 x2 + u. Regressing y on x1 only:
Cov(x1 , x2 )
plim β̂1 = β1 + β2 .
Var(x1 )
Sign of the bias
Bias = β2 × sign Cov(x1 , x2 ). Example (treatment intensity vs. survival, omitted health):
sicker patients get more intensive treatment, so Cov(Intensity, Health) < 0, and health raises
survival (β2 > 0) ⇒ bias <0⇒ OLS underestimates the treatment eect.
3
