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Homework2Key - Economics 4261 Introduction to Econometrics...

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Economics 4261 Introduction to Econometrics Fall 2010 HOMEWORK 2 : SIMPLE LINEAR REGRESSION Question 1 Consider two random variables X and Y , such that X, Y is distributed bivariate normal. Suppose μ X 2 . 7 , μ Y 3. Further, σ X . 36 and σ Y . 25. The correlation between X and Y is ρ . 8. (a) How is X Y distributed? (b) Calculate Pr X Y 0. (c) Calculate Pr Y 3 . 3 X 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) X Y N μ X μ Y , σ 2 X 2 ρσ X σ Y σ 2 Y Hence, X Y N . 3 , . 36 2 2 . 8 . 36 . 25 . 25 2 . (b) This is a right-tailed test. Since X Y N . 3 , 0 . 0481 , it follows that Z . 3 0 . 0481 1 . 368 . The p-value for the one-sided test is 0 . 0857. Thus, the probability is 8 . 57%. (c) First we need Y X x , which is given by Y X x N μ Y ρ σ Y σ X x μ X , σ 2 Y 1 ρ 2 . Plugging in the numbers, we find, Y X 3 N 2 . 7 . 8 . 25 . 36 3 2 . 7 , . 25 2 1 . 64 , or, Y X 3 N 3 . 1 ¯ 6 , 0 . 0225 . The z -test is z 3 . 3 3 . 1 ¯ 6 0 . 0225 . 889 . This is also a right-tailed test, where p 0 . 8133, so the answer is 1 . 8133 0 . 187, or 18 . 7%. [email protected] http://www.econ.umn.edu/ ~ will3324 Page 1 of 9
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Economics 4261 Introduction to Econometrics Fall 2010 Question 2 Suppose the pdf for X, Y is given by f x, y : x y 3 for 0 x 1 and 0 y 2, and f x, y : 0 otherwise. (a) Verify this is a valid pdf. (b) What is the marginal density with respect to Y ? (c) Find Pr X 2 Y . (d) What is E X Y 1 ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) We just need to verify that the pdf integrates to unity. The cdf is given by X Y x 3 y 3 dy dx. Thus our proposed candidate is, F x, y : x 2 y 6 xy 2 6 , for x, y 0 , 1 0 , 2 . Clearly F 0 , 0 0. Also, note that F 1 , 2 1 3 2 3 1. We are good to go. (b) To obtain the marginal distribution with respect to Y , we integrate over X . g y X f x, y dx 1 0 x 3 y 3 dx x 2 6 xy 3 1 0 2 y 1 6 for y 0 , 2 . (c) To solve this question, it might be beneficial to draw a rectangle and see that Pr X 2 Y 1 0 1 2 x 0 f x, y dy dx 1 0 1 2 x 0 x 3 y 3 dy dx 1 0 xy 3 y 2 6 1 2 x 0 dx 1 0 x 2 6 x 2 24 dx x 3 18 1 0 x 3 73 1 0 1 18 1 72 4 72 1 72 5 72 . [email protected] http://www.econ.umn.edu/ ~ will3324 Page 2 of 9
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Economics 4261 Introduction to Econometrics Fall 2010 (d) To calculate E X Y 1 , we use the following identity: E X Y y 1 0 x h x y dx, where h x y : f x, y g y . The marginal distribution, g y is given in (b) by g y 2 y 1 6 . Thus, E X Y y 1 0 x x y 3 2 y 1 6 dx, where y 1. Hence, E X Y y 1 0 x x 1 3 2 1 6 dx 1 0 x 6 x 1 9 dx 2 3 1 0 x x 1 dx 2 3 1 0 x 2 x dx 2 3 1 3 x 3 1 2 x 2 1 0 2 3 1 3 1 2 2 3 2 6 3 6 2 3 5 6 10 18 5 9 [email protected] http://www.econ.umn.edu/ ~ will3324 Page 3 of 9
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Economics 4261 Introduction to Econometrics Fall 2010 Question 3 We want to examine the relationship between age of a vehicle in years (x) and it’s selling price (y). We propose that
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