Understanding the standardized residuals in time-series analysis I am recently learning about time-series analysis and the model diagnostics. I am facing difficulties in understanding the following points:

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*Mathematically, I understand that standardizing the residuals is dividing them by the conditional or unconditional volatility. But, I am not able to intuitively understand, why we actually need the standardizing of residuals (because in practice I am reading that "residuals" should follow the white noise process - then why the scaling?) ? Like why we use standardized residuals in every tests (like Ljung-Box Test, Jarque-Bera test, etc.) or in plotting the autocorrelation and partial autocorrelation plots instead of simple residuals?

*Next is, we plot the autocorrelation and partial autocorrelation plots for the standardized residuals and their squares to study and compare them with white noise. I am not able to capture the motive of comparing the ACF or PACF plots for the square of standardized residuals too! What is the significance of analyzing the 2 plots for the squared standardized residuals?

Any pointers will be really helpful!
 A: Consider an ARMA(1,1)-GARCH(1,1) model$\color{blue}{^{*}}$
\begin{aligned}
x_t &= \varphi_0+\varphi_1 x_{t-1}+\varepsilon_t+\theta_1\varepsilon_{t-1}, \\
\varepsilon_t &= \sigma_t z_t, \\
\sigma_t^2 &= \omega+\alpha_1\varepsilon_{t-1}^2+\beta_1\sigma_{t-1}^2, \\
z_t &\sim i.i.d.(0,1)
\end{aligned}
for some zero-mean, unit-variance distribution $d$. The model's distributional assumption ($i.i.d.(0,1)$) is on $z_t$, not $\varepsilon_t$. $z_t$ are known as standardized errors (innovations, shocks, errors, residuals). In models where $\sigma_t^2$ is constant (no GARCH part), $\varepsilon_t$ is proportional to $z_t$ (just their variances differ) and you need not standardize but rather can work directly with $\varepsilon_t$.
You examine ACF and PACF of $\hat z_t\color{blue}{^{**}}$ as the i.i.d. assumption rules out nonzero autocorrelation in $z_t$. You examine ACF and PACF of $\hat z_t^2$ as the i.i.d. assumption rules out autoregressive conditional heteroskedasticity in them.
$\color{blue}{^{*}}$ This is not restrictive but taken as an example to make the discussion concrete.
$\color{blue}{^{**}}$Or $\hat\varepsilon_t$ if $\sigma_t^2=\sigma^2 \ \forall \ t$ here and further. Hats denote fitted values.
