The last lessons have spent a lot of time describing the slope and intercept terms (and...
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The last lessons have spent a lot of time describing the slope and intercept terms (and their variances) of the one-variable sample regression function. We also know that for any particular value of the independent variable (call it Xo), that the predicted value of Yo is Y=B+, X. (This is sometimes called a "point prediction.") a) (10) Prove that is an unbiased estimator of E[Y₁|Xo]. b) (10) Derive the formula for the variance of . Show at least two steps in this derivation. Hint 1: You are looking for Var(Y) = Var(+₁X). This is the variance of a sum of two random variables. What is the general formula for such a sum? (Go back to week 2 lectures, if you need a reminder.) Use that formula now. b. Hint 2: If you did hint 1 correctly, you will see you need the formula for cov(,). Take it on faith that this can be found to be cov(₁,₁)=-X var(B₁). (You might find it interesting that the two estimators have a negative correlation. A steeper slope tends to imply a lower intercept, and vice versa.) The last lessons have spent a lot of time describing the slope and intercept terms (and their variances) of the one-variable sample regression function. We also know that for any particular value of the independent variable (call it Xo), that the predicted value of Yo is Y=B+, X. (This is sometimes called a "point prediction.") a) (10) Prove that is an unbiased estimator of E[Y₁|Xo]. b) (10) Derive the formula for the variance of . Show at least two steps in this derivation. Hint 1: You are looking for Var(Y) = Var(+₁X). This is the variance of a sum of two random variables. What is the general formula for such a sum? (Go back to week 2 lectures, if you need a reminder.) Use that formula now. b. Hint 2: If you did hint 1 correctly, you will see you need the formula for cov(,). Take it on faith that this can be found to be cov(₁,₁)=-X var(B₁). (You might find it interesting that the two estimators have a negative correlation. A steeper slope tends to imply a lower intercept, and vice versa.)
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Related Book For
Foundations of Finance The Logic and Practice of Financial Management
ISBN: 978-0132994873
8th edition
Authors: Arthur J. Keown, John D. Martin, J. William Petty
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