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:

1. 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?
2. 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!

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.
• @ChintanRajvir, for any time series process $x_t$ (not necessarily being the standardized error $z_t$), $\text{Corr}(x_t,x_{t-j})$ and $\text{Corr}(x_t^2,x_{t-j}^2)$ are not necessarily related. You can have $x_t$ follow an ARMA(p,q) model with constant conditional variance as well as $x_t$ with constant conditional mean but following a GARCH(s,r). Simulate $x_t$ as ARMA(p,q) with constant conditional variance and inspect its ACF and the ACF of $x_t^2$. Then simulate $x_t$ as GARCH(s,r) with constant conditional mean and inspect its ACF and the ACF of $x_t^2$. You will see. – Richard Hardy Feb 26 at 10:44