hw1-fall09 - a = 2 4 a 1 a 2 a 3 3 5 , A = 2 4 a 11 a 12 a...

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Homework 1 Econometrics: ECO 721 Fall 2009 Due Thursday, September 17 You must turn in this assignment before class begins on the due date. In this and all future assignments, you are to turn in neatly written answers to all questions. Photocopies are not acceptable. Loose pages will not be accepted. You must use a staple or strong clip, i.e., no 1. Let A = 1 3 2 1 ± , B = 1 3 2 4 ± : Verify that i. ( AB ) 0 = B 0 A 0 ii. AB 6 = BA 2. Let X = 2 6 6 6 4 1 x 1 z 1 1 x 2 z 2 . . . . . . . . . 1 x N z N 3 7 7 7 5 : Show that i. X 0 i =N = 2 4 1 x z 3 5 , where i is a N ± 1 vector of ones, and x and z denote the means of x and z respectively. ii. X 0 X=N = 2 4 1 x z x 1 N P x 2 i 1 N P x i z i z 1 N P x i z i 1 N P z 2 i 3 5 : 3. Expand the following matrix products where all the matrices are square and nonsingular. i. X = [ AB + ( EF ) 1 ][( CD ) 0 + GH ] 0 ii. X = [( A + B ) 0 + ( EF ) 1 ][( CD ) 1 + GH ] 4. Let X = 2 6 6 6 4 1 x 1 z 1 1 x 2 z 2 . . . . . . . . . 1 x N z N 3 7 7 7 5 : Show that i. X 0 i =N = 2 4 1 x z 3 5 , where i is a N ± 1 vector of ones, and x and z denote the means of x and z respectively. ii. X 0 X = 2 4 N P x i P z i P x i P x 2 i P x i z i P z i P x i z i P z 2 i 3 5 : 1
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5. Let
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Unformatted text preview: a = 2 4 a 1 a 2 a 3 3 5 , A = 2 4 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 3 5 and x = 2 4 x 1 x 2 x 3 3 5 . Show that i. @ ( a x ) =@x = a ii. @ ( Ax ) =@x = A 6. A matrix A is idempotent if AA = A: Show that I & X ( X X ) & 1 X , where X n k , is idempotent. 7. Use the de&nitions V ( X ) = E ( X & E ( X )) 2 and Cov ( X;Y ) = E [( X & E ( X ))( Y & E ( Y ))] and properties of the expectation operator, along with the assumptions that E ( X ) = 0 ; E ( y ) = 0 , to demonstrate that i. V ar ( aX + bY ) = a 2 V ( X ) + 2 abCov ( X;Y ) + b 2 V ( Y ) : ii. Cov ( aX + bY;cX + dY ) = acV ( X ) + ( ad + bc ) Cov ( X;Y ) + bdV ( Y ) . 2...
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hw1-fall09 - a = 2 4 a 1 a 2 a 3 3 5 , A = 2 4 a 11 a 12 a...

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