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I'm building a model using the auto.arima function in R, and it gives the following output:

arima(x = revtrend, order = c(2, 1, 2))
Coefficients:
   ar1     ar2     ma1      ma2
1.1894 -0.1975  0.4387  -0.5613

Can someone help me write this as an actual mathematical model? I believe I'm getting a bit lost in the notation. Thank you!

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  • $\begingroup$ See these threads (you may need to narrow down the search further by adding terms like "output", "auto.arima" or other). $\endgroup$ May 22, 2023 at 10:51

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Let $B(.)$ denote the backshift operator, i.e. $B y_t = y_{t-1}$. Then we can write a general non-seasonal ARIMA(p, d, q) as

$$ \underbrace{(1 - \phi_1 B - ... - \phi_p B^p)}_{\text{AR(p)}} \ \underbrace{(1-B)^d}_{\text{d differences}} \ y_t = \underbrace{(1+\theta_1 B + ... + \theta_q B^q) \epsilon_t}_{\text{MA(q)}} $$

and $\epsilon_t \overset{iid}{\sim}N(0, \sigma)$

You just need to substitute the parameter estimates in with $p=2, d=1, q=2$. If you want to remove the backshift operator, simply expand everything out and notice that $B^k y_t = y_{t-k}$

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