# 2 10 points the last lessons have spent a lot of time

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(10) What percentage of the variation of your Y variable is explained by X?2)(10 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 Y0is 0010ˆˆˆYX. (This is sometimes called a “point prediction.”) If we haven’t covered it yet in class, takeas given that the estimators 01ˆˆandare unbiased.a)(10) Prove that 0ˆYis an unbiased estimator of Y0.
0001001001000ˆˆˆˆˆ[|][][][][|]E YXEXEEXXE YX3)(20 points) Start with the definition of goodness of fit,. Show mathematically how this is exactly equal to the square of the sample correlation coefficient, r. I never fail to be surprised by this result.
4)(20 points, 4 points for each piece.) Compare the following two regressions:i.01ˆˆiiiYXeii.01ˆˆˆ(4)iiiYXEquation i. is exactly the regression we’ve been working with thus far, so all the formulas we’ve derived thus far apply immediately. In equation ii all the values of the independent variable in the sample have been multiplied by 4. How does this change, if at all, the values of 1ˆ,0ˆ, SST, and r2? For the sake of clarity, please denote the OLS values for equation ii with a *, and compare them to the “regular” OLS estimates we derived in class.
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