You work for Xanadu, a luxury resort in the tropics. The daily temperature in the region is beautiful year-round, with a mean around 76 (Fahrenheit!) and no conditional mean dynamics. Occasional pressure systems, however, can cause bursts of temperature volatility. Such volatility bursts generally don’t last long enough to drive away guests, but the resort still loses revenue from fees on activities that are less popular when the weather isn’t perfect. In the middle of such a period of high temperature volatility, your boss gets worried and asks you make a forecast of volatility over the next 10 days. After some experimentation, you find that daily temperature yt follows yt|Ωt−1 ∼ N(µ,σ2 t), where σ2 t follows a GARCH(1,1) process, σ2 t = ω +αϵ2 t−1 +βσ2 t−1.

(a) Derive the optimal forecast of GARCH(1,1) process, σ2 t+h,t.

(b) Estimation of your model using historical daily temperature data yields ˆ µ =76, ˆ ω = 3, ˆ α = 0.6, and ˆ β = 0. If yesterday’s temperature was 92 degrees, generate point forecasts for each of the next 10 days’ conditional variance.