<p>If you are writing <span class="math-container">$(Z'X)^{-1}X'Y$</span>, you must be considering a case in which there is one instrument and one endogenous regressor. Thus order holds (&quot;order&quot; meaning there are at least as many instruments as endogeneous regressors). &quot;Rank&quot; would mean <span class="math-container">$Z'X$</span> is invertible.</p>
<p>If you are considering a general case with more instruments than endogenous regressors, we need <span class="math-container">$Z'X$</span> to be full rank. If <span class="math-container">$X$</span> is <span class="math-container">$N\times K$</span> and <span class="math-container">$Z$</span> is <span class="math-container">$N \times \ell$</span> then <span class="math-container">$Z'X$</span> is <span class="math-container">$\ell \times k$</span>. Order would mean <span class="math-container">$\ell \ge k$</span>. Rank would mean the rank of <span class="math-container">$Z'X$</span> is <span class="math-container">$k$</span>.</p>

CC BY-SA 4.0
2022-11-20T21:17:13.577Z
2022-11-20T21:17:13.577Z
Michael Gmeiner
<p>For an equation in a simultaneous system to be identified two conditions must hold: i) the order condition, and ii) the rank condition.</p>
<p>b_IV=(Z'X)^(-1) Z'Y</p>
<ul>
<li>How to demonstrate in matrix form that both order and rank conditions
need to hold for identification with instrumental variables.</li>
</ul>
<p>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?</p>

CC BY-SA 4.0
2022-11-19T22:03:13.653Z
I advise you to write math in LaTeX.
Nicolas Torres
<econometrics><mathematical-economics><statistics>
Demonstrate order and rank conditions for identification with instrumental variables
Dr. Z
# Demonstrate order and rank conditions for identification with instrumental variables

(Asked by: username_0 on 2022-11-19)

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?

**Tags:** econometrics, mathematical-economics, statistics
> **Comments:**
> **username_1**: I advise you to write math in LaTeX.  

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## 1 answers

### Answer by username_2 on 2022-11-20. Score: 1

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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