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140A Key Topics

140A Key Topics - Key Topics Introduction to Econometrics...

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Key Topics Introduction to Econometrics Mathematical Concepts Understand and verify the following properties ° P J j =1 x j p j = x 1 p 1 + x 2 p 2 + ± ± ± + x J p J ° P J j =1 ( x j + z j ) p j = P J j =1 x j p j + P J j =1 z j p j ° ° P J j =1 x j p j ± 2 = P J j =1 P J k =1 x j x k p j p k = P J j =1 x 2 j p 2 j + P J j =1 P J k =1 j 6 = k x j x k p j p k You should know this from prior coursework, but it is covered in the Random Variables lecture Probability Concepts Understand that for discrete random variables X , which takes J distinct values, and U , which takes K distinct values, ° E ( X ) = P J j =1 x j ± P ( X = x j ) := ° X ° V ar ( X ) := E ( X ² ° X ) 2 = P J j =1 ( x j ² ° X ) 2 ± P ( X = x j ) ° Cov ( X; U ) := E [( X ² ° X ) ( U ² ° U )] = P J j =1 P K k =1 ( x j ² ° X ) ( u k ² ° U ) ± P ( X = x j ; U = u k ) ° E ( U j X = x j ) = P K k =1 u k ± P ( U = u k j X = x j ) ° V ar ( U j X = x j ) := P K k =1 [ u k ² E ( U j X = x j )] 2 ± P ( U = u k j X = x j ) You should know this from prior coursework, but it is covered in the Random Variables lecture Understand and verify the following implications ° E ( U j X ) = 0 means E ( U j X = x j ) = 0 for all ( x 1 ; : : : ; x J )
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° E ( U j X ) = 0 ) ² E ( U ) = 0 Cov ( X; U ) = 0 ° E ( U 2 j X ) = ± 2 ) E ( U 2 ) = ± 2 This is covered in the Random Variables lecture Statistical Concepts
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