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Pg. 51 of Green Book (correlation among 3 variables)

sak

Joined
8/16/09
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18
The question is: 3 random variables x, y, z. Correlation between x and y is 0.8 and between x and z is 0.8. Find maximum and minimum correlation between y and z.

I don't think we have sufficient information for solving this. All correlation does is provides us information about linear relation between two variables and has no information regarding the amount of variation of one variable when the other changes i.e. slope.

So, the solution appears incorrect to me. What am I missing here?

Thanks
 
That question has already been asked in this forum. I don't remember the answer, however, it is clear that there must be some realation. For example, if the correlations were 1 instead of 0.8, the correlation between y and z should also be 1.
 
My problem is not how to solve it but the solution in the Green Book using cos(theta) = 0.8 seems incorrect to me as I do not agree that angle between two vectors can be decided by correlation between them as has been assumed in the solution.
 
Think of each variable as a vector. If you centre and standarize each vector (to each entry subtract the mean and divide by stadnard deviation) Then the product of those two vectors (which would be the formula for correlation) would be the cos of the angle.
 
Remember that the a correlation matrix is positive semi-definite, so its determinant must be larger than or equal to zero. From this, you can easily work out the appropriate range for the missing correlation.
 
rothmann said:
Remember that the a correlation matrix is positive semi-definite, so its determinant must be larger than or equal to zero. From this, you can easily work out the appropriate range for the missing correlation.
Click to expand...
I don't think this is a sufficient condition in general, all principal minors need to be be nonnegative. (In this case it may be sufficient).
 
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