$1.$ Let us consider the multiple linear regression model y $=$ X $+$ u E$($u$)=0$

in the presence of endogeneity, that is MLR$\mathrm{.}4$ and even MLR$\mathrm{.}4$ is violated. Let Z denote a matrix of instruments, which includes the exogenous regressors from X and at least as many instruments zj to deal with the endogenous regressors in X$.$ Let the optimal instruments be defined by X$=$ Z$($Z $$Z$)1$Z $$X

$($a$)$ Show $$ IVopt $=($X$$X$)1$X$$ y $=($X$$X$)1$X$$ y $=$ $2$SLS Briefly comment what we understand by optimal instruments and how we obtain them in practice.

$($b$)$ Show that when we have exact identification, that $$ IVopt $=($X$$X$)1$X$$ y $=($Z $$X$)1$Z $$ y $=$ $$ IV Briefly comment what we understand by exact identification