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?

  • 6
    $\begingroup$ How do you make sense of "$X^{-1}$" when $X$ is not a square matrix? $\endgroup$
    – whuber
    Commented Dec 5, 2018 at 14:20

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.