Abstract
An efficient particle Markov chain Monte Carlo methodology is proposed for the rolling-window estimation of state space models. The particles are updated to approximate the long sequence of posterior distributions as we move the estimation window. To overcome the well-known weight degeneracy problem that causes the poor approximation, we introduce a practical double-block sampler with the conditional sequential Monte Carlo update where we choose one lineage from multiple candidates for the set of current state variables. Our proposed sampler is justified in the augmented space through theoretical discussions. In the illustrative examples, it is shown to be successful to accurately estimate the posterior distributions of the model parameters.
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Notes
In the discussion, the notation [k, l] (\(k<l\)) works as “an index” that indicates the interval from k to l.
The marginal density of \({\hat{\pi }}\) is \(\pi (x_{s-1:t-K-1}, x^{\dagger }_{t-K:t}, \theta \mid y_{s-1:t})\) as shown in Proposition 5.1 with \(x^{\dagger }_{t-K:t}=(x_{t-K}^{k_{t-K}^*},\ldots ,x_t^{k_t^*})\).
The marginal density of \({\check{\pi }}\) is \(\pi (x_{s:s+K-1}^{\dagger }, x_{s+K:t}, \theta \mid y_{s:t})\) as shown in Proposition 5.3 with \(x^{\dagger }_{s:s+K-1}=(x_{s}^{k_{s}^*},\ldots ,x_{s+K-1}^{k_{s+K-1}^*})\).
We use the notation \({\hat{p}}(y_t \mid x_{t-K-1}^n,y_{s-1:t-1},\theta ^n)\) since it is an unbiased estimator of \(p(y_t \mid x_{t-K-1}^n,y_{s-1:t-1},\theta ^n)\) as we shall show in Proposition 5.2.
Note that we need to generate \(x_{s-1}^{n,m}\) to compute \({\hat{p}}\) in (32).
The data are downloaded at http://realized.oxford-man.ox.ac.uk/data/download.
We also tried using other values of M but the computation time is the shortest with \(M= 300\).
Also see Supplementary Material C for the comparison of the computation time of the practical double-block sampler with those of the MCMC and the particle MCMC.
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Acknowledgements
We thank the Editor and two anonymous referees for their helpful comments. All computational results in this paper are generated using Ox metrics 7.0 (see Doornik (2009)). This work was supported by JSPS KAKENHI Grant numbers 25245035, 26245028, 17H00985, 15H01943, 19H00588.
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Awaya, N., Omori, Y. Particle rolling MCMC with double-block sampling. Jpn J Stat Data Sci 6, 305–335 (2023). https://doi.org/10.1007/s42081-022-00170-2
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DOI: https://doi.org/10.1007/s42081-022-00170-2