Consider the standard simple regression model y = Bo + Bx + u under standard assumptions...

 

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Consider the standard simple regression model y = Bo + B₁x + u under standard assumptions The usual OLS estimators B, and 3, are unbiased for their respective population parameters. Let B, be the estimator of B, obtained by assuming the intercept is zero. (i) (ii) (iii) (iv) Find E(3₁) in terms of the x, Bo, and B₁. Verify that B₁ is unbiased for B, when the population intercept (B) is zero. Are there other cases where B₁ is unbiased? ≥ Σ²₁(x¡ − x)², with Find the variance of B₁. (Hint: The variance does not depend on Bo.) Show that Var(B₁) ≤ Var(B₁). [Hint: For any sample of data, Σ strict inequality unless x = 0.] Comment on the tradeoff between bias and variance when choosing between 3, and B₁. Consider the standard simple regression model y = Bo + B₁x + u under standard assumptions The usual OLS estimators B, and 3, are unbiased for their respective population parameters. Let B, be the estimator of B, obtained by assuming the intercept is zero. (i) (ii) (iii) (iv) Find E(3₁) in terms of the x, Bo, and B₁. Verify that B₁ is unbiased for B, when the population intercept (B) is zero. Are there other cases where B₁ is unbiased? ≥ Σ²₁(x¡ − x)², with Find the variance of B₁. (Hint: The variance does not depend on Bo.) Show that Var(B₁) ≤ Var(B₁). [Hint: For any sample of data, Σ strict inequality unless x = 0.] Comment on the tradeoff between bias and variance when choosing between 3, and B₁.