Note I am considering the standard $t$ distribution $(\mu=0,\sigma=1)$
The method of moments for $\nu>2$ is derived in this question
My question is, if the true (population) value of $\nu$ is $2$, and we don't know this, then what is the MOM estimator estimating?
My confusion is that in this case the MOM estimator uses higher order moments that don't exist, so does this mean that if $\nu=2$ the MOM estimator will never get this right?
More generally, if we use a MOM estimator based on moments that only exist for certain parameter values, what happens if the true parameter values are not in that range?