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bubai@matt.ksu.ksu.edu (Priyabroto) writes:

>Hi,

>I had a small question:

>Let Y = Xb + u   (in matrix notation)

>such that Cov(X,u) is not equal to zero.

>Now, we use Z as an instrumental variable where:

>Cov(X,Z) is not equal to zero but,
>Cov(Z,u) is equal to zero.

>Then, WHY is it that b hat(instr. var) = (Z'X)inverse * Z'y?

I don't understand the question.  This is the _definition_ of the IV
estimator (assuming that Z and X have the same dimensions).  The
statistical motivation for the estimator comes from the fact that

Inverse(Z'X)*Z'y = b + Inverse(Z'X/n)*(Z'u/n)

and the last term converges to zero in large samples due to the law of
large numbers and the assumption that E(Z'u) = 0.

>As we know b hat = (X'X)inverse * X'y.

This is the definition of the least squares estimator, which is not
the same as the IV estimator, and which does not converge to b in
large samples when X and u are correlated.

Perhaps you are asking how to show that the IV estimator and the least
squares estimator are the same thing.  This cannot be done in general.
The least squares estimator can be thought of as a special case of the
IV estimator obtained by setting Z = X.

--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209