In the Multiplicative Error Model (MEM) specification by Engle in "New Frontiers for ARCH models", he wrote that MEM specified an error that is multipled times the mean and the specification is:
$$ x_t = \mu_t \epsilon_t $$
$$ \epsilon_t | \Im_{t-1} \sim D(1,\phi_t^2) $$
I am not sure I understand what epsilon is supposed to be, is it a normal of mean 1, but then what is the variance supposed to be ?
Engle mentions that the MEM is good to model the process which have to be positive. From my understanding this is only true if $\epsilon_t$ is positive, so unless $D$ is a log-normal distribution or another distribution guaranteeing positivity, you cannot guarantee that $x_t$ itself will be positive, is my understanding correct ?