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**Time Series: "Residuals" of ARMA model**

To check whether the white noise {a

_{t}} are uncorrelated, we usually look at the residuals (which are sample estimates of the white noise {a

_{t}}) and residual plots. But I just don't understand the meaning of "residuals" in the context of ARMA model...

In the above definition, the residual "a

_{t}hat" is in terms of the white noise terms

a

_{t-1}, a

_{t-2}, ..., a

_{t-q}. But we know that the white noise terms are

*unobservable*(residuals are observable, white noise terms are not), and there is no way we can know the exact values of a

_{t-1}, a

_{t-2}, ..., a

_{t-q}, right? Now if we don't know everything on the right hand side, how can we calculate the residuals "a

_{t}hat"? I just don't understand how residuals of ARMA model can be calculated based on this definition.

I tried searching the internet, but couldn't find much.

Hopefully someone can explain. Thank you!