The data file name is “SP500.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.