Question $1$: Conceptual Understanding $($a$)$ Define volatility clustering and illustrate it with an example from a real f inancial time series $($e$.$g$.,$ daily stock returns downloaded from a source such as Yahoo Finance$).($b$)$ Identify the proposer of the ARCH model and the year of its introduction, and provide a concise explanation of what the ARCH model entails and its purpose in time series analysis. $($c$)$ Evaluate the limitations of using sample variance as a measure of volatility in financial time series.