$[8$ Points Each$]$ You are a managing director at a boutique investment bank in Los Angeles.

Each day your two interns give you a one$-$day$-$ahead forecast of the Euro$/$Dollar exchange rate.

At the end of three months, you calculate each intern's series and find that the mean errors are

zero, and the error variances and covariances are hat$({)}_A^2=15\mathrm{.}76,$hat$({)}_B^2=9\mathrm{.}17,$ and hat$({)}_{AB}=8\mathrm{.}55.$

a$)$ If you had to choose between these two forecasts, which would you choose and why?

b$)$ If you could combine the two forecasts, what would the optimal weights be according to

the variance$-$covariance method? What's the resulting forecast variance?

c$)$ Is it guaranteed that a combined forecast using the optimal weights from b$)$ will have

lower MSE? Why or why not?