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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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 Spring '09
 Partha
 Econometrics

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