Prediction, Filtering and Smoothing in Non-Linear and Non-Normal Cases Using Monte Carlo Integration

SUMMARY
A simulation-based non-linear filter is developed for prediction and smoothing in non-linear and/or nonnormal
structural time-series models. Recursive algorithms of weighting functions are derived by applying
Monte Carlo integration. Through Monte Carlo experiments, it is shown that (1) for a small number of
random draws (or nodes) our simulation-based density estimator using Monte Carlo integration (SDE)
performs better than Kitagawa's numerical integration procedure (KNI), and (2) SDE and KNI give less
biased parameter estimates than the extended Kalman filter (EKF). Finally, an estimation of per capita
final consumption data is taken as an application to the non-linear filtering problem.
1. INTRODUCTION
For a non-linear filtering problem the most heuristic and easiest approximation is to use the
Taylor series expansion and apply the conventional linear recursive Kalman filter algorithm
directly to the expanded non-linear measurement and transition equations (see, for example,
Wishner et al., 1969; Gelb, 1974). When applying the Taylor series expansion to the non-linear
measurement and transition equations, the structure of the approximated error terms is
completely different from that of the original ones and therefore we encounter some problems
with the approximated errors:
(1) The expectation of the error terms is not necessarily zero;
(2) The state vector is correlated with the errors;
(3) The error term in the measurement equation is correlated with the error term in the
transition equation; and
(4) The error terms are not normal (see Tanizaki, 1991).