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Following the rationale from Econometric Methods with Applications in Business and Economics by Heij et al., the instrumental variables estimator $b_{IV}$ for the linear regression model

$y = X\beta + \varepsilon$

with instrumental variables $Z$ (with more variables than $\#\beta$) is given by

$b_{IV} = (X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'y$.

The book proposes calculating this $b_{IV}$ using two stage least squares:

  1. Regress each column of $X$ on $Z$, resulting in $\hat{X} = Z(Z'Z)^{-1}Z'X$
  2. Regress $y$ on $\hat{X}$, with parameter estimates $b_{IV} = (\hat{X}'\hat{X})^{-1}\hat{X}'y$

Am I wrong by thinking that calculating $b_{IV} = (X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'y$ directly wil yield the same results as two stage least squares, since

$b_{IV} = (X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'y = (\hat{X}'\hat{X})^{-1}\hat{X}'y$?

And if it will yield the same results, why does the book give any attention to two stage least squares, and doesn't just leave it at something like

"you can calculate the instrumental variables estimator using the equation $b_{IV} = (X'Z(Z'Z)^{-1}Z'X)^{-1}X'Z(Z'Z)^{-1}Z'y$."

?

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I haven't gone through your math against a good text, and my 2SLS theory is a little stale.

But I think you are right. Obviously, 2SLS involves doing a stage one with the instrumental variables to develop a new (transformed) set of variables in the form of $\hat{X}$ from the instruments. And step two is simply OLS, as I recall, with that.

Provided the construction of the $\hat{X}$ matrix can be done as simply algebraic manipulations, then it is just plugging this into the second stage formula.

If that is correct - and I don't see why not - the explanation may be to help remind the motivation of doing 2SLS. The goal of that first stage is to get you a set of independent variables that have the right qualities - e.g., uncorrelated with the error term or filling in for a missing effect.

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    $\begingroup$ Thank you for your answer! I think you are right. The lecture slides that accompany this book simply compare the $b_{IV}$ above (which is derived from the GMM framework) with the 2SLS estimator, and show that they are equal. I think the book also tries to make this point, but does not conclude the argumentation. $\endgroup$ – pabk Oct 23 '17 at 8:42

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