Problem $1$: In$-$Sample $($IS$)$ Market Prediction $(50$ points$)$

Objective: In this exercise, you will predict the time series of S&P$500$ returns usign the volatility index $($VIX$).$

Instructions:

Install yfinance.

Import the necessary libraries.

Download daily adjusted closing prices for tickers $^$GSPC and $^$VIX since $1980.$

Calculate monthly returns of S&P$500$ and end$-$of$-$month levels of VIX.

Regress S&P$500$ returns in month t$+1$ on VIX at the end of month t$.$

Report the regression results using the summary$()$ method.

In a markdown cell or a comment in a code cell, interpret the regression results. Does VIX predict market return?

Problem $2$: Out$-$Of$-$Sample $($OOS$)$ Market Prediction $(50$ points$)$

Objective: In this problem, you will test the predictive power of VIX out of sample.

Instructions:

Repeat steps $1-4$ from the previous problem.

Create the historical expanding window average of S&P$500$ returns with a minimum window of $60$ months.

In a for loop, create the expanding window prediction of S&P$500$ returns using VIX as a predictor.

Plot the two predictions in one figure.

Calculate and print the sum of squared residuals for both predictions.