Advanced Econometrics
Complete Course Summary
The Bootstrap: Theory, Re
nements & Applications,
Weak Instruments & Linear IV Regression,
Panel Data Models: Static & Dynamic
Lecture Notes, Tutorials & Worked Exam Exercises
,Advanced Econometrics Complete Course Summary 1
Contents
I The Bootstrap: Theory & Methods 3
1 The Bootstrap: Basic Idea & Algorithm 3
1.1 Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Bootstrap without Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Regression Bootstraps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Consistency of the Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Con
dence Intervals & Hypothesis Testing with the Bootstrap 4
2.1 Bootstrap Percentile-t Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Asymptotic Re
nement & Hypothesis Testing 4
3.1 Asymptotic Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Nonparametric vs. Parametric Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Hypothesis Testing with Asymptotic Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Iterated Bootstrap for Extra Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 Why 999 Bootstrap Replications and not 1000? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Power, Subsampling & Reliability of the Bootstrap 6
4.1 Bootstrap Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 Subsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.3 Computational Cost of the Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Bootstrap Consistency & GMM Overidenti
cation 7
5.1 Bootstrap Consistency: Probability an Observation is Not Resampled . . . . . . . . . . . . . . . . . 7
5.2 Discretion at Necessary Steps (Including Resampling Bias) . . . . . . . . . . . . . . . . . . . . . . . 7
5.3 GMM Overidenti
cation (J-test) Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Time Series Bootstrap 8
6.1 Time Series Model with ARMA Error Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2 Dynamic Linear Autoregressive (AR) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Heteroskedasticity-Robust (Wild) Bootstrap 8
7.1 Intermediate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7.2 Wald Test Studentized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.3 Speci
cation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.4 GMM Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Worked Tutorial Exercises Bootstrap for Regression Models 9
II Weak Instruments & Linear IV Regression 10
9 Linear IV Regression Model 10
9.1 Model & Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9.2 Consistency of 2SLS Under Weak Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.3 Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.4 Dynamic Linear Panel Data Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
10 Weak-Instrument Robust Tests 11
10.1 LIML is Robust to Weak Instruments, 2SLS is Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
11 Limited Information Maximum Likelihood (LIML) & the Stock-Yogo Rule 12
11.1 LIML Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
11.2 Anderson-Rubin (AR) Statistic Detailed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
11.3 Score / LM Statistic (Kleibergen-Moreira) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
11.4 Likelihood Ratio (LR) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
11.5 Why This is Better than AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12 Stock-Yogo Rule & Score/LM Statistic Continued 14
12.1 Score/Lagrange Multiplier (LM) Statistic Near H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
,Advanced Econometrics Complete Course Summary 2
12.2 Rule of Thumb: Stock-Yogo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
13 Worked Tutorial Exercises Weak Instruments & IV 15
III Panel Data Models 16
14 Static Panel Data: Fixed Eects & First Dierences 16
14.1 Panel Data Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
14.2 FD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.3 Instrumental Variables Approach (AH: FD + IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.4 First Dierence GMM E
cient (Anderson-Hsiao) Optimal Weight Matrix . . . . . . . . . . . . . . . 17
14.5 Two-Step GMM E
cient Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.6 Fixed Eects: Additive vs. Multiplicative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15 Testing Panel Data Assumptions with the Bootstrap 17
15.1 Bootstrap Method & Panel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15.2 Asymptotic Re
nement for Panel Data (Bootstrap over i) . . . . . . . . . . . . . . . . . . . . . . . . 18
15.3 Serial Correlation Test: Testing Model Adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
15.4 Weak-Instrument Robust Methods (AB Fixed-Panel Estimator) . . . . . . . . . . . . . . . . . . . . . 18
16 Weak Identi
cation in Dynamic Panel GMM 18
16.1 Weak Instruments in Dynamic Panel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
17 Worked Tutorial Exercises Panel Data Models 19
, Advanced Econometrics Complete Course Summary 3
Part I
The Bootstrap: Theory & Methods
1 The Bootstrap: Basic Idea & Algorithm
1.1 Main Idea
Instead of assuming a true population distribution F0 , we:
1. Treat the observed sample as the population.
2. Resample from it.
3. Re-estimate the statistic many times.
4. Use the empirical distribution of these estimates.
The bootstrap replaces
rst-order, asymptotic and bias that asymptotic theory omits: asymptotic re
nement.
1.2 Bootstrap without Re
nement
Suppose yi = f (xi , β) + εi , i = 1, . . . , n. We want to know: Var(β̂)?
