1. Estimate the simple GARCH(1,1) model on the S&P500 daily log returns using the maximum likelihood estimation (MLE) technique. First estimate

σ2 t+1

= ω + αR2 t

+ βσ2 t , with Rt = σt zt , and zt ∼ N(0, 1)

Let the variance of the first observation be equal to the unconditional variance, Var(Rt ). Set the starting values of the parameters to α = 0.1, β = 0.85, and

ω = Var(Rt )(1−α−β) ≈ 0.012 ∗ 0.05 = .000005. (Excel Hint: The number

π is calculated in Excel using the function pi()). Re-estimate the equation using variance targeting, that is, set ω = Var(Rt )(1−α −β), and use Solver to find

α and β only. Check how the estimated parameters and persistence differ from the variance model in Chapter 1.