Consider Exercise 11.10. (a) Find out if the estimator hat{eta}_1 found in Exercise 11.10 is unbiased or not in the no-intercept model. (b) Suppose that V(epsilon_i)=sigma^2. Find out the MSE of hat{eta}_1.
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- The problem involves checking if the estimator \(\hat{\beta_1}\) is unbiased in a no-intercept model. - The problem also requires finding the Mean Squared Error (MSE) of \(\hat{\beta_1}\) given that the variance of the error terms \(\epsilon_i\) is \(\sigma^2\). Show more…
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Consider the standard simple regression model y = ̠₀ + ̠₁x + u under the Gauss-Markov Assumptions SLR.1 through SLR.5. The usual OLS estimators ̢̠₀ and ̢̠₁ are unbiased for their respective population parameters. Let ̠̃₁ be the estimator of ̠₁ obtained by assuming the intercept is zero (see Section 2.6). (a) Find E(̠̃₁) in terms of the xᄒ, ̠₀, and ̠₁. Verify that ̠̃₁ is unbiased for ̠₁ when the population intercept (̠₀) is zero. Are there other cases where ̠̃₁ is unbiased? (b) Find the variance of ̠̃₁. (Hint: the variance does not depend on ̠₀.) (c) Show that Var(̠̃₁) ≤ Var(̢̠₁). [Hint: for any sample of data, ̣ᄂᄒ₁ xᄒ" ≥ ̣ᄂᄒ₁(xᄒ - x̄)", with strict inequality unless x̄ = 0.]
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