Consider a stochastic volatility model, with t N(0, 1) and t N(0, 2 ). For this problem we fix = 0.9702,

Consider a stochastic volatility model,

with t ∼ N(0, 1) and ηt ∼ N(0, σ2 η
). For this problem we fix φ = 0.9702, ση = 0.178, and β = 0.5992 (Pitt and Shephard, 1999: section 4.2).

a. Simulate a data set y1, . . . , yT , using θ0 = 0 and T = 200. State the random variate seed that you use. For example, if you use R, use set.seed(1963).
Plot yt versus t as a time series.

b. Let `(θt) = ln f(yt | θt) denote the log-likelihood factor for θt . Find (µt , σt)
for the quadratic approximation `(θt) ≈ −1/(2σ
2 t )(θt −µt)
2 (see the previous problem).

c. Implement the adapted auxiliary particle filter from the previous problem.
As in Section 4.3, let Dt = (y1, . . . , yt) denote the data up to time t. Plot the posterior means θt = E(θt | Dt) against time, and add posterior medians and 25% and 75% quantiles.

y = e exp(0/2) and 0+1 = 0, +11,