Considering the issue of volatility modelling, compare the volatility estimation using the VIX index and the volatility of the S&P500 index, using an econometric model. (Using R to calculate question d and e!!!!!!!!!!!!!)

(d.) Calculate the correlation between the VIX and the squared returns of the S&P500 index, and the fitted volatility using the GARCH model. Calculate the correlation over the entire time sample. Calculate the correlation on a year by year basis and plot the annual correlation.

(e.) Considering the correlation values, is the VIX a good representative of the volatility of the index? If not, why? Propose a way to test this in a quantitative way.