AR/ARMA model - test residual for independence or lack of correlation Shumway and Stoffer (2017), a great book that I highly recommend, define an AR(p) model as $x_t=\phi_1 x_{t-1}+\phi_2 x_{t-2}+...+\phi_p x_{t-p}+w_t$, where $w_t$~$wn(0,\sigma_w^2)$; $w_t$ is not assumed to be Normally distributed. Notice that means $w_t$ is white noise (uncorrelated) not i.i.d. noise (independent). The ARMA process is also defined with white noise residuals. Yet on page 139 as part of model diagnostics they state:
"If the model fits well, the standardized residuals should behave as an i.i.d. sequence with mean 0 and variance 1.... Unless the time series is Gaussian, it is not enough that the residuals are uncorrelated".
In my mind, there is a clear inconsistency here. We've assumed that the model has uncorrelated errors, not independent errors; it therefore seems necessary to test for correlation only.

*

*Is there an inconsistency here?

*I checked a number of textbooks and I've only seen ARMA models defined with white noise errors. This makes me wonder, in R when fitting a model with the sarima command of the astsa package (written by the authors of the above mentioned book) what is assumed of the error term? Independence or lack of correlation?


I don't have the 4th edition of the book, but the answer is No, there's no contradiction because authors clearly state that they sometimes use more restrictive condition of IID, instead of a simply white noise. For instance, in Example 1.8 of 3rd edition they say

We will, at times, also require the noise to be independent and
identically distributed (iid) random variables with mean 0 and
variance $\sigma^2_w$.

The reason to use more restrictive condition can vary from didactic (easier to explain and derive certain results) to meaningful, such as obtaining sharper results under limited circumstances.
