lecture3

1. Introduction to Econometrics

• Definition: Econometrics involves statistical methods for estimating economic relationships, testing theories, and evaluating policies.

• Focus: It addresses issues related to non-experimental or observational economic data, which is different from controlled experiments in natural sciences.

• Use Cases: It is employed for testing economic theories or examining critical relationships for business decisions and policy analysis.

2. Key Concepts in Econometrics

2.1 Econometrics Questions

• An example is determining how wages respond to educational attainment (Returns to education).

• Policy questions can include:

  • Effects of minimum schooling requirements on wages.

  • Impact of varying educational choices on earnings inequality.

2.2 Simple Linear Regression Model

2.2.1 Components of the Model

• Dependent Variable (y): e.g., hourly wage.

• Independent Variable (x): e.g., years of education.

• Error Term (u): Reflects unobservable factors affecting y other than x.

• Mathematical Formulation:

  • Equation: y = β0 + β1x + u.

2.3 Assumptions in Regression Model

• Assumption 1: Mean of error term E(u) = 0.

• Assumption 2: Error term (u) is mean independent of x; E(u|x) = E(u) (u and x uncorrelated).

2.4 Population Regression Function (PRF)

• Given Assumption 2, we can derive E(y|x) = β0 + β1x, which states that the conditional expectation of y given x is a linear function.

• Example: E(y|x) relates to the mean wage for individuals at specific education levels.

3. Estimation of the Model Parameters

3.1 Ordinary Least Squares (OLS) Estimator

• Used to estimate parameters β0 and β1 by minimizing the sum of squared residuals.

• The formulas for OLS estimators are:

  • βˆ1 = Σ(xi - x¯)(yi - y¯) / Σ(xi - x¯)² (slope)

  • βˆ0 = y¯ - βˆ1x¯ (intercept).

3.2 Estimating Residuals and Fitted Values

• Fitted Values: yˆi = βˆ0 + βˆ1xi.

• Residuals: uˆi = yi - yˆi.

4. Properties of the OLS Estimator

4.1 Algebraic Properties

• Sum of residuals: Σuˆi = 0.

• Covariance between regressors and residuals: Σxi uˆi = 0.

• The mean of fitted values equals the mean of y.

4.2 Goodness-of-Fit

• R-squared (R²): Measures the proportion of variability in the dependent variable explained by the independent variable.

  • R² = SSE/SST = 1 - SSR/SST.

• Example interpretation: An R² of 0.1648 implies that education explains about 16% of the variation in wages.

5. Assessing The OLS Estimators

5.1 Unbiasedness

• Theorem: E(βˆ0) = β0 and E(βˆ1) = β1 under certain assumptions (SLR.1 to SLR.4).

5.2 Variances of the OLS Estimators

• Variance formulas:

  • Var(βˆ1) = σ² / Σ(xi - x¯)².

  • Var(βˆ0) = (σ² / (n-2)) * Σ(x²) / Σ(xi - x¯)².

5.3 Estimating Error Variance

• Estimated σ² = SSR / (n-2).

  • Standard error estimation uses residuals: √σˆ².

6. Example: Wage Effects of Education

6.1 Data Analysis Steps

1. Formulate economic question: How does wage respond to education?

2. Set up empirical model: wage = β0 + β1 schooling + u.

3. Collect data on wages and educational attainment.

4. Compute OLS estimates based on sample averages.

5. Develop regression line: wage = -3.569 + 0.8597 schooling.

6. Interpret: Adding another year of schooling increases predicted wages by approximately $0.86.