Classical Linear Regression Model Quiz

Classical Linear Regression Model (CLRM)

• The CLRM model is a linear relationship between variables
• Assumption 1: The model is linear in parameters (e.g., Y₁ = B₁ + B₂X₁ + Uᵢ)
• Assumption 2: Explanatory variables (X) are uncorrelated with the error term (U) and non-stochastic
• Assumption 3: The expected value of the error term (U) is zero, given the explanatory variable (X), meaning E(U | X) = 0
• Assumption 4: The variance of the error term (U) is constant across all observations (homoscedasticity), var(Uᵢ) = σ²
• Assumption 5: There's no correlation between error terms for different observations, cov(Uᵢ, Uⱼ) = 0 for i ≠ j
• Assumption 6: The regression model is correctly specified
• Assumption 7: Error terms follow a normal distribution (Uᵢ ~ N(0, σ²))

Properties of OLS Estimators

• Gauss-Markov Theorem: OLS estimators have minimum variance in the class of linear unbiased estimators (BLUE)
• Property 1: OLS estimators (b₁ and b₂) are linear functions of the dependent variable (Y)
• Property 2: OLS estimators are unbiased; E(b₁) = B₁ and E(b₂) = B₂
• Property 3: The OLS estimator of the error variance (σ²) is unbiased, E(σ²) = σ²
• Property 4: OLS estimators are efficient; their variances are smaller than any other linear unbiased estimator for B₁ and B₂
• Sampling distributions of b₁ and b₂ are normal if the error terms are normally distributed (b₁ ~ N(B₁, σ²_b₁), b₂ ~ N(B₂, σ²_b₂))