How can I prove with $Var(\hat{u})_t= E(\hat{u}^2_t)= (1- h_t)\sigma^2_0$ that MM estimator $\hat{\sigma}^2 \equiv \frac{1}{n} \sum_{t=1}^n \hat{u}_t^2$ is consistent? I can may assume that a LLN applies to the average in that equation.
Notice that $1- h_t$ is the typical diagonal element of $M_x$
(Ex 3.13 Davidson & Mackinnon )