# Demonstrate order and rank conditions for identification with instrumental variables

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?

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

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 $$k$$.