Introduction to the DLM: The First-Order Polynomial Model

Abstract

Many important underlying concepts and analytic features of dynamic linear models are apparent in the simplest and most widely used case of the first-order polynomial model. By way of introduction to DLMs, this case is described and examined in detail in this Chapter. The first-order polynomial model is the simple, yet non-trivial, time series model in which the observation series Y _t is represented as Y _t = μ _t + ν _t , μ _t being the current level of the series at time t, and ν _t ∼ N[0, V _t ] the observational error or noise term. The time evolution of the level of the series is a simple random walk μ _t = μ _t−1 + ω _t , with evolution error ω _t ∼ N[0, W _t ]. This latter equation describes what is often referred to as a locally constant mean model. Note the assumption that the two error terms, observational and evolution errors, are normally distributed for each t. In addition we adopt the assumptions that the error sequences are independent over time and mutually independent. Thus, for all t and all s with t ≠ s, ε _t and ε _s are independent, ω _t and ω _s are independent, and ν _t and ω _s are independent. Further assumptions at this stage are that the variances V _t and W _t are known for each time t. Figure 2.1 shows two examples of such Y _t series together with their underlying μ _t processes. In each the starting value is μ₀ = 25, and the variances defining the model are constant in time, V _t = V and W _t = W, having values V = 1 in both cases and evolution variances (a) W = 0.05, (b) W = 0.5. Thus in (a) the movement in the level over time is small compared to the observational variance, W = V/20, leading to a typical locally constant realisation, whereas in (b) the larger value of W leads to greater variation over time in the level of the series.