How to compute error terms in moving average time series model? [duplicate]

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.

1 Answer

" Are they just residuals? " . Yes !

Read my response to Moving-average model error terms

• 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? Jul 22 '19 at 13:38
• the coefficients are found by optimization i.e. trial & error through iteration. Jul 22 '19 at 13:50
• I see. But the trial and error stage is done first? Then we proceed to estimate other parameters? Jul 22 '19 at 13:51
• trial and error is done for all parameters simultaneously Jul 22 '19 at 13:52
• But I thought there is a method to choose them efficiently... Jul 22 '19 at 14:18