HW 5 NEW.docx - 1(20 points The last lessons have spent a...

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1) (20 points) 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 X 0 ), that the predicted value of Y 0 is ˆ ˆ ˆ Y X . (This is sometimes called a “point prediction.”) a) (10) Prove that 0 ˆ Y is an unbiased estimator of E[Y 0 |X 0 ]. 0 0 1 0 0 1 0 0 1 0
b) (10) Derive the formula for the variance of 0 ˆ Y . Show at least two steps in this derivation. a. Hint 1: You are looking for 0 0 1 0 ˆ ˆ ˆ ( ) ( ) Var Y Var X . 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.

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