### Distinguished University Professor, Department of Economics, University of Pittsburgh

*Jean-Francois Richard graduated in Physics and received a Ph.D. in Economics from the Catholic University of Louvain in 1973. His academic career includes positions at the London School of Economics (1973-1974), Catholic University of Louvain (1974-1986), Duke University (1986-1991) and University of Pittsburgh since 1991. He served as research director of C.O.R.E. (Center for Operations Research and Econometrics, Louvain) from 1983 to 1986 and chair of the Economics department (Pittsburgh) from 2000 to 2006. He is a Fellow of the Econometric Society and of the Journal of Econometrics. A detailed CV is available from http://www.econ.pitt.edu/fantin/*

## Efficient Filtering in State-Space Representations

##### Friday, June 26, 2:30 – 3:30 PM

##### 1175 Benedum Hall

The objective of filtering is to track the dynamic evolution of unobservable state variables using noisy measurements of observables. This requires the computation of integrals over the unobserved state. When models are linear and Gaussian, these integrals obtain analytically from the Kalman filter ; departures entail integrals that must be approximated numerically. A broad range of numerical techniques have been proposed to accomplish filtering (as well as likelihood estimation) given departures from linearity and/or gausssianity. Here we propose an enhancement of the Particle Filter (PF) developed by Gordon et al. (1993) and Kitagawa (1996).

The PF achieves filtering via the construction of discrete approximations to the densities that appear in the predictive and updating stages of the filtering process. While the PF is conceptually simple and easy to program, it suffers from a major and well-documented shortcoming : the discrete supports of period-t approximations are determined using period-(t-1) information, and thus ignore crucial information provided by period-t measurements. That is to say, the supports are not adapted. This shortcoming can produce very large biases in the filtered estimates of the state as well as in the likelihood estimates; the elimination of these biases can require prohibitively large numbers of particles.

Here we propose a generalization of the PF designed to achieve full adaption. The generalization features two key extensions : the density approximations it constructs are continuous and integral are calculated using the Efficient Importance Sampling (EIS) methodology designed by Richard and Zhang (2007). As we illustrate in the context of the benchmark bearings-only tracking problem, the EIS filter can produce dramatic reductions in Mean Squared Errors (MSEs) – by several order of magnitude – relative to existing techniques. In order to achieve reductions in MSEs on the scale delivered by the EIS filter, the PF would require prohibitive computational costs.

This research is a collaborative effort between D.N. DeJong, H. Dharmarajan, R. Liesenfeld, and J.F. Richard.