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ANS-HW+1,+Summer+2008

# ANS-HW+1,+Summer+2008 - Department of Economics...

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Department of Economics Prof. Derek DeLia Econometrics 220:322:B6 Summer 2008 ANSWERS TO HOMEWORK #1 Q1 : a. Your equation should look something like this e R b I b F b D b a Q t t t t t + + + + + = 4 3 2 1 . You may use different symbols if you want. b. The variables are Q t , D t , F t , I t , and R t . c. The parameters are a, b1, b2, b3, and b4. The term e represents the error in the equation. d. We expect b1<0, b2>0, b3>0, and b4<0. The sign for a is ambiguous. Q2 : Expected revenue is 950 ) 5 . 9 ( 100 ) ( 100 = = X E Expected cost is 5 . 117 ) 400 ( 1 . 0 ) 225 ( 2 . 0 ) 100 ( 2 . 0 ) 25 ( 5 . 0 ) ( 2 = + + + = X E Note that ( 29 2 2 ) ( ) ( X E X E Expected profit is 950-117.5 = 832.5, or \$832.50. Q3 : ) ( 2 ε E can be interpreted as the variance of ε . To see why, notice that ( 29 [ ] ) ( ) ( ) ( 2 2 ε ε ε ε E E E V = - = . Q4 : a. The covariance is 2.5 and the correlation coefficient is 0.84. b. The covariance is 25 and the correlation coefficient is 0.84. Q5 : [ ] [ ] )) ( ( )) ( ( )) ( ))( ( ( ) , cov( Y E Y k X E X E kY E kY X E X E kY X - - = - - = [ ] ) , cov( )) ( ))( ( ( Y X k Y E Y X E X kE = - - = k k Y X r k Y X k Y V X V k Y X k kY V X V kY X kY X r Y X ) , ( ) , cov( ) ( ) ( ) , cov( ) ( ) ( ) , cov( ) , ( 2 = = = = σ σ In Q3 part b, k=10. Page 1 of 6

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Q6 : a. ) ( 5 ) , cov( X V Y X = , 1 ) , ( = Y X r b. ) ( 5 ) , cov( X V Y X - = , 1 ) , ( - = Y X r c. This part generalizes parts a and b [ ] [ ] ) ( )) ( ( )) ( ( )) ( ))( ( ( ) , cov( X bV X E X b X E X E Y E Y X E X E Y X = - - = - - = b b X V b X bV X V b X V X bV Y V X V Y X Y X r = = = = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , cov( ) , ( 2 Since Y is linear in each case, the linear correlation between X and Y is always perfect. If
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