Interpretation of the ARIMA coefficients in a time series

Question

I am trying to understand the coefficients retrieved from running auto.arima in R on my monthly time series of the annual change in House prices. When doing so, I obtain the following outcome:

Series: AC.HousePrices 
ARIMA(1,1,1)(0,0,1)[12] with drift         

Coefficients:
         ar1      ma1     sma1   drift
      0.3243  -0.6592  -0.7892  -6e-04
s.e.  0.1733   0.1333   0.1161   4e-04

sigma^2 estimated as 0.0008257:  log likelihood=275.22
AIC=-540.44   AICc=-539.96   BIC=-526.07

To be honest I do not understand why I have two sets of parameters (p,d,q) and (P,D,Q)? The first set (1,1,1) seems to indicate that the series is first-order autoregressive model, nonstationary and with a simple exponential smoothing with drift? What are the second set of values (0,0,1)[12], is it telling me that my series looks yearly seasonal [12]?

Check out existing threads on ARIMA and SARIMA models. You will find a lot of information there. Check also Hyndman & Athanasopoulos "Forecasting: principles and practice" [Section 8.9 Seasonal ARIMA models](https://www.otexts.org/fpp/8/9). — Richard Hardy, Jul 12 '16 at 17:14
Several related threads: [ARIMA SARIMA model mathematical formula](http://stats.stackexchange.com/questions/90476/arima-sarima-model-mathematical-formula), [How do I write a mathematical equation for ARIMA (0,2,1) x (0,0,1) period 12](http://stats.stackexchange.com/questions/61510/how-do-i-write-a-mathematical-equation-for-arima-0-2-1-x-0-0-1-period-12?rq=1), [How To Write Seasonal ARIMA model mathematically](http://stats.stackexchange.com/questions/82197/how-to-write-seasonal-arima-model-mathematically?rq=1). Let us know if you still need more clarification, and what precisely. — Richard Hardy, Jul 12 '16 at 19:03
hey @Richard, as expected it indicates a Seasonal ARIMA. The links have been proved useful thanks! — Economist_Ayahuasca, Jul 14 '16 at 18:23
Good. It would be nice to close it somehow, I just don't know what is the best way to do that. If there was an answer, you could accept it; but there is none (you could write one yourself, though). Or maybe we could mark it as duplicate to some other question you found useful? — Richard Hardy, Jul 14 '16 at 19:29

score 2 · Accepted Answer · answered Aug 01 '16 at 14:04

The second part of the ARIMA model (P,D,Q) corresponds to the seasonal component (12 indicates the number of periods per season). In particular, the seasonal component (0,0,1) indicates a spike at lag 12 in the ACF but no other significant spikes, and The PACF will show exponential decay in the seasonal lags; that is, at lags 12, 24, 36. See explanation here.

Interpretation of the ARIMA coefficients in a time series

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