ACF of absolute or squared residuals? When fitting a model (e.g. ARIMA but really any model for this matter) I've been taught to examine the residuals for autocorrelation. This is true with any model - you expect the residuals to be white noise, i.e. i.i.d if the model fits the data (see image below).
However, taking the ACF of the residuals does not prove they are independent because you can have a series that is uncorrelated but not independent. For such series, taking the ACF of absolute values might show strong correlation after lag 0.
So when examining the residuals to determine the model fits the data and all that is left is white noise, should one also look at the ACF of absolute or squared residuals too?


Your intuition is correct. I.i.d. errors is the ideal, and autocorrelation in their absolute values or squares violates that. For some models and some of their uses, uncorrelated errors may be good enough.
E.g. in a regression with time series data, uncorrelated but dependent errors could yield consistent and unbiased estimates and prediction intervals that have correct coverage unconditionally (on average over time). This may be good enough. Meanwhile, i.i.d. errors could yield consistent, unbiased and efficient estimates and prediction intervals with correct coverage conditional on all available information. This is better.
Examining autocorrelation in absolute values and/or squares of residuals is standard practice when one suspects presence of autoregressive conditional heteroskedasticity. This is common e.g. in financial variables such as stock prices or their logarithmic returns. Model such as GARCH are used to account for such autocorrelation.
