Write ARIMA model from auto.arima output

Question

I want to write down the mathematical model for the following arima.test output:

> auto.arima(x)
Series: x 
ARIMA(1,0,0)(1,0,0)[12] with non-zero mean 

Coefficients:
         ar1    sar1       mean
      0.4706  0.5623  6304.4752
s.e.  0.1332  0.1262   828.9899

sigma^2 estimated as 3550092:  log likelihood=-538.48
AIC=1084.96   AICc=1085.68   BIC=1093.33

I understand this is a seasonal model with period 12, and p, d, and q are respectively 1, 0, and 0, for both the series and the seasonal parts.

However, I fail to interpretate the meaning of the coefficients and how to write this model together, what kind of formula I need to apply.

Answer 1

This link explains non-seasonal ARIMA, and this link explains seasonal ARIMA in R. (Note that other software packages have different conventions, in particular in terms of intercepts and of the sign for MA components.) Putting everything together, your ARIMA(1,0,0)(1,0,0)[12] model is, using the backshift operator $B$:

$$ (1-\phi_1B)(1-\Phi_1B^{12})y_t = c+\epsilon_t, $$

or

$$ (1-\phi_1B)(y_t-\Phi_1y_{t-12}) = c+\epsilon_t, $$

or

$$ y_t-\phi_1y_{t-1}-\Phi_1y_{t-12}+\phi_1\Phi_1y_{t-13} = c+\epsilon_t, $$

where your parameter estimates are

$$ \hat{\phi}_1=0.4706, \quad\hat{\Phi}_1=0.5623,\quad \hat{c}=6304.4752,\quad \epsilon_t\sim N(0,\sigma^2)\text{ with }\hat{\sigma}^2=3550092.$$