Likelihood evaluation and state estimation for nonlinear state space models

We apply Harrison and Stevens\u27 (1976) state space model with switching to model additive outliers in an autoregressive process. In the special case when the size of the outlier is fixed, the forward-backward algorithm can be used to estimate the parameters of the model. In the more natural situation when the outliers are of random size, the forward-backward algorithm is not efficient. We use a variance reduction technique called importance sampling to obtain accurate estimates of the likelihood function. Outliers in the time series can be identified with this algorithm. The algorithm is also used to measure the quality of the approximation proposed by Shumway and Stoffer (1991) for state space models with switching in the observation equation. Examples are provided where the approximation works quite well and where it fails. In addition, the importance sampling algorithm is shown to be superior to another Monte Carlo approach known as the Gibbs sampler when estimating the conditional means of the unobserved states. The state space model with switching is applied to the problem of estimating the risk premium in the forward foreign exchange rate. Finally, a numerical integration approach based on Monte Carlo techniques is developed for general state space models. The algorithm updates the densities of the state variables recursively. The algorithm is tested on a number of data sets. The running times suggest that the approach will be very competitive against algorithms based on Markov chain Monte Carlo methods