Lecture09-2010 - Moments of More that One Random Variable...

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Moments of More that One Random Variable Charles B. Moss July 14, 2010 I. Covariance and Correlation A. Defnition 4.3.1 The covariance between two random variables X and Y can be defned as Cov ( X,Y )=E[(X E[X])(Y E[Y])] = E [XY XE [Y] E[x]Y+E[X][Y]] =E[XY] E[X]E[Y] E [X] E [Y] + E [X] E [Y] (1) 1. Note that this is simply a generalization oF the standard vari- ance Formulation. Specifcally, letting Y X yields Cov ( XX ) = E [XX] E[X]E[X] =E[X 2 ] (E [X]) 2 (2) 2. ±rom a sample perspective, we have V ( X )= 1 N N i =1 x 2 i ¯ x 2 Cov ( 1 N N i =1 x i y i ¯ x ¯ y (3) 3. Together the variance and covariance matrices are typically written as a variance matrix 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture IX Fall 2010 Σ= " V ( X )C o v ( X,Y ) Cov ( Y,X ) V ( Y ) # = " σ xx σ xy σ yx σ yy # (4) Note that Cov ( )= σ xy = σ =Cov( ). 4. Substituting the sample measures into the variance matrix yields S = " s xx s xy s s # = " 1 N N i =1 x i x i ¯ x ¯ x 1 N N i =1 x i y i ¯ x ¯ y 1 N N i =1 y i x i ¯ y ¯ x 1 N N i =1 y i y i ¯ y ¯ y # = 1 N " i =1 x i x i N i =1 x i y i N i =1 y i x i N i =1 y i y i # " ¯ x ¯ x ¯ x ¯ y ¯ y ¯ x ¯ y ¯ y # (5) The sample covariance matrix can then be written as S = 1 N " x 1 ··· x N y 1 y N # " ¯ x ¯ y # h ¯ x ¯ y i . (6) 4. In terms of the theoretical distribution, the variance matrix can be written as " R −∞ R −∞ ( x μ x ) 2 f ( x, y ) dx dy R −∞ R −∞ ( x μ x )( y μ y ) f ( x, y ) dx, dy R −∞ R ( x μ x y μ y ) f ( x, y ) dx dy R −∞ R −∞ ( y μ y ) 2 dx dy # (7) 5. Example 4.3.2. Compute the covariance for the data pre-
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Lecture09-2010 - Moments of More that One Random Variable...

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