Bootstrap analog: F (x, θ̂), sample yi∗ (xi ), bootstrap sample: y1∗ , . . . , yn∗ .
∗b ∗b
We can easily generate F (x, θ̂): the b-th bootstrap sample {y1 , . . . , yn } gives one realization of the sample
∗b
mean θ̂ . We can approximate Var(θ̂) by:
B
1 X ∗b
d (θ̂)
Var = (θ̂ − θ̄∗ )2
B
b=1
Bootstrap Algorithm (General)
1. Given data {u1 , . . . , un }, draw a bootstrap sample {u∗1 , . . . , u∗n } of size n.
∗ ∗ ∗
2. Calculate the estimate: θ̂ = θ̂(u1 , . . . , un )
3. Repeat steps 1 and 2 B times.
θ̄∗ = B1 b θ̂∗b
P
a.
b. T ∗ = (θ̂∗ − θ̂)/σ̂ (for CI)
c. T ∗b = . . .
Conduct inference:
∗
1. Bias: B̂boot = θ̄ − θ̂ approximates −θ
q E[θ̂]
1
P ∗b
2. Standard error: SEboot (θ̂) = B−1 b (θ̂ − θ̄∗ )2
3. 2-sided equal-tail CI: [θ̂ − z1−α/2 SEboot , θ̂ + z1−α/2 SEboot ], where z(·) is a critical value based on the
standard normal.
1.3 Regression Bootstraps
Pairs bootstrap (a.k.a. nonparametric bootstrap): resample pairs (yi , xi ) with replacement.
Residual bootstrap: resample residuals ûi , build yi∗ = x′i β̂ + u∗i , then keep xi
xed and re-estimate.
1.4 Consistency of the Bootstrap
When errors are heteroskedastic, the true asymptotic covariance of OLS is:
avar(β̂) = (X ′ X)−1 X ′ diag(σi2 )X(X ′ X)−1 ̸= σ 2 (X ′ X)−1
Pairs/residual bootstrap variance is consistent under heteroskedasticity? Pairs bootstrap: yes; residual bootstrap:
only under homoskedasticity.
Wild bootstrap: residual bootstrap generalized to allow for heteroskedasticity:
u∗i = ûi · vi , vi i.i.d. with E[vi ] = 0, E[vi2 ] = 1
Complete Course Summary
The Bootstrap: Theory, Re
nements & Applications,
Weak Instruments & Linear IV Regression,
Panel Data Models: Static & Dynamic
Lecture Notes, Tutorials & Worked Exam Exercises
,Advanced Econometrics Complete Course Summary 1
Contents
I The Bootstrap: Theory & Methods 3
1 The Bootstrap: Basic Idea & Algorithm 3
1.1 Main Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Bootstrap without Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Regression Bootstraps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Consistency of the Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Con
dence Intervals & Hypothesis Testing with the Bootstrap 4
2.1 Bootstrap Percentile-t Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Asymptotic Re
nement & Hypothesis Testing 4
3.1 Asymptotic Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Nonparametric vs. Parametric Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Hypothesis Testing with Asymptotic Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Iterated Bootstrap for Extra Re
nement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 Why 999 Bootstrap Replications and not 1000? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Power, Subsampling & Reliability of the Bootstrap 6
4.1 Bootstrap Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 Subsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.3 Computational Cost of the Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 Bootstrap Consistency & GMM Overidenti
cation 7
5.1 Bootstrap Consistency: Probability an Observation is Not Resampled . . . . . . . . . . . . . . . . . 7
5.2 Discretion at Necessary Steps (Including Resampling Bias) . . . . . . . . . . . . . . . . . . . . . . . 7
5.3 GMM Overidenti
cation (J-test) Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Time Series Bootstrap 8
6.1 Time Series Model with ARMA Error Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.2 Dynamic Linear Autoregressive (AR) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Heteroskedasticity-Robust (Wild) Bootstrap 8
7.1 Intermediate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7.2 Wald Test Studentized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.3 Speci
cation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.4 GMM Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8 Worked Tutorial Exercises Bootstrap for Regression Models 9
II Weak Instruments & Linear IV Regression 10
9 Linear IV Regression Model 10
9.1 Model & Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
9.2 Consistency of 2SLS Under Weak Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.3 Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.4 Dynamic Linear Panel Data Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
