Models with unknown parameters

Abstract

In the previous chapters we presented some basic DLMs for time series analysis, assuming that the system matrices Ft, Gt, Vt, and Wt were known. This was done to more easily study their behavior and general properties. In fact, in time series applications the matrices in the DLM are very rarely completely known. In this chapter we let the model matrices depend on a vector of unknown parameters, ? say. The unknown parameters are usually constant over time, but we also give some examples where they may have a temporal evolution. Anyway, the dynamics of ?t will be such as to maintain the linear, Gaussian structure of DLMs. In a classical framework one typically starts by estimating ?, usually by maximum likelihood. If the researcher is only interested in the unknown parameters, the analysis terminates here; if, on the other hand, he is interested in smoothing or forecasting the values of the observed series or those of the state vectors, the customary way to proceed is to use the estimated value of ? as if it were a known constant, and apply the relevant techniques of Chapter 2 for forecasting or smoothing.