Bayesian Forecasting and Dynamic Models pp 37-74 | Cite as

# 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 μ_{0} = 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.

## Keywords

Discount Factor Time Series Model Constant Model Observational Variance Exponentially Weight Move Average## Preview

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