Abstract
This paper focuses on sequential Monte Carlo approximations of smoothing distributions in conditionally linear and Gaussian state spaces. To reduce Monte Carlo variance of smoothers, it is typical in these models to use Rao-Blackwellization: particle approximation is used to sample sequences of hidden regimes while the Gaussian states are explicitly integrated conditional on the sequence of regimes and observations, using variants of the Kalman filter/smoother. The first successful attempt to use Rao-Blackwellization for smoothing extends the Bryson-Frazier smoother for Gaussian linear state space models using the generalized two-filter formula together with Kalman filters/smoothers. More recently, a forward-backward decomposition of smoothing distributions mimicking the Rauch-Tung-Striebel smoother for the regimes combined with backward Kalman updates has been introduced. This paper investigates the benefit of introducing additional rejuvenation steps in all these algorithms to sample at each time instant new regimes conditional on the forward and backward particles. This defines particle-based approximations of the smoothing distributions whose support is not restricted to the set of particles sampled in the forward or backward filter. These procedures are applied to commodity markets which are described using a two-factor model based on the spot price and a convenience yield for crude oil data.
Highlights
State space models are bivariate stochastic processes {(Yi, Zi)}i≥1 where the state sequence (Zi)i≥1 is a Markov chain which is only partially observed through the sequence (Yi)i≥1
By integrating over all possible paths, a1:i−1, ai is sampled in {1, . . . , J}. This particle rejuvenation step allows to explore states which are not in the support of the particles produced by the forward filter and improves the accuracy and the variance of the original FFBS algorithm, see Section 3 for numerical illustrations. Another modification of the FFBS algorithm based on a Markov chain Monte Carlo (MCMC) sampling step was introduced in ([21], Section 5.2)
For the Chicago Mercantile Exchange (CME) West Texas Intermediate crude oil (WTI) Crude Oil, backwardation effect is more frequent than contango effect so that α1 should be greater than α2
Summary
State space models are bivariate stochastic processes {(Yi, Zi)}i≥1 where the state sequence (Zi)i≥1 is a Markov chain which is only partially observed through the sequence (Yi)i≥1. The first fully Rao-Blackwellized SMC smoother which should lead to consistent approximations when the number of particles grows to infinity was proposed by [3] and extends the Bryson-Frazier smoother for Gaussian linear state space models using the generalized two-filter formula with Rao-Blackwellization steps for the forward and the backward filters. This two-filter approach combines a forward filter with a backward information filter which are approximated numerically using SMC for the regime sequence and Kalman filtering techniques for the hidden linear states. A detailed derivation of the algorithms is provided in the “Appendix: Technical lemmas”
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