Nonlinear and Non-Gaussian State-Space Modeling with Monte by Tanizaki H.

By Tanizaki H.

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The following procedure is taken for rejection sampling smoother: (i) pick one of αj,t+1|T , j = 1, 2, · · · , N with probability 1/N and one of αi,t−1|t−1 , i = 1, 2, · · · , N with probability ∗ qij,t , (ii) generate a random draw z from the proposal density P∗ (·) and a uniform random draw u from the interval between zero and one, (iii) take z as αm,t|T if u ≤ ω(z) and go back to (ii) otherwise, (iv) repeat (i) – (iii) N times for m = 1, 2, · · · , N , and (v) repeat (i) – (iv) T times for t = T − 1, T − 2, · · · , 1.

Denote the acceptance probability by ω(x) = Px (x)/aP∗ (x), where a is defined as a ≡ supx Px (x)/P∗ (x) and the assumption of a < ∞ is required for rejection sampling. Under the setup, rejection sampling is implemented as: (i) generate a random draw of x (say, x0 ) from P∗ (x) and (ii) accept it with probability ω(x0 ). The accepted random draw is taken as a random draw of x generated from Px (x). In the case where both Px (x) and P∗ (x) are normally distributed as N (µ, σ 2 ) and N (µ∗ , σ∗2 ), it is easily shown that σ∗2 > σ 2 is required for the condition a < ∞, which implies that P∗ (x) has to be distributed with larger variance than Px (x).

Simulation II (Stochastic Volatility Model): The system is represented as: yt = exp and αt = δαt−1 + ηt for 0 ≤ δ < 1. 5 is less computational then the quasi-smoother in this section. To obtain the smoothing random draws, the quasi-smoother requires the filtering random draws while the Markov chain Monte Carlo procedure does not utilize them. 13 However, we should keep in mind that the Markov chain Monte Carlo approach uses a large number of random draws from the convergence property of the Gibbs sampler and that the quasi-filter and the quasi-smoother do not give us the exact solution.

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