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Representation similar to $Z'X(X'X)^{-1}X'Z$ frequently appear to e.g. 2SLS.

I think that $Z'X(X'X)^{-1}X'Z = Z'XX^{-1}X'^{-1}X'Z = Z'(XX^{-1})(X'^{-1}X')Z = Z'Z$. So why it seems that in the context of e.g. 2SLS, the representation can not be simplified? What is the mistake I made?

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    $\begingroup$ How do you make sense of "$X^{-1}$" when $X$ is not a square matrix? $\endgroup$
    – whuber ♦
    Dec 5, 2018 at 14:20

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Because $X$ is not a square matrix; it has more rows than columns. Typically very very many more.

The matrix $X'X$ is invertible if the columns of $X$ are linearly independent, but the rows of $X$ cannot be linearly independent because there are too many of them in a space of small dimension.

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