A Restricted Resampling Scheme For Sequential Monte Carlo

Sequential Monte Carlo (SMC) methods have become a very popular class of numerical methods for the solution of optimal estimation problems in non-linear non-Gaussian scenarios. They generate sequentially weighted Monte Carlo samples of the unobservable state variables, and use these weighted samples for statistical inference. Thanks to the availability of ever-increasing computational power, these methods are already used in real-time applications appearing in fields as diverse as computer vision, signal processing, tracking, robotics, econometrics and finance. The important advan-tages of SMC over MCMC are its less possibility to become trapped in a local mode, high efficiency in online estimation and adaptation to parallel computing.Resampling is an important step in standard SMC method which multiplies sam-ples with large weight while prunes away samples with small weight in a probability manner. It solves the problem of weight degeneracy in sequential importance sampling (SIS) algorithm. But when the signal to noise ratio is high, for example in some dy-namic stochastic generalized equilibrium (DSGE) models, the variance of random term in state equation is usually assumed to be many orders of magnitude larger than that of observation equation, frequent resampling will rapidly impoverish diversity of samples and thus has a negative impact on the estimation results.Fearnhead and Clifford (2003) points out that having multiple copies of a particle is wasteful when state space is discrete as it will lead to repetitive calculation. So they come up with a new resampling method that guarantees no multiple copies of particles in the final set. However, for problems where the state space is continuous, there is an advantage in having multiple copies of particles as new descendants cannot be exactly the same. Therefore we propose a new resampling scheme for continuous state space models that limits the maximum number of copies each particle can generate to be R (R>1). This resampling scheme tries to balance between the need of diversity (i.e., having multiple distinct samples) and the need of focus (i.e., giving more presence to those samples with large weight).Numerical experiments show that when signal to noise ratio is high, SMC with our resampling scheme can reduce the mean squared errors of posterior estimates and improve the accuracy of likelihood estimates. We also apply this new filter in a particle Markov chain Monte Carlo framework to carry out Bayesian inference of a DSGE model. Numerical results indicate that this new resampling scheme can improve the efficiency of parameter estimation.