Summary
Joya da Silva Patricio Gomes
Advanced Econometrics
Email:
Student Number: 2806884
December 13, 2024
,Contents
Week 1 1
Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Recap: Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Recap: Linear AR(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Difficulties with Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Difficulties with Nonlinear Models continued . . . . . . . . . . . . . . . . . . 4
Stationarity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Week 2 7
Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Challenges in Analyzing Complex Models . . . . . . . . . . . . . . . . . . . 7
Probabilistic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Value-at-Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Impulse Response Functions (IRFs) . . . . . . . . . . . . . . . . . . . . . . . . 9
Dynamic Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Stochastic Properties of Dynamic Probability Models . . . . . . . . . . . . . 10
Stationarity, Dependence and Ergodicity . . . . . . . . . . . . . . . . . . . . . 10
Stability of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Fading Memory and Dependence Structures . . . . . . . . . . . . . . . . . . 11
Bounded Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Week 3 12
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Filters and Data-Generating Processes (DGPs) . . . . . . . . . . . . . . . . . 12
Invertibility of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Invertibility in Perturbed Dynamic Equations . . . . . . . . . . . . . . . . . . 14
Multivariate Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Extremum Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Criterion Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
M-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Existence and Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Week 4 18
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2
,Advanced Econometrics Summary
Consistency of Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Stochastic Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Identifiable Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Strong consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Consistency under Misspecification . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Asymptotic Normality of Estimators . . . . . . . . . . . . . . . . . . . . . . . 21
Extremum Estimators and Asymptotic Normality . . . . . . . . . . . . . . . 21
Well-Behaved Functions and Asymptotic Normality . . . . . . . . . . . . . . 22
Approximate Statistical Inference Using Asymptotic Normality . . . . . . . 22
Estimating the Asymptotic Variance . . . . . . . . . . . . . . . . . . . . . . . 23
Week 5 23
Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Method Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Least Squares and the Weighted L2-Norm . . . . . . . . . . . . . . . . . . . . 24
MLE and Kullback-Leibler Divergence . . . . . . . . . . . . . . . . . . . . . . 24
Specification Tests with Pseudo-True Parameters . . . . . . . . . . . . . . . . 25
Estimator Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Model Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Ensemble Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Week 6 26
Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Dynamic Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
The Importance of Exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Instrumental Variables and A/B Testing . . . . . . . . . . . . . . . . . . . . . 29
CONTENTS 3
, Week 1
Chapter 1
Recap: Simple Linear Regression
The Linear Regression Model
The linear regression model is specified as:
yt = α + βxt + ϵt , (1)
where:
• yt is the dependent variable (also known as the endogenous variable or target).
• xt is the independent variable (also known as the exogenous variable or predictor).
• α is the intercept term.
• β is the slope parameter, which measures the effect of a one-unit change in xt on yt .
• ϵt is the error term, representing unexplained variability.
Assumptions in Linear Regression
For the Ordinary Least Squares (OLS) method to provide meaningful estimates, certain
assumptions must be satisfied:
• Linearity: The relationship between yt and xt is linear.
• Exogeneity: The error term is uncorrelated with the regressors, i.e., E(ϵt | xt ) = 0.
• Homoscedasticity: The variance of the error term is constant, i.e., Var (ϵt | xt ) = σ2 .
• No Perfect Multicollinearity: The regressors are not perfectly collinear.
• Independence: The observations are independently and identically distributed (i.i.d).
Ordinary Least Squares (OLS) Estimation
The OLS method estimates the parameters α and β by minimizing the sum of squared
residuals:
T
(α̂, β̂) = arg min ∑ (yt − α − βxt )2 . (2)
α,β t=1
1
Joya da Silva Patricio Gomes
Advanced Econometrics
Email:
Student Number: 2806884
December 13, 2024
,Contents
Week 1 1
Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Recap: Simple Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Recap: Linear AR(1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Difficulties with Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Difficulties with Nonlinear Models continued . . . . . . . . . . . . . . . . . . 4
Stationarity problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Week 2 7
Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Challenges in Analyzing Complex Models . . . . . . . . . . . . . . . . . . . 7
Probabilistic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Value-at-Risk (VaR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Impulse Response Functions (IRFs) . . . . . . . . . . . . . . . . . . . . . . . . 9
Dynamic Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Stochastic Properties of Dynamic Probability Models . . . . . . . . . . . . . 10
Stationarity, Dependence and Ergodicity . . . . . . . . . . . . . . . . . . . . . 10
Stability of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Fading Memory and Dependence Structures . . . . . . . . . . . . . . . . . . 11
Bounded Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Week 3 12
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Filters and Data-Generating Processes (DGPs) . . . . . . . . . . . . . . . . . 12
Invertibility of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Invertibility in Perturbed Dynamic Equations . . . . . . . . . . . . . . . . . . 14
Multivariate Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Extremum Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Criterion Function Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
M-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Z-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Existence and Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Week 4 18
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2
,Advanced Econometrics Summary
Consistency of Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Stochastic Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Identifiable Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Strong consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Consistency under Misspecification . . . . . . . . . . . . . . . . . . . . . . . 21
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Asymptotic Normality of Estimators . . . . . . . . . . . . . . . . . . . . . . . 21
Extremum Estimators and Asymptotic Normality . . . . . . . . . . . . . . . 21
Well-Behaved Functions and Asymptotic Normality . . . . . . . . . . . . . . 22
Approximate Statistical Inference Using Asymptotic Normality . . . . . . . 22
Estimating the Asymptotic Variance . . . . . . . . . . . . . . . . . . . . . . . 23
Week 5 23
Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Method Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Least Squares and the Weighted L2-Norm . . . . . . . . . . . . . . . . . . . . 24
MLE and Kullback-Leibler Divergence . . . . . . . . . . . . . . . . . . . . . . 24
Specification Tests with Pseudo-True Parameters . . . . . . . . . . . . . . . . 25
Estimator Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Model Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Ensemble Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Week 6 26
Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Structural Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Dynamic Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
The Importance of Exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Instrumental Variables and A/B Testing . . . . . . . . . . . . . . . . . . . . . 29
CONTENTS 3
, Week 1
Chapter 1
Recap: Simple Linear Regression
The Linear Regression Model
The linear regression model is specified as:
yt = α + βxt + ϵt , (1)
where:
• yt is the dependent variable (also known as the endogenous variable or target).
• xt is the independent variable (also known as the exogenous variable or predictor).
• α is the intercept term.
• β is the slope parameter, which measures the effect of a one-unit change in xt on yt .
• ϵt is the error term, representing unexplained variability.
Assumptions in Linear Regression
For the Ordinary Least Squares (OLS) method to provide meaningful estimates, certain
assumptions must be satisfied:
• Linearity: The relationship between yt and xt is linear.
• Exogeneity: The error term is uncorrelated with the regressors, i.e., E(ϵt | xt ) = 0.
• Homoscedasticity: The variance of the error term is constant, i.e., Var (ϵt | xt ) = σ2 .
• No Perfect Multicollinearity: The regressors are not perfectly collinear.
• Independence: The observations are independently and identically distributed (i.i.d).
Ordinary Least Squares (OLS) Estimation
The OLS method estimates the parameters α and β by minimizing the sum of squared
residuals:
T
(α̂, β̂) = arg min ∑ (yt − α − βxt )2 . (2)
α,β t=1
1
