Question

Consider the standard simple regression model y = ?0 + ?1 x + 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). (i) Find E(??1) in terms of the xi, ?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? (ii) Find the variance of ??1. (Hint: The variance does not depend on ?0.) (iii) Show that Var(??1) ? Var(??1). [Hint: For any sample of data, ?_{i=1}^n xi^2 ? ?_{i=1}^n (xi - x?)^2, with strict inequality unless x? = 0.] (iv) Comment on the tradeoff between bias and variance when choosing between ??1 and ??1.

          Consider the standard simple regression model y = ?0 + ?1 x + 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).

(i) Find E(??1) in terms of the xi, ?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?

(ii) Find the variance of ??1. (Hint: The variance does not depend on ?0.)

(iii) Show that Var(??1) ? Var(??1). [Hint: For any sample of data, ?_{i=1}^n xi^2 ? ?_{i=1}^n (xi - x?)^2, with strict inequality unless x? = 0.]

(iv) Comment on the tradeoff between bias and variance when choosing between ??1 and ??1.

Consider the standard simple regression model y = ?0 + ?1 x + 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).

(i) Find E(??1) in terms of the xi, ?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?

(ii) Find the variance of ??1. (Hint: The variance does not depend on ?0.)

(iii) Show that Var(??1) ? Var(??1). [Hint: For any sample of data, ?i=1^n xi^2 ? ?i=1^n (xi - x?)^2, with strict inequality unless x? = 0.]

(iv) Comment on the tradeoff between bias and variance when choosing between ??1 and ??1.

Added by Kara H.

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