I have a $T \times T$ idempotent and symmetric matrix $M_X$, defined as $M_X$ = $I_T - X(X'X)^{-1}X'$, where $X$ is a $T \times K$ matrix of $T$ observations of $K$ regressors.
For the following simple regression model:
$y_t$ = $β_1$ + $β_2x_t$ + $ε_t$
Where the vector of REAL errors (not model residuals) is $ε$, I want to prove that the expectation of the square of the product of the idempotent matrix and the error vector is equal to the expectation of a quadratic form of the two, i.e. proving that:
$E((M_Xε)'(M_Xε))$ = $E(ε'M_Xε)$
I am not sure why this is the case - I suspect it has to do with the property of idempotence, but trying it with sample values seems to suggest otherwise.