10 Weak-Instrument Robust Tests 11
10.1 LIML is Robust to Weak Instruments, 2SLS is Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
11 Limited Information Maximum Likelihood (LIML) & the Stock-Yogo Rule 12
11.1 LIML Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
11.2 Anderson-Rubin (AR) Statistic Detailed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
11.3 Score / LM Statistic (Kleibergen-Moreira) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
11.4 Likelihood Ratio (LR) Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
11.5 Why This is Better than AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12 Stock-Yogo Rule & Score/LM Statistic Continued 14
12.1 Score/Lagrange Multiplier (LM) Statistic Near H0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
,Advanced Econometrics Complete Course Summary 2
12.2 Rule of Thumb: Stock-Yogo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
13 Worked Tutorial Exercises Weak Instruments & IV 15
III Panel Data Models 16
14 Static Panel Data: Fixed Eects & First Dierences 16
14.1 Panel Data Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
14.2 FD Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.3 Instrumental Variables Approach (AH: FD + IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.4 First Dierence GMM E
cient (Anderson-Hsiao) Optimal Weight Matrix . . . . . . . . . . . . . . . 17
14.5 Two-Step GMM E
cient Weight Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
14.6 Fixed Eects: Additive vs. Multiplicative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15 Testing Panel Data Assumptions with the Bootstrap 17
15.1 Bootstrap Method & Panel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
15.2 Asymptotic Re
nement for Panel Data (Bootstrap over i) . . . . . . . . . . . . . . . . . . . . . . . . 18
15.3 Serial Correlation Test: Testing Model Adequacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
15.4 Weak-Instrument Robust Methods (AB Fixed-Panel Estimator) . . . . . . . . . . . . . . . . . . . . . 18
16 Weak Identi
cation in Dynamic Panel GMM 18
16.1 Weak Instruments in Dynamic Panel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
17 Worked Tutorial Exercises Panel Data Models 19
, Advanced Econometrics Complete Course Summary 3
Part I
The Bootstrap: Theory & Methods
1 The Bootstrap: Basic Idea & Algorithm
1.1 Main Idea
Instead of assuming a true population distribution F0 , we:
1. Treat the observed sample as the population.
2. Resample from it.
3. Re-estimate the statistic many times.
4. Use the empirical distribution of these estimates.
The bootstrap replaces
rst-order, asymptotic and bias that asymptotic theory omits: asymptotic re
nement.
1.2 Bootstrap without Re
nement
Suppose yi = f (xi , β) + εi , i = 1, . . . , n. We want to know: Var(β̂)?
Bootstrap analog: F (x, θ̂), sample yi∗ (xi ), bootstrap sample: y1∗ , . . . , yn∗ .
∗b ∗b
We can easily generate F (x, θ̂): the b-th bootstrap sample {y1 , . . . , yn } gives one realization of the sample
∗b
mean θ̂ . We can approximate Var(θ̂) by:
B
1 X ∗b
d (θ̂)
Var = (θ̂ − θ̄∗ )2
B
b=1
Bootstrap Algorithm (General)
1. Given data {u1 , . . . , un }, draw a bootstrap sample {u∗1 , . . . , u∗n } of size n.
∗ ∗ ∗
2. Calculate the estimate: θ̂ = θ̂(u1 , . . . , un )
3. Repeat steps 1 and 2 B times.
θ̄∗ = B1 b θ̂∗b
P
a.
b. T ∗ = (θ̂∗ − θ̂)/σ̂ (for CI)
c. T ∗b = . . .
Conduct inference:
∗
1. Bias: B̂boot = θ̄ − θ̂ approximates −θ
q E[θ̂]
1
P ∗b
2. Standard error: SEboot (θ̂) = B−1 b (θ̂ − θ̄∗ )2
3. 2-sided equal-tail CI: [θ̂ − z1−α/2 SEboot , θ̂ + z1−α/2 SEboot ], where z(·) is a critical value based on the
standard normal.
1.3 Regression Bootstraps
Pairs bootstrap (a.k.a. nonparametric bootstrap): resample pairs (yi , xi ) with replacement.
Residual bootstrap: resample residuals ûi , build yi∗ = x′i β̂ + u∗i , then keep xi
xed and re-estimate.
1.4 Consistency of the Bootstrap
When errors are heteroskedastic, the true asymptotic covariance of OLS is:
avar(β̂) = (X ′ X)−1 X ′ diag(σi2 )X(X ′ X)−1 ̸= σ 2 (X ′ X)−1
Pairs/residual bootstrap variance is consistent under heteroskedasticity? Pairs bootstrap: yes; residual bootstrap:
only under homoskedasticity.
Wild bootstrap: residual bootstrap generalized to allow for heteroskedasticity:
u∗i = ûi · vi , vi i.i.d. with E[vi ] = 0, E[vi2 ] = 1
