8.10 ARIMA vs ETS

It is a common myth that ARIMA models are more general than exponential smoothing. While linear exponential smoothing models are all special cases of ARIMA models, the non-linear exponential smoothing models have no equivalent ARIMA counterparts. There are also many ARIMA models that have no exponential smoothing counterparts. In particular, every ETS model is non-stationary, while ARIMA models can be stationary.

The ETS models with seasonality or non-damped trend or both have two unit roots (i.e., they need two levels of differencing to make them stationary). All other ETS models have one unit root (they need one level of differencing to make them stationary).

The following table gives some equivalence relationships for the two classes of models.

ETS model ARIMA model Parameters
ETS(A,N,N) ARIMA(0,1,1)  $\quad~~\,\theta_1 = \alpha-1$
ETS(A,A,N) ARIMA(0,2,2) \begin{align*} \theta_1 &= \alpha+\beta-2\phantom{\phi}\\ \theta_2 &= 1-\alpha \end{align*}
ETS(A,Ad,N) ARIMA(1,1,2) \begin{align*} \phi_1&=\phi\\ \theta_1 &= \alpha+\phi\beta-1-\phi\\ \theta_2 &= (1-\alpha)\phi \end{align*}
ETS(A,N,A) ARIMA(0,0,m)(0,1,0)m
ETS(A,A,A) ARIMA(0,1,m+1)(0,1,0)m
ETS(A,Ad,A) ARIMA(1,0,m+1)(0,1,0)m

For the seasonal models, there are a large number of restrictions on the ARIMA parameters.