# Interpretation of the ARIMA coefficients in a time series

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) 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), is it telling me that my series looks yearly seasonal ?

## 1 Answer

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.