Consider the restricted and unrestricted linear models, show that $\beta^*-\hat\beta=(X^\prime M_ZX)^{-1}X^\prime M_Z M_XY$.

Question

Consider the restricted and unrestricted linear models:

$Y=X\beta+Z\gamma+u$, $u\overset{iid}\sim N(0,\sigma^2I)$ $\quad$ $(1)$

$Y=X\beta+u$, $u\overset{iid}\sim N(0,\sigma^2I)$ $\quad$ $(2)$

Denote the estimator from $(1)$ and $(2)$ as $\beta^*$ and $\hat\beta$ respectively, show that

$\beta^*-\hat\beta=(X^\prime M_ZX)^{-1}X^\prime M_Z M_XY$,

where $M_X=I-X(X^\prime X)^{-1}X^\prime$,$M_Z=I-Z(Z^\prime Z)^{-1}Z^\prime$.

It is well known that $\hat\beta=(X^\prime X)^{-1}X^\prime Y$. But how to estimate $\beta^*$?