Econometrics

1. Foundations of Econometrics

What is Econometrics?

• Econometrics applies statistical methods to economic data to test hypotheses and estimate causal relationships.

• Key challenge: Distinguishing correlation from causation.

Regression Analysis as a Tool

• Regression allows us to model relationships:

  Yi=β0+β1Xi+εiY_i = \beta_0 + \beta_1 X_i + \varepsilon_iYi​=β0​+β1​Xi​+εi​

  • YiY_iYi​: Dependent variable (outcome).

  • XiX_iXi​: Independent variable (predictor).

  • εi\varepsilon_iεi​: Error term (captures unobserved factors).

• Key Question: Does XXX cause YYY? Or is the relationship spurious due to omitted variables, reverse causality, or measurement error?

2. Probability and Statistical Foundations

Understanding Random Variables

• Discrete vs. Continuous:

  • Discrete: Limited set of outcomes (e.g., number of students in a class).

  • Continuous: Infinite possible values (e.g., income levels).

• Probability Distributions:

  • PDF (Probability Density Function): Shows the likelihood of different outcomes.

  • CDF (Cumulative Distribution Function): Shows the probability of observing a value ≤ a given point.

• Expectation & Variance:

  • Expected Value (Mean): E[Y]=∑PiYiE[Y] = \sum P_i Y_iE[Y]=∑Pi​Yi​

  • Variance: Measures spread of distribution.

  • Standard Deviation: Square root of variance.

• Covariance & Correlation:

  • Covariance: Measures how two variables move together.

  • Correlation: Standardized covariance (bounded between -1 and 1).

3. Types of Data

• Cross-sectional: Many units at a single time.

• Time series: Single unit observed over multiple time periods.

• Panel data: Combines both (e.g., state-level unemployment rates over 10 years).

4. Ordinary Least Squares (OLS) and Assumptions

OLS estimates parameters by minimizing the sum of squared residuals.

Key Assumptions (Gauss-Markov)

1. Linearity: Model correctly specifies the relationship.

2. Random Sampling: Observations are independent.

3. No Perfect Multicollinearity: No exact linear relationships among predictors.

4. Zero Conditional Mean of Errors: E[ε∣X]=0E[\varepsilon | X] = 0E[ε∣X]=0 (no omitted variable bias).

5. Homoskedasticity: Error variance is constant across values of XXX.

6. Normality of Errors (for inference): ε∼N(0,σ2)\varepsilon \sim N(0, \sigma^2)ε∼N(0,σ2).

5. Threats to Internal Validity

Internal validity refers to whether a study correctly identifies a causal effect. Threats arise when the estimated relationship between XXX and YYY is biased.

1. Omitted Variable Bias (OVB)

Occurs when a variable that affects both XXX and YYY is left out of the model.

How to Spot It:

• Ask: Is there a missing factor that could be driving both XXX and YYY?

• If the omitted variable is correlated with XXX, OLS estimates are biased.

• Example:

  • Regression: Income=β0+β1Education+ε\text{Income} = \beta_0 + \beta_1 \text{Education} + \varepsilonIncome=β0​+β1​Education+ε

  • Omitted Variable: Ability

  • If ability increases both education and income, the effect of education is overstated.

How to Address It:

• Include the omitted variable (if measurable).

• Use fixed effects to control for unobservable factors.

• Use an Instrumental Variable (IV).

2. Reverse Causality

Occurs when YYY actually causes XXX instead of the other way around.

How to Spot It:

• Ask: Could the dependent variable be influencing the independent variable?

• Example:

  • Regression: Crime Rate=β0+β1Police Presence+ε\text{Crime Rate} = \beta_0 + \beta_1 \text{Police Presence} + \varepsilonCrime Rate=β0​+β1​Police Presence+ε

  • Reverse Causality: High crime rates cause an increase in police presence.

How to Address It:

• Lagged variables: Use past values of XXX to predict current YYY.

• Instrumental Variables (IV).

3. Measurement Error

Occurs when the independent variable XXX is measured with error.

How to Spot It:

• Ask: Is XXX reported or measured inaccurately?

• Example:

  • If people underreport their income in surveys, bias may result.

Types of Measurement Error:

• Classical Measurement Error (random error): Reduces precision, but does not bias estimates.

• Non-classical Measurement Error (systematic error): Biases estimates.

How to Address It:

• Use instrumental variables or better data sources.

4. Misspecified Functional Form

Occurs when a model assumes a linear relationship when the true relationship is nonlinear.

How to Spot It:

• Check scatter plots: Do relationships appear non-linear?

• Example:

  • Quadratic relationships: Income and happiness might have a diminishing return.

How to Address It:

• Add polynomial terms (e.g., X2X^2X2).

• Use log transformations.

5. Outliers and Leverage Points

Extreme values can distort estimates.

How to Spot It:

• Look at histograms or scatter plots.

• Example:

  • A single billionaire in an income regression may distort results.

How to Address It:

• Winsorizing (replace extreme values with threshold values).

• Robust regression methods.

6. Sample Selection Bias

Occurs when the sample is not representative of the population.

How to Spot It:

• Ask: Does the sample systematically exclude certain groups?

• Example:

  • Studying only employed people when analyzing income ignores those who can’t work.

How to Address It:

• Use Heckman selection models.

6. Methods to Address Internal Validity Issues

1. Randomized Control Trials (RCTs)

• Gold standard for causal inference.

• Randomly assigns treatment and control.

2. Difference-in-Differences (DiD)

• Compares treatment & control groups before and after a policy change.

• Key Assumption: Parallel Trends (control group is a good counterfactual).

3. Fixed Effects (FE)

• Controls for unobserved characteristics that do not change over time.

• Used in panel data (e.g., state-by-year analysis).

4. Instrumental Variables (IV)

• Used when XXX is endogenous (correlated with the error term).

• Example: Using distance to school as an instrument for education.

5. Regression Discontinuity (RD)

• Uses a cutoff rule (e.g., students above a certain GPA get scholarships)