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Currently I am studying time series Moving Average model MA(q) $$X_t -\mu= \epsilon_t + \theta_1\epsilon_{t-1} + \theta_2 \epsilon_{t-2} + ... + \theta_q \epsilon_q$$ where $\theta_1,...,\theta_q$ are parameters and $\epsilon_t$ is a white noise with mean $0$ and constant variance.

I have been puzzling on how to compute $\epsilon_t.$ Are they just residuals?

This post illustrated how to compute $\epsilon_t$ in $MA(1)$ model. But I do not see how the same technique can be carried forward to $MA(q)$ model.

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" Are they just residuals? " . Yes !

Read my response to Moving-average model error terms

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  • $\begingroup$ In this case, what is a procedure of fitting data with MA(q) model? Which variable should we find first? As residual is the difference between true value and predicted value by forecasting model, I suppose residual is calculated the last. Since we can obtain sample acf, by theoretical formula of acf of MA(q) model, we can find variable $\theta_1,...,\theta_q.$ Am I right? $\endgroup$ – Idonknow Jul 22 '19 at 13:38
  • $\begingroup$ the coefficients are found by optimization i.e. trial & error through iteration. $\endgroup$ – IrishStat Jul 22 '19 at 13:50
  • $\begingroup$ I see. But the trial and error stage is done first? Then we proceed to estimate other parameters? $\endgroup$ – Idonknow Jul 22 '19 at 13:51
  • $\begingroup$ trial and error is done for all parameters simultaneously $\endgroup$ – IrishStat Jul 22 '19 at 13:52
  • $\begingroup$ But I thought there is a method to choose them efficiently... $\endgroup$ – Idonknow Jul 22 '19 at 14:18

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