Does seasonal differencing in SARIMA model take care of additive/ multiplicative seasonality?

Question

I am exploring the use of ARIMA and Seasonal ARIMA models (SARIMA). In some of my datasets, I can clearly observe seasonality in the ACF and PACF plots (the lines at seasonal lags clearly cutting the confidence interval region).

In order to make such data stationary, and to account for seasonality in my model, is seasonal differencing enough? For example, would a first order seasonal differencing for monthly data (as shown below) work?

ARIMA (0, 0, 0) × (0, 1, 0)12

Or should I try a more traditional method of finding the seasonal index by using a smoother (MA/Exponential Smoothing) and then using an additive or multiplicative method to calculate seasonal index? How would these methods affect my model?

Answer 1

Is seasonal differentiating enough?

In short, "it depends". I mean, there might be a need for more differentiating and not only seasonal. However, it is entirely possible that your time series (TS) need only one seasonal differentiating.

There is such a thing as a SARIMA(0,1,0). Consider the full ARMA (p,d,q) x (P,D,Q)[s]:

$\Phi_{P}\left(B^{s}\right) \phi_p(B) \nabla_{s}^{D} \nabla^{d} x_{t}=\Theta_{Q}\left(B^{s}\right) \theta_q(B) w_{t}$

where we define the following operators (which are like functions, that we treat as variables until we apply them in the $x_t$),

$B^{n}(x_t) = x_t - x_{t-n}$

$\Delta^d(x_t) = (1-B)^dx_t$

$\nabla_{s}^{D} x_{t}=\left(1-B^{s}\right)^{D} x_{t}$

$\phi_p(B)=1-\phi_{1} B-\phi_{2} B^{2}-\cdots-\phi_{p} B^{p}$

$\theta_q(B)=1+\theta_{1} B+\theta_{2} B^{2}+\cdots+\theta_{q} B^{q}$

$\Phi_{P}\left(B^{s}\right)=1-\Phi_{1} B^{s}-\Phi_{2} B^{2 s}-\cdots-\Phi_{P} B^{P s}$

$\Theta_{Q}\left(B^{s}\right)=1+\Theta_{1} B^{s}+\Theta_{2} B^{2 s}+\cdots+\Theta_{Q} B^{Q s}$

Setting only D = 1, and the rest of the parameters as 0, we have:

$\Delta^{1}_{s}x_t = w_t$

$x_t - x_{t-s} = w_t$

That means, that you only need to estimate the parameters from the white noise $w_t$ to know the properties of your TS.

PS: I'm, of course assuming that the expected value or mean of $x_t$ is 0. Otherwise, for the whole process change: $x_t$ to $x_t - \mu$.

How will changing to Exponential Smoothing impact my model?

A lot. It is a different model. The ETS family of models, which MA and Exp. Smooth. are one of them, have a different mindset to dealing with seasonality. In the SARIMA family we differentiate the series and add the corresponding lag parameters basically; in the ETS family they change the model structure completely.

My recommendation to you would be to compare the two approaches through a metric fit for both models, such as AIC, BIC, accuracy, etc.