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MIT14_30s09_lec13

# MIT14_30s09_lec13 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 13 Konrad Menzel March 31, 2009 Covariance The covariance of X and Y is a measure of the strength of the relationship between the two random variables. Definition 1 For two random variables X and Y , the covariance is defined as Cov( X, Y ) = E [( X E [ X ])( Y E [ Y ])] First, by just applying the definitions, we get Property 1 Cov( X, X ) = Var( X ) Property 2 Cov( X, Y ) = Cov( Y, X ) Furthermore, we have the following result which is very useful to calculate covariances: Property 3 Cov( X, Y ) = E [ XY ] E [ X ] E [ Y ] This is a generalization of the analogous property of variances, and the proof uses exactly the same kind of argument. Let’s do an example to see how this result is useful: Example 1 Suppose X and Y have joint p.d.f. 8 xy if 0 x y 1 f XY ( x, y ) = 0 otherwise What is the covariance Cov( X, Y ) ? - Let’s calculate the components which enter according to the right- hand side of the equation in property 7: 1 1 E [ XY ] = xyf XY ( x, y ) dxdy = 8 x 2 y 2 dydx −∞ −∞ 0 x 1 1 1 8 = 8 x 2 y 2 dy dx = x 2 (1 x 3 ) dx 0 x 0 3 1 8 1 8 x 3 x 6 = ( x 2 x 5 ) dx = 3 0 3 3 6 0 8 1 1 4 = = 3 3 6 9 1
Also, by the same steps as above, 1 1 1 8 E [ Y ] = 8 x y 2 dy dx = x (1 x 3 ) dx 3 0 x 0 1 8 1 8 x 2 x 5 8 3 4 = ( x x 4 ) dx = = · = 3 0 3 2 5 0 3 10 5 and 1 1 1 8 E [ X ] = 8 x 2 ydy dx = x 2 (1 x 2 ) dx 0 x 0 2 1 1 1 8 = 4 ( x 2 x 4 ) dx = 4 = 3 5 15 0 Putting all pieces together, and applying property 7, 4 8 4 4 · 25 32 · 3 4 Cov( X, Y ) = E [ XY ] E [ X ] E [ Y ] = · = = 9 15 5 225 225 We already showed that for two independent random variables X and Y , the variance of the sum equals the sum of variances. Here’s a generalization to random variables which are not necessariy independent: Property 4 Var( X + Y ) = Var( X ) + Var( Y ) + 2Cov( X, Y ) The idea behind the proof is to apply properties 3 and 7 to get Var( X + Y ) = E [( X + Y ) 2 ] E [ X + Y ] 2 = E [ X 2 + 2 XY + Y 2 ] E [ X ] 2 2 E [ X ] E [ Y ] E [ Y ] 2 = E [ X 2 ] E [ X ] 2 + E [ Y 2 ] E [ Y ] 2 + 2 ( E [ XY ] E [ X ] E [ Y ]) = Var( X ) + Var( Y ) + 2Cov( X, Y ) Property 5 For random variables X, Y, Z , Cov( X, aY + bZ + c ) = a Cov( X, Y ) + b Cov( X, Z ) Property 6 Cov( aX + b, cY + d ) = ac Cov( X, Y ) Since by the last property, the covariance changes with the scale of X and Y , we would like to have a standardized measure which gives us the strength of the relationship between X and Y , and which is not affected by changing, say, the units of measurement of the two variables. The most frequently used measure of that kind is the correlation coeﬃcient:

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MIT14_30s09_lec13 - MIT OpenCourseWare http/ocw.mit.edu...

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