# Seasonal ARIMA with Exogenous varibales [closed]

Can somebody help me out with Seasonal ARIMA equation for model (1,0,1) (1,0,1)

## closed as unclear what you're asking by Andy, Xi'an, Glen_b♦, Peter Flom♦Dec 26 '14 at 11:53

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• It's not 100% clear what help you're after, but it seems possible that your question may already have been answered, since questions about models for seasonal ARMA models have been asked before. – Glen_b Dec 26 '14 at 8:54
• I am replicating this particular model from SPSS in excel.so i need the final formula/equation for Y(t). which is of the form Y(t)= c + u(t)[(1+Θ1Lfreq)(1+θ1L)]........ – USer123 Dec 26 '14 at 9:05
• see, for example, this answer – Glen_b Dec 26 '14 at 9:22

## 1 Answer

ARIMA(1,0,1) would be

$$(1-\phi_1 L)x_t=(1+\theta_1 L)\varepsilon_t.$$

Then add the seasonal part to obtain SARIMA(1,0,1)(1,0,1)

$$(1-\Phi_1 L^{\text{freq}})(1-\phi_1 L)x_t=(1+\Theta_1 L^{\text{freq}})(1+\theta_1 L)\varepsilon_t$$

where $\text{freq}$ is the frequency of the time series (e.g. 4 for quarterly data, 12 for monthly data).

Also look here for the general case and here as well.

• What is εt, how do I get it? – USer123 Jan 8 '15 at 5:08
• $\varepsilon_t$ is an unobserved error term. In different contexts $\varepsilon_t$'s may be called disturbances, innovations or shocks. Since you do not observe them, you can only derive them from the data using the formula above. For that you need to have the estimated coefficient values first. – Richard Hardy Jan 8 '15 at 8:21
• @Taylor, I rolled back your edit because I believe it introduced a mistake. Check the definitions of period and frequency. – Richard Hardy Oct 19 '17 at 5:21
• @Taylor, hmm, interesting. This is totally opposite to what I think is standard in the time series literature and beyond (like physics or casual use). I have not encountered such use of period vs. frequency before, so I was surprised. I think it is safe to leave it as is as it is pretty common in my experience (and most of my experience is from the statistical time series analysis, applied and theoretical). Thanks for the heads-up, though. – Richard Hardy Oct 19 '17 at 14:30
• @RichardHardy yeah I'm starting to notice it your way more now too. I guess if you thought of your data as number of observations per season, that's the right units for frequency, but if you thought of your data on the faster time frame, it would be the number of time points per cycle, which is period. Weird...never noticed this. – Taylor Oct 19 '17 at 20:46