Econ 120B, Fall 2003, Answers to Homework #1Exercise 3.11The slope of the line connecting the first and second point is(Y2 − Y1)/(X2 − X1).The slope of the line connecting the second and third points is(Y3 − Y2)/(X3 − X2).By proceeding similarly, the slope of the line connecting pointt − 1 withtis givenby(Yt − Yt−1)/(Xt − Xt−1). Therefore,β˜is the average of the slopes of the straightlines connecting successive data points. The scatter diagram for this is givenbelow.The average of all the slopes in the scatter diagram is given byβ˜ = n −11_____ 2Σn HAIXt − Xt−1Yt − Yt−1________JAKTo compute the expected value ofβ˜, we have,Xt − Xt−1Yt − Yt−1________ = Xt − Xt−1α + βXt + ut − α − βXt−1 − ut−1__________________________ = β + Xt − Xt−1ut − ut−1________Hence,β˜ = β + n −11_____ 2Σn HAIXt − Xt−1ut − ut−1________JAKXtis nonrandom andE(ut) = E(ut−1) = 0. Therefore,E( β˜) = β, which means thatβ˜isunbiased. By the Gauss-Markov Theorem, OLS estimates are most efficientamong unbiased linear estimators. This implies that any other such estimator, inparticularβ˜, is inefficient (or at least is no more efficient) than the OLS estimate.1Y o o o o o o
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