Flashcards on Introduction to Simple Regression Analysis
Introduction to Simple Regression Analysis Guide
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Linear Regression Inference & Diagnostics
61 cards
Card 1
Question: What are the estimated OLS slope and intercept for the regression of Y on X given ΣX=15, ΣY=46, ΣXY=113, ΣX²=55, n=5 (with mean-deviation results Σxy=
Answer: Slope β̂ = Σxy/Σx² = -25/10 = -2.5. Intercept α̂ = Ȳ - β̂·X̄ = 9.2 - (-2.5)(3) = 16.7.
Card 2
Question: Why can we compute β̂ using mean-deviation forms (x = X - X̄, y = Y - Ȳ)?
Answer: Because Σx = Σy = 0, so Σxy = ΣXY - nX̄Ȳ and Σx² = ΣX² - nX̄²; using deviations simplifies the algebra and yields the same β̂.
Card 3
Question: In the sample regression Y_i = α̂ + β̂X_i + û_i, what does the û_i represent?
Answer: û_i is the estimated sample error (residual) for observation i — the deviation of observed Y_i from the fitted value α̂ + β̂X_i.
Card 4
Question: What does the statement 'Σ x ū₁ = 0' indicate about the relationship between the explanatory variable and the (sample) error term in deviation form?
Answer: It indicates that the explanatory variable (in deviations) is uncorrelated with the sample error term: the sum of products of x and residuals equals z
Card 5
Question: Why is Σ y ū₁ not equal to Σ ȳ_i ū_i (i.e., why is Σ y ū₁ ≠ Σ ȳ_i ū_i)?
Answer: Because y denotes mean-deviation values (Y - Ȳ) while ȳ_i denotes the sample mean Ȳ; multiplication with residuals differs when using deviations versu
Card 6
Question: What is the expression for the variance of the estimator β (var(β̄)) in terms of the disturbances u_i and coefficients k_i?
Answer: var(β̄) = E(∑ k_i u_i)^2 = σ^2 ∑ k_i^2 (since E(u_i u_j)=0 for i≠j).
Card 7
Question: How does ∑ k_i^2 relate to the observed x_i in the simple regression variance formula for β̂?
Answer: Recall that ∑ k_i^2 = 1 / ∑ x_i^2, so var(β̂) = σ^2 / ∑ x_i^2.
Card 8
Question: Write the standard error of the slope estimator β̂ (s.e.(β̂)).
Answer: s.e.(β̂) = √Var(β̂) = √(σ^2 / ∑ x_i^2).
Card 9
Question: Express the estimator α̂ in terms of α, (β̂−β), X̄ and ū (equation 3.31 rearranged).
Answer: α̂ − α = −(β̂ − β) X̄ + ū, so α̂ = α − (β̂ − β) X̄ + ū.
Card 10
Question: Derive the variance of α̂ using var(α̂−α) = E[−(β̂−β)X̄ + ū]^2. What is the resulting expression before substituting var(β̂)?
Answer: var(α̂) = X̄^2 var(β̂) + E(ū^2) − 2E[(β̂−β)X̄ ū]; the cross term expectation is zero, and E(ū^2)=σ^2/n, so var(α̂) = X̄^2 var(β̂) + σ^2/n.