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For an equation in a simultaneous system to be identified two conditions must hold: i) the order condition, and ii) the rank condition.

b_IV=(Z'X)^(-1) Z'Y

  • How to demonstrate in matrix form that both order and rank conditions need to hold for identification with instrumental variables.

I know that we need det|Z'X|≠0 for (Z'X)-1 to exist. I think it has something to do with (Z'X)-1. The Z'X matrix has to be square and of full rank?

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  • $\begingroup$ I advise you to write math in LaTeX. $\endgroup$ Commented Nov 19, 2022 at 22:03

1 Answer 1

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If you are writing $(Z'X)^{-1}X'Y$, you must be considering a case in which there is one instrument and one endogenous regressor. Thus order holds ("order" meaning there are at least as many instruments as endogeneous regressors). "Rank" would mean $Z'X$ is invertible.

If you are considering a general case with more instruments than endogenous regressors, we need $Z'X$ to be full rank. If $X$ is $N\times K$ and $Z$ is $N \times \ell$ then $Z'X$ is $\ell \times k$. Order would mean $\ell \ge k$. Rank would mean the rank of $Z'X$ is $k$.

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