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I saw 2 types of formulas for ARIMA(0, 2, 2):

$$Ŷ_t = 2Y_{t-1} - Y_{t-2} + (α + β - 2)e_{t-1} + (1 - α)e_{t-2}$$ and $$y_t = -θ_1e_{t-1} - θ_2e_{t-2} + e_t \quad\quad (with \space y_t \space being \space 2nd \space difference \space of \space Y_t)$$

I'm confortable with the second formula which conform to the ARIMA(p,d,q) definition. is the first equivalent to second? what are α and β in the first formula? how is that the coeffient of $e_{t-1}$ term depends on both α and β?

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They are equivalent models with different parameterizations.

The first form arises when showing that an ETS(A,A,N) model (a.k.a Holt's trend method) is equivalent to an ARIMA(0,2,2) model. The $\alpha$ and $\beta$ are the smoothing parameters from the ETS model. See https://otexts.com/fpp3/arima-ets.html

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  • $\begingroup$ Thank you for the answer and comprehensive information $\endgroup$ – techie11 Jun 2 at 13:28

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