Based on the response to this post, MLE is used instead of Least Squares for ARIMA models because the errors in the MA(q) part of the model are unobserved.

I don't get that: don't we have empirical values for the error terms: $e_1,e_2,....$ by using $Y_t - \hat{Y_t},Y_{t-1} - \hat{Y}_{t-1},....$ which can be updated recursively as the model is tuned?

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    $\begingroup$ An "empirical value" is not the same as the actual error, it's only an estimate of the same. Think about what happens at $t=0$; you don't have a $Y_{t-1}\dots Y_{t-q}$ to use to calculate $\hat{Y}_0$ or the associated errors, so you can't calculate them exactly, you can only estimate them as part of the MLE procedure - and that estimation means all the subsequent errors ($e_{t \geq 0}$) are also only estimates. $\endgroup$ – jbowman Jan 24 '18 at 18:50
  • $\begingroup$ @jbowman can't we just initialize $\hat{Y}_0$ the way we do for a simple exponential smoothing model and then update it after the fact? $\endgroup$ – Reinstate Monica Jan 24 '18 at 21:44
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    $\begingroup$ Sure, you can, but that initialized value isn't the TRUE value. The issue is that the true values are unobserved, and you have to deal with that somehow. If you knew them, you could use least squares to fit $y_i$ as a linear function of $e_{i-1}, \dots, e_{i-q}$, but you don't, so you can't. The r.h.s. of your regression equation is full of unobserved quantities. Hence the use of MLE; it provides maximum likelihood estimates of all the $e_i$, even $e_{-1}\dots e_{-q}$. But you can use other techniques to estimate the parameters as well. $\endgroup$ – jbowman Jan 24 '18 at 22:55

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