ARMA models and residual series

Assuming that model is correct, why does the residual series of an ARMA model resemble a white noise process?

• it's the definition of the ARMA process, you want certain properties of residuals such as no autocorrelation in them, so they look like white noise Jun 13 '18 at 14:50
• If you are referring to the residuals to the fit of an ARMA model to data, then the answer depends on how you are performing the fit. As @Aksakal points out, the residuals in the model itself are white noise by definition.
– whuber
Jun 13 '18 at 14:55
• If outliers (pulses,seasonal pulses.level shifts,local time trends) are untreated the ACF tests for "whiteness" are fundamentally flawed as their untreated presence downwards biases the ACF providing an "Alice in Wonderland effect". Similarly if there are changes in ARMA parameters over time or if the estimated error variance is not homogeneous over time. Jun 13 '18 at 17:06

I think you might be referring to the MA(p) terms, which enter the ARMA in an autoregressive way.

For example, in an ARMA(1,1) model we have

$$Y_t = \alpha + \phi Y_{t-1} + \epsilon_t + \theta\epsilon_{t-1}$$

Here, the residual of the model are NOT $$Y_t - \alpha - \phi Y_{t-1} = \epsilon_t + \theta\epsilon_{t-1}$$

Which you're right wouldn't be a white noise process if $\theta \neq 0$, but residuals are instead:

$$Y_t - \alpha - \phi Y_{t-1} -\theta\epsilon_{t-1} = \epsilon_t$$

Which is, by definition of $\epsilon_t$, a white noise process

Why white noise residuals are desired:

It is a general principle of time series models (including ARMA) that you would like to capture all systematic dynamics in the data. This means, every variation that can be explained by input variables, auto-regression, moving-average etc. should be explained.

If after your model building, the residuals are purely random, and are not correlated with each other i.e. show no systematic pattern, then you are generally satisfied, because you reached a point where you cannot go further. White noise is completely unpredictable.

And a bit more pragmatic answer:

The residuals of an ARMA process are by defintion white noise. If you fit your data to an ARMA model, and you see that residuals are white noise, then this indicates you that the model fits the data well.

• "The residuals of a sufficient ARMA model" ...... Jun 13 '18 at 19:45
• @IrishStat, you are right, that sentence was a little confusing. I changed "ARMA model" to "ARMA process". Jun 13 '18 at 21:05