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I have an auto.arima model output with ARIMA(000)(110)[4] with sigma^2=0.005, so I assume the model fits well the data. But I'm trying to understand the model itself... If I did all the maths correctly, the change at time t depends on the change at the same time in 2 previous years, being expressed as:

xt = (α1+1) xt-4 - α1xt-8 + wt.

Is it correct?

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That's right. Assuming you don't have a constant in the model, and using your notation, then the model with one seasonal difference and one seasonal AR term is

$(1 - \alpha_1 L^4)(1 - L^4)x_t = \epsilon_t$

which is

$x_t - \alpha_1 x_{t-4} - x_{t-4} + \alpha_1 x_{t-8} = \epsilon_t$

or, as you wrote,

$x_t = (1 + \alpha_1)x_{t-4} - \alpha_1 x_{t-8} + \epsilon_t$

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