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

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?

• See this, for example. Mar 31, 2017 at 5:26
• Possible duplicate of How do I write a mathematical equation for ARIMA (0,2,1) x (0,0,1) period 12 Mar 31, 2017 at 11:18
• I don't see how this question is a duplicate of the suggested thread at all. This question asks if seasonal differencing will make such data stationary. I don't even see that mentioned in the linked thread, much less answered. I'm voting to leave open. Mar 31, 2017 at 17:24

# 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.

• Your answer assumes that readers know the complete framework behind ARMA, including the precise meaning of each letter-symbol in the equation. You need to define each (Greek) letter with a separate one-sentence definition. Your answer is much too complicated and targets merely 'insiders' (already experts) in this field. Please improve the clarity of your answer and appeal to the general reader of this forum - thanks! Jun 24, 2021 at 22:00
• Thank you very much, @MatchMakerEE! I hope that my edit of the answer helped in clarifying. If not, I can try it again! Jun 26, 2021 at 23:23