Time Series: "Residuals" of ARMA model

I still don't get it...

I believe the white noise {a_t} are unobservable (this term is used in all of my textbooks) because the true value of the parameters, phi's and theta's, are unknown. We can never calculate the exact value of {a_t}. We can only calculate the (observable) residuals "a_t hat", but based on the above definition of residual, to calculate a_t hat, we need to know the values of the (unobservable) white noise terms
a[sub:1pifj3j2]t-1[/sub:1pifj3j2], a[sub:1pifj3j2]t-2[/sub:1pifj3j2], ..., a[sub:1pifj3j2]t-q[/sub:1pifj3j2] which seem to be going in circles because they are defining the residuals in terms of the white noises? What sense does it make?

So how can we actually compute the values of "residuals" of ARMA model?

Thanks!

The definition of "residual" above seems to be just going in circles to me. We want to calculate the residual which is an estimate of the white noise term, but the definition of residual itself depends on the white noise terms which are unobservable. How come?


I understand that the parameters (phi and theta) can be estimated (e.g. by MLE). But when you look at the definition of "residual" above, they are defining the residual in terms of the white noises
a[sub:3ux8kd5d]t-1[/sub:3ux8kd5d], ..., a[sub:3ux8kd5d]t-q[/sub:3ux8kd5d] which doesn't make any sense to me. On the right hand side of the definition, everything is known or can be calculated except for the white noise terms. But since we don't know the values of the white noise terms, there is no way we can calculate any residual, so I'm really confused as to how residuals of ARMA model are calculated in practice.

Hopefully someone can clarify this.
Any help is much appreciated!