Flexible Least Squares for Approximately Linear Systems

The problem of filtering and smoothing for a system described by approximately linear dynamic and measurement relations has been studied for many decades. Yet the potential problem of misspecified dynamics, which makes the usual probabilistic assumptions involving normality and independence questionable at best, has not received the attention it merits. This study proposes a probability-free filter that meets this misspecification problem head on, referred to as Generalized Flexible Least Squares for Approximately Linear Systems (GFLS-ALS). A Fortran program implementation is provided for GFLS-ALS, and references to simulation and empirical results are given. Although GFLS-ALS has close connections with the standard Kalman filter, it is concretely demonstrated that there are also important conceptual and computational distinctions. The Kalman filter provides a unique estimate for the state sequence, conditional on maintained probability assumptions for discrepancy terms. In contrast, the GFLS-ALS filter provides a family of state sequence estimates, each of which is vector-minimally incompatible with the prior dynamical and measurement specifications. The GFLS-ALS filter was incorporated into the statistical package GAUSS/TSM in 1997.Annotated pointers to related work can be accessed at http://www2.econ.iastate.edu/tesfatsi/flshome.htm