Although terms such as “trend” and “seasonal” are intuitively appealing, they are mental constructs as we cannot observe them directly. Therefore, we use a structural modeling approach that treats them as *unobserved components* (Harvey, 1989; Harvey and Shephard, 1993). In the empirical work we used the STAMP (Structural Time Series Analyser, Modeller and Predictor) software in conjunction with GiveWin; for details, see Koopman et al. (2000).

We define the components at time *t *as follows: trend = µ_{t}; slope = β; seasonal component = γ_{t}; and irregular component = ε. We assume that the process is observed at unit time intervals (*t, t+1,…) *and that there are *s *such intervals in a year (e.g. s=12 for monthly data). We then allow each component to evolve over time according to the specifications:

µ_{t }=µ_{t −1}+β_{t −1 }+η_{t } (1)

β_{t }=β_{t −1}+ς_{t } (2)

and

γ_{t }+γ_{t −1 }+....+γ_{t +−1 }=ω_{t } (3)

The quantities η_{t}, ζ, and ω_{t} represent zero mean, random shifts in the corresponding component. We assume such shifts to be independent of one another and uncorrelated over time; we also assume that they are independent of the “irregular” component, ε_{t}, seen in Equation (4) below. Equations (1)–(3) are known as the *state *or *transition *equations since they describe the underlying states of the process, or the transition of the components from one time period to the next.

Equations (1) and (2) provide a general framework for describing the evolution of the trend. If the process being modeled does not require all of these components, they can be dropped from the specification. The components are tested in sequential fashion as follows (Harvey, 1989, pp. 248-56):

- Does the slope disturbance term have positive variance? [Zero variance corresponds to the slope being fixed over time]
- Does the level disturbance have positive variance? [Zero variance corresponds to the mean level being fixed over time]
- If the slope disturbance is dropped, does the slope differ from zero?

If all three statistical tests produced negative outcomes, the trend term would be reduced to a constant.

When the time series is seasonal, we check:

- Does the seasonal disturbance term have positive variance? [Zero variance corresponds to a stable seasonal pattern]
- If the seasonal disturbance is dropped, are the seasonal components significantly different from zero? [Is there any seasonal pattern?]

If we drop the seasonal disturbance term, we are left with a “classical” model with fixed seasonal components. If the seasonal pattern is rejected completely, we reduce the model purely to its trend components.

The observed series is related to the states of the system by the *observation (or measurement) equation*:

*y*_{t }=µ+γ_{t }+ε_{t } (4)

where ε_{t} denotes the ‘irregular’ component.

The irregular component has zero mean and is assumed to be serially uncorrelated (i.e., not predictable) and independent of the disturbances in the state equations.

Estimation proceeds by maximum likelihood (Harvey, 1989, pp. 125-128). Operational details are provided in Koopman et al. (2000, section 8.3). The key parameters are the four variances corresponding to the disturbance terms [σ_{ε}, σ_{η}, σ_{ζ}^{2}and σ_{ω}^{2}]. Note that we assume these variances are constant over time; the time series may need to be transformed to justify this assumption, at least to a reasonable degree of approximation. The four variance terms control the form of the model, allowing each of level, slope and seasonal to be stochastic or fixed; slope and seasonal elements may be present or absent. Table D1 illustrates the principal variations. If fixed components are included in a model, the corresponding terms appear in the state equations (e.g. fixed seasonal coefficients) but the variance term is zero. If the components are stochastic, the same terms appear in the model, but the variance is strictly positive. The most general form is the Basic Structural Model (BSM), in which all components are stochastