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IsildursHeir
11-10-2007, 07:35 PM
I am new to this forum and don't know if creating a new thread for help in a problem is an allowed scenario. If it is not, request the mods to please direct me to the correct way of the same. Apologies in advance :)

I have struggled a lot with this but cant seem to find a solution. Will really appreciate if anybody can help.

Suppose that y=a+2x+z+u, where E(u|x,z) = 0. Let q be a random variable with the following properties:
i) Cov(q,x) = 0.7,
ii) Cov(q,z) = 0.5,
iii) Cov(q,u) = 0.5
We run a regression of y on 1 and x but use q as an instrument for x. let b(hat) be the coefficient of x in this IV regression. What is the probability limit of b(hat)?

Golden Rule
11-10-2007, 09:09 PM
Note that
xhat = [Cov(x,q)/Var(q)] * q

and that
bhat = Cov(xhat,y)/Var(xhat)

Then just remember that Cov(X+Y,Z) = Cov(X,Z)+Cov(Y,Z) and that
Cov(aX,Y) = a*Cov(X,Y) and Cov(a,X) = 0 for constant a.
Just substitute for y and xhat into the bhat expression, keep applying those formulas again and again, simplify, and you're done.

IsildursHeir
11-19-2007, 07:09 PM
Thanks for the reply. It was helpful:)