Lecture09-2010

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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-
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online