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ans_hw1_eco321_2010f - ECO321 by H Morita Economic...

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ECO321 by H. Morita Economic Statistics II Solution - HW1 Part I [9 points] Suppose that you collected the following sample. X 3 4 5 6 Y 8 7 4 3 (1) Construct a scatterplot. [1 point] Answer: (2) In the regression Y i = β 0 + β 1 X i + u i , find the estimate of intercept ( hatwide β 0 ) and the esti- mate of slope ( hatwide β 1 ). [2 points] Answer: Since n = 4, X = (3 + 4 + 5 + 6) / 4 = 4 . 5, and Y = (8 + 7 + 4 + 3) / 4 = 5 . 5, V ar ( X ) = 1 n - 1 n summationdisplay i =1 ( X i - X ) 2 = bracketleftbig (3 - 4 . 5) 2 + (4 - 4 . 5) 2 + (5 - 4 . 5) 2 + (6 - 4 . 5) 2 bracketrightbig / 3 = 1 . 667 Cov ( X, Y ) = 1 n - 1 n summationdisplay i =1 ( X i - X )( Y i - Y ) = [(3 - 4 . 5)(8 - 5 . 5) + (4 - 4 . 5)(7 - 5 . 5) + (5 - 4 . 5)(4 - 5 . 5) + (6 - 4 . 5)(3 - 5 . 5)] / 3 = - 3 Thus, ˆ β 1 = Cov ( X, Y ) V ar ( X ) = - 3 1 . 667 = - 1 . 8 ˆ β 0 = Y - ˆ β 1 X = 5 . 5 - ( - 1 . 8)(4 . 5) = 13 . 6 (3) Find SSR , ESS , and TSS . [3 points] Answer: We found ˆ Y i = ˆ β 0 + ˆ β 1 X i = 13 . 6 - 1 . 8 X i and ˆ u i = Y i - ˆ Y i , so, X 3 4 5 6 Y 8 7 4 3 ˆ Y 8.2 6.4 4.6 2.8 ˆ u -.2 .6 -.6 .2 1
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TSS = n summationdisplay i =1 ( Y i - Y ) 2 = (8 - 5 . 5) 2 + (7 - 5 . 5) 2 + (4 - 5 . 5) 2 + (3 - 5 . 5) 2 = 17 SSR = n summationdisplay i =1 ( ¯ u i ) 2 = ( - . 2) 2 + ( . 6) 2 + ( - . 6) 2 + ( . 2) 2 = . 8 ESS = n summationdisplay i =1 ( hatwide Y i - Y ) 2 = (8 . 2 - 5 . 5) 2 + (6 . 4 - 5 . 5) 2 + (4 . 6 - 5 . 5) 2 + (2 . 8 - 5 . 5) 2 = 16 . 2 Note that TSS = ESS + SSR .
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