topic 3 notes Econometrics

Preface

• Notes designed to complement ECON20110/30370 lectures.

• Focus on mathematical aspects of econometrics.

• Lecture slides emphasize application and intuition.

• Report typos or errors via email.

Table of Contents

7 Endogeneity and Causal Inference

• 7.1 Causal Models and Endogeneity

  • Distinction between descriptive and causal analysis.

  • Causal analysis essential for evaluating interventions.

  • Counterfactual approach to identify causal relationships.

Descriptive Analysis vs Causal Analysis

• Regression can aim for prediction or causal relationships.

  • Prediction Focus:

    • OLS used as a descriptive tool.

    • Orthogonality condition holds, enabling consistent estimators.

    • Issues like omitted variable bias do not affect predictions.

  • Causal Analysis Focus:

    • Identify and measure causal influences.

    • Essential to control for confounding factors (ceteris paribus).

7.2 Exogeneity and Consistent Estimation of Causal Effects

• Key model where outcome variable Y is influenced by multiple variables.

  • Identify if E(ε) = 0 and cov(X_l, ε) = 0.

  • Exogeneity condition crucial for consistent estimation of causal effects.

7.3 Endogeneity

• Endogeneity arises when explanatory variables are correlated with the error term.

  • Challenges conventional regression assumptions.

  • Biased OLS estimates when orthogonality fails.

7.4 Sources of Endogeneity

• Common Sources:

  1. Omitted variable bias.

  2. Wrong functional form.

  3. Measurement error.

  4. Simultaneous causality bias.

  5. Sample selection bias.

7.5 Omitted Variable Bias (OVB)

• OVB occurs when a key determinant is omitted, affecting observed relationships.

  • OVB Formula:

    • Derived under conditions of OVB; critical to understand the direction of bias.

7.5.1 The OVB Formula — Simple Setting

• Example: Representing wages as a function of years of education and ability.

  • Impact of omitting ability on estimated effect of education.

7.6 Functional Form Misspecification

• Arises when the model fails to summarize the true causal relationship.

  • Need for careful evaluation of model specifications.

7.7 Measurement Error

• Distortion from inaccuracies in data collection (e.g., survey errors).

  • Impacts reliability of regression analyses.

7.7.1 Measurement Error in X

• Measurement error in X affects estimates through bias.

7.7.2 The Classical Measurement Error in X Problem — Attenuation Bias

• Underestimates true effects due to measurement error, particularly concerned in regression models.

7.7.3 Measurement Error in Y

• Measurement error in Y increases error variance without introducing bias if uncorrelated with X.

7.8 Simultaneity Bias

• Occurs when independent variables influence each other.

  • Example: Demand and supply equations; challenges the traditional causal model assumptions.

7.9 Missing Data and Sample Selection Bias

• Different effects of missing data:

  1. Missing Data Completely at Random: No bias induced.

  2. Missing Data Conditional on X: Unbiased but less precise.

  3. Sample Selection Bias: Correlation with Y leads to biased estimates.

7.10 Possible Solutions to Endogeneity

• Strategies include excluding omitted variables, using instrumental variables (IV), and performing RCTs.

Addressing Endogeneity with Instrumental Variables (IV)

8.1 Endogeneity and the Problem of Inconsistent OLS

• OLS implications when variables exhibit endogeneity.

8.2 Instrumental Variables: Validity Requirements

• Necessary conditions for IV to be effective include relevance, exogeneity, and exclusion.

8.3 Two-Stage Least Squares (2SLS)

• Method for obtaining consistent estimates in the presence of endogeneity.

• Used for estimating causal models using the IV technique.

8.4 Important Considerations and Results for 2SLS

• The quality of IV selection and the model structure affects results.

Appendix 8.A The Over-Identifying Restrictions Test

• Tests to validate the instruments used in IV analysis, emphasizing the need for theoretical grounding alongside empirical results.