Is log of ETS(MNM) a ETS(ANA) model?

Question

Given two time series models $ETS(MNM)$ and $ETS(ANA)$. If we replace $y_t $ by $\ln y_t $ in $ETS(MNM)$, is it true the model will become $ETS(ANA)$?

Answer 1

They are not exactly the same, however the log transformation makes somewhat $ANA$ model $MNM$. Let's write the equations for the both models.

$$ ETS(MNM) = \begin{cases} y_t = l_{t-1}\cdot s_{t-12}\cdot (1+u_t) \\ l_t = l_{t-1}\cdot (1+\alpha u_t)\\ s_t = s_{t-12}\cdot (1+\gamma u_t) \end{cases} $$

$$ ETS(ANA) = \begin{cases} y_t = l_{t-1} + s_{t-12} + u_t \\ l_t = l_{t-1}+\alpha u_t\\ s_t = s_{t-12} + \gamma u_t \end{cases} $$ Remember that $\ln(ab) = \ln a + \ln b$. Hence, $$ \ln y_t = \ln(l_{t-1}) + \ln(s_{t-12}) + \ln(1 + u_t)$$ Same for other equations. As we can see, the models are not exactly the same, however it became additive ($ANA$).