So on wikipedia [here][1] under *Examples*, it is mentioned that *ARIMA(0,2,2)* is given by:

$$
X_t = 2X_{t-1} - X_{t-2} + (\alpha + \beta -2)\epsilon_{t-1} + (1 - \alpha)\epsilon_{t-2} + \epsilon_t \ \ \ \ \ (1)
$$

**My question is:** how this equation has been derived?

We know that *ARIMA(0,2,2)* means $d=2$ (second order differencing) and $q=2$ is the MA (moving-average) order. 

If $MA(q)$ is [given by][2]:
$$
X_t = \mu + \epsilon_t + \theta_1\epsilon_{t-1} + ... + \theta_q\epsilon_{t-q}
$$

therefore (?):

$$
X_t = \mu + \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2\epsilon_{t-2} \\
X_{t-1} = \mu + \epsilon_{t-1} + \theta_1\epsilon_{t-2} + \theta_2\epsilon_{t-3}
\\
X_{t-2} = \mu + \epsilon_{t-2} + \theta_1\epsilon_{t-3} + \theta_2\epsilon_{t-4}
$$

After removing $\epsilon_{t-3}, \epsilon_{t-3}$, for the second order differencing we compute $X_t - 2X_{t-1} + X_{t-2}$: 

$$
X_t - 2X_{t-1} + X_{t-2} = (\theta_1 - 2)\epsilon_{t-1} + (\theta_2 - 2\theta_1 + 1)\epsilon_{t-2} + \epsilon_t
$$

and the coefficients of the error terms differ from those in *(1)*.

  [1]: https://en.wikipedia.org/wiki/Autoregressive_integrated_moving_average
  [2]: https://en.wikipedia.org/wiki/Moving-average_model