I am looking at the derivation for instrument variables and there is a particular step which I am not sure how it is achieved.
$$ \frac{(Z^TZ)^{-1}Z^TY}{(Z^TZ)^{-1}Z^TX} $$
is simplified to
$$ (Z^TX)^{-1}Z^TY $$
when Z and X are both n x a and Y is n x 1
How do you get from first to second?
This expression $$ \frac{(Z^TZ)^{-1}Z^TY}{(Z^TZ)^{-1}Z^TX} $$ is well defined only if $Z$ and $X$ are $n \times 1 $ vectors, thus $Z'Z = \sum z_i^2$ and $Z^TX = \sum z_i x_i$, hence the derivation is straightforward $$ \frac{(Z^TZ)^{-1}Z^TY}{(Z^TZ)^{-1}Z^TX} = \frac{\sum z_i ^ 2 \sum z_i y_i }{ \sum z_i ^ 2 \sum z_i x_i} = ( \sum z_i x_i ) ^ {-1} \sum z_i y_i = ( Z^TX )^{-1} Z^TY . $$
Where $(Z^TZ)^{-1}Z^TY$ is non scalar, then the expression should be $$ ( (Z^TZ)^{-1}Z^TX ) ^{-1} (Z^TZ)^{-1}Z^TY = (Z^TX)^{-1} Z^T Z (Z^T Z)^{-1} Z^TY = ( Z^TX ) ^{-1}Z^TY $$