Annex A: State Space Models (SSMs)

SSMs have been used for time series modelling across many disciplines including population. SSMs are a type of hierarchical model and their hierarchical structure effectively incorporates the modelling of two time series:

• a state (often known as a process) time series that is unobserved and attempts to reflect the true, but hidden, state of migration (e.g. the underlying trend in migration); and
• an observation time series that consists of observations of, or measurements (e.g. the actual migration counts based on either survey (such as the IPS) or admin-based sources (such as Home Office data)).

For example, the actual population of migrants over time would be the state time series, while the incomplete and imprecise counts of migrants sampled in a survey would be the observation time series.

Process variation represents the processes that change the population size through time, while observation error reflects differences between the hidden state (i.e. the trend) and the observed data due to randomness or imprecision in the sampling or survey methodology. These two components act at different levels of the model hierarchy, and the SSM framework allows them to be modelled separately.

When an SSM model is fitted to a time series, the process and observation parameters can be estimated as well as the hidden states (e.g. the trend). These estimates of the hidden states generally reflect the true state of nature better than the original observations.