If 𝑅𝑡 follows this GARCH(1,1) process, What are the formulas of its conditional and unconditional variances.

The GARCH(1,1) process can be expressed as:

𝑅𝑡 = 𝜇 + 𝜀𝑡

𝜀𝑡 = 𝜎𝑡𝜀̃𝑡

𝜎2𝑡 = 𝑤 + 𝛼(𝜀2𝑡−1 − 𝜎2𝑡−1) + 𝛽𝜎2𝑡−1

where 𝜇 is the mean of the process, 𝜀𝑡 is the error term at time t, 𝜎𝑡 is the conditional standard deviation at time t, 𝜀̃𝑡 is a standardized error term, 𝑤 is the unconditional variance of the process, 𝛼 and 𝛽 are the parameters of the GARCH(1,1) model.

The conditional variance at time t is given by:

Var(𝜀𝑡|𝑭𝑡−1) = 𝜎2𝑡

The unconditional variance is given by:

Var(𝜀𝑡) = 𝑤/(1−𝛼−𝛽)

where 𝑭𝑡−1 represents the information set at time t-1.