*This preview shows
pages
1–4. Sign up
to
view the full content.*

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **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 . 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. Lets 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 ) ? - Lets 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 x 1 1 1 8 = 8 x 2 y 2 dy dx = x 2 (1 x 3 ) dx x 3 1 8 1 8 x 3 x 6 = ( x 2 x 5 ) dx = 3 3 3 6 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 x 1 8 1 8 x 2 x 5 8 3 4 = ( x x 4 ) dx = = = 3 3 2 5 3 10 5 and 1 1 1 8 E [ X ] = 8 x 2 ydy dx = x 2 (1 x 2 ) dx x 2 1 1 1 8 = 4 ( x 2 x 4 ) dx = 4 = 3 5 15 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. Heres 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 not affected by changing, say, the units of measurement of the two variables....

View
Full
Document