Consider the fat-tailed nonlinear state-space model studied in Carvalho, Johannes, Lopes, and Polson (2010) and given by yt = t+v t t t

=

t 1

(1 + 2 t 1) +w t where t N(01) t N(01)and t IG( 2 2)

(a) Assume that vwand areknown. Propose, implement, and com pare SMC approaches for ltering without parameter learning.

Figure 6.5 Results of applying the PL algorithm of Carvalho, Johannes, Lopes, and Polson (2008) with M = 500 particles in the AR(1) plus noise model.

(a) Time trajectories of the mean (solid line) and quantiles (2.5% and 97.5%) of the posterior distribution of the AR coe cient .

(b) Time trajectories of the mean (solid line) and quantiles (2.5% and 97.5%) of the posterior distribution of the observational variance v.

(c) Time trajectories of the mean (solid line) and quantiles (2.5% and 97.5%) of the posterior distribution of w.

(d) Time traces of the true state t (solid line) and posterior mean of the particle approximation to the distribution of t

(b) Assume that is known but vw and are unknown. Propose, im plement, and compare SMC approaches for ltering and parameter learning.