# D ans since m i 1 f i y i ˆ β ˆ β 1 x i m i ˆ β

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(d) (Ans) Since M i = 1 - F i , Y i = ˆ β 0 + ˆ β 1 X i M i + ˆ β 2 F i + ˆ β 3 X i F i + ˆ u i = ˆ β 0 + ˆ β 1 X i + ˆ β 2 F i + ( ˆ β 3 - ˆ β 1 ) X i F i + ˆ u i = 21 . 3 + 2 . 5 X i - 5 . 2 F i + 0 . 2 X i F i + ˆ u i The first equation corresponds to Seo’s regression equation, and the second equation represents Sung’s regression equation. By comparing the coefficients we can find Seo’s estimators from Sung’s estimators. To see the equations clearer, let’s investigate each case when F i = 1 and F i = 0. When F i = 1 (and M i = 0) , Seo’s equation becomes Y i = ˆ β 0 + ˆ β 2 + ˆ β 3 X i + ˆ u i 4

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which should be consistent with Sung’s equation when F i = 1 . Y i = (21 . 3 - 5 . 2) + 2 . 7 X i + ˆ u i When F i = 0 (and M i = 1) , Seo’s equation is Y i = ˆ β 0 + ˆ β 1 X i + ˆ u i which should be also consistent with Sung’s equation when F i = 0 . Y i = 21 . 3 + 2 . 5 X i + ˆ u i Therefore, the estimators that Seo has are ˆ β 0 = 21 . 3 ˆ β 1 = 2 . 5 ˆ β 2 = - 5 . 2 ˆ β 3 = 2 . 7 . 4. (a) (Ans) Recall the Taylor’s approximation formula is f ( x ) ; f ( x 0 ) + ( x - x 0 ) f 0 ( x 0 ). Let f ( x ) = ln(1 + x ) and x 0 = 0 . Then we have ln(1 + x ) ; f (0) + ( x - 0) f 0 (0) = x . (b) (Ans) ln(1 + 0 . 01) ; 0 . 00995 < 0 . 01 = x ln(1 + 0 . 1) ; 0 . 0953 < 0 . 1 = x ln(1 + 0 . 3) ; 0 . 2624 < 0 . 3 = x As we see the approximation works better when x is closer to zero. 5
(c) (Ans) For any concave function f ( x ) , the linearly approximated tangent line at point x lies above f ( x ) . Since ln(1 + x ) is a concave function, ln(1 + x ) x . (d) (Ans) The expected change in Y i when X i increases from 10 to 10.1 is, 2 { ln(10 . 1) - ln 10 } . Using calculator we can see that this value is about 0 . 0199 . On the othe hand, by using the approxmation ln(1 + x ) ; x, 2 { ln(10 . 1) - ln 10 } = 2 ln(1 . 01) = 2 × 0 . 01 = 0 . 02 which is a little bit larger than the actual value. 5. (a) (Ans) lim r 0 x r - 1 r = lim r 0 x r ln x 1 = ln x (b) (Ans) The Box-Cox transformed model is boiled down to the linear model or linear log regression model with particular parameter values. For instance, when β 2 = 1, it becomes a linear regression model. When β 2 0, Y i β 0 + β 1
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