Linear Innovations State Space Models with Random Seed States

Exponential smoothing was used in Chap. 5 to generate the one-step-ahead prediction errors needed to evaluate the likelihood function when estimating the parameters of an innovations state space model. It relied on a fixed seed state vector to initialize the associated recurrence relationships, something that was rationalized by recourse to a finite start-up assumption. The focus is now changed to stochastic processes that can be taken to have begun prior to the period of the first observed time series value, and which, as a consequence, have a random seed state vector. The resulting theory of estimation and prediction is suitable for applications in economics and finance where observations rarely cover the entire history of the generating process.

The Kalman filter (Kalman 1960) can be used in place of exponential smoothing. Like exponential smoothing, it generates one-step-ahead prediction errors, but it works with random seed states. It is an enhanced version of exponential smoothing that is used to update the moments of states and associated quantities by conditioning on successive observations of a time series. It will be seen that it was devised for stationary time series and that it cannot be adapted for nonstationary time series without major modifications.

An alternative to the Kalman filter is an information filter, which also conditions on successive observations. However, instead of having a primary focus on the manipulation of moments of associated random quantities, it relies on linear stochastic equations. By using an information filter, the problems encountered with the Kalman filter for nonstationary data conveniently disappear. An information filter can be applied to both stationary and nonstationary time series without modification. The version presented here is an adaptation of the Paige and Saunders (1977) information filter to the linear innovations state space model context.