A well-organized report with a reasonable balance between text, figures, tables and results. Sometimes, too much raw R output is shown, and it would be better to number the sections, figures and tables for easy reference.
The relationship to past projects is explained. Whereas other projects have analyzed volatility indirectly via a study of financial prices, this one studies an index of volatility directly. The benefit or disadvantage of this different approach is not clearly explained or explored.
There is some novelty (at least, compared to previous 531 projects) in focusing on a volatility time series. However, the project would benefit from including a formal mathematical definition of volatility and specifying the calculation methodology used to construct 30-day volatility. Details of this construction might affect the statistical properties of the resulting index.
“strong evidence against stationarity, indicating the presence of a unit root” is flawed logic. Evidence against stationarity implies non-stationarity which may or may not be well described by a unit root model.
ACF/PACF. The oscillating ACF cannot be generated by AR(1). It may be surprising that the ACF goes negative.
It is surprising that STL shows periodicity. 30-day periodocity is not commonly a major aspect of financial data.
The peaks in the periodogram for the residuals are an artifact. The periodic behavior is because the cyclic component is constructect to filter out certain frequencies, and the peaks here are from what is left over.
The figure shows that some roots are very close to the unit circle, and the AR/MA roots just left of the imaginary axis almost cancel. The assessment that everything looks good is over-confident.
AIC values comparing different amounts of differencing are not comparable.
Fitting GARCH to volatility is surprising. This model was developed for modeling the returns themselves. Here, you are studying the volatility of volatility, and this should be explained and justified.
“The ACF of the ARIMA residuals still shows some significant lags that exceed the confident bounds. […] This totally suggests that the ARIMA model might miss some dynamic structure in the data, like in the volatiltiy clustering.” But, volatility clustering does not show up in the ACF, so this inference looks wrong. However, ACF peaks can be due to outliers that can be resolved by GARCH, so in some ways this point can be correct.
The conclusion, “our project connects modelling results to financial risk management, ensuring statistical insights translate into actionable economic decisions” is over-stated. There is no explicit explanation of how the results translate into actionable economic decisions.
Minor point: The ARMA-GARCH residual plots have an error in the x-axis tick mark values.