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Homework2Key - Economics 4261 Introduction to Econometrics...

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Unformatted text preview: Economics 4261 Introduction to Econometrics Fall 2010 HOMEWORK 2 : SIMPLE LINEAR REGRESSION Question 1 Consider two random variables X and Y , such that p X,Y q is distributed bivariate normal. Suppose X 2 . 7 , Y 3. Further, X . 36 and Y . 25. The correlation between X and Y is . 8. (a) How is X Y distributed? (b) Calculate Pr p X Y 0. (c) Calculate Pr p Y 3 . 3 | X 3 q .................................................................................................................. (a) X Y N X Y , 2 X 2 X Y 2 Y Hence, X Y N . 3 , . 36 2 2 p . 8 qp . 36 qp . 25 q . 25 2 . (b) This is a right-tailed test. Since X Y N r . 3 , . 0481 s , it follows that Z . 3 ? . 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 Y Y X p x X q , 2 Y p 1 2 q . Plugging in the numbers, we find, Y | X 3 N 2 . 7 . 8 . 25 . 36 p 3 2 . 7 q , . 25 2 p 1 . 64 q , or, Y | X 3 N r 3 . 1 6 , . 0225 s . The z-test is z 3 . 3 3 . 1 6 ? . 0225 . 889 . This is also a right-tailed test, where p . 8133, so the answer is 1 . 8133 . 187, or 18 . 7%. kwilliams@umn.edu http://www.econ.umn.edu/ ~ will3324 Page 1 of 9 Economics 4261 Introduction to Econometrics Fall 2010 Question 2 Suppose the pdf for p X,Y q is given by f p x,y q : p x y q{ 3 for 0 x 1 and 0 y 2, and f p x,y q : 0 otherwise. (a) Verify this is a valid pdf. (b) What is the marginal density with respect to Y ? (c) Find Pr p X 2 Y q . (d) What is E r X | Y 1 s ? .................................................................................................................. (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 p x,y q : x 2 y 6 xy 2 6 , for p x,y q P r , 1 s r , 2 s . Clearly F p , q 0. Also, note that F r 1 , 2 s 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 p y q X f p x,y q dx 1 x 3 y 3 dx x 2 6 xy 3 1 2 y 1 6 for y P r , 2 s . (c) To solve this question, it might be beneficial to draw a rectangle and see that Pr p X 2 Y q 1 1 2 x f p x,y q dy dx 1 1 2 x x 3 y 3 dy dx 1 xy 3 y 2 6 1 2 x dx 1 x 2 6 x 2 24 dx x 3 18 1 x 3 73 1 1 18 1 72 4 72 1 72 5 72 ....
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This note was uploaded on 02/24/2011 for the course ECON 4261 taught by Professor Staff during the Spring '08 term at Minnesota.

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Homework2Key - Economics 4261 Introduction to Econometrics...

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