04 Jun Question 1) 25 points) The last lessons have sp
Question
1) 25 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 X0), that the predicted value of Y0 is . (This is sometimes called a point prediction.) If we havent covered it yet in class, take as given that the estimators are unbiased.
a) (10) Prove that is an unbiased estimator of Y0.
b) (15) Derive a formula for the variance of . Show at least two steps in this derivation.
a. Hint 1: You are looking for . 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 . Take it on faith that this can be found to be . (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.)
2) (25 points) Compare the following two regressions:
a.i.
a.ii.
Equation i. is exactly the regression weve been working with thus far, so all the formulas weve derived thus far apply. In equation ii the independent variable has been multiplied by 3. How does this change, if at all, the values of ,, , R2,
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