Inequality Constrained State Space Models

The standard Kalman filter cannot handle inequality constraints imposed on the state variables, as state truncation induces a non-linear and non-Gaussian model. We propose a Rao-Blackwellised particle filter with the optimal importance function for forward filtering and the likelihood function evaluation. The particle filter effectively enforces the state constraints when the Kalman filter violates them. We find substantial Monte Carlo variance reduction by using the optimal importance function and Rao-Blackwellisation, in which the Gaussian linear sub-structure is exploited at both the cross-sectional and temporal levels.