You are given the weekly $($Wednesday to Wednesday$)$ closing prices of the S&P $500$ stock market index, labelled SP$500,$ covering the period $02$ January $1991-30$ September $2020.$ The data file name is $"$SP$500.$xls$",$ which is uploaded to Canvas along with this file.

a$)$ Discuss the statistical properties of the series by $($i$)$ calculating relevant summary statistics of the returns, and $($ii$)$ plotting the returns, as well as their histograms and quantile$-$quantile $($QQ$)$ diagrams.

$[5\%]$

b$)$ Plot the ACF for returns, squared returns, and absolute returns, and then discuss whether any of these plots provide an indication of the predictability of the series.

$[10\%]$

c$)$ Describe the ARCH$-$GARCH family of models and explain why they are useful in explaining the volatility of stock market returns.

$[15\%]$

d$)$ Build three univariate GARCH$-$type models which nest ARCH $($e$.$g$.,$ GARCH, PGARCH, etc.$)$ to estimate the volatility of returns, explaining the motivation for their use. Test for the differences between the models $($e$.$g$.,$ parameter significance and Likelihood Ratio $($LR$)$ tests$),$ and discuss how their volatility estimates and residuals differ.

$[20\%]$

Notes: $($i$)$ Conduct all your statistical tests for this question at the $5\%$ significance level, and $($ii$)$ Support your discussion for this question with appropriate mathematical equations and references in the relevant area$($s$)$ of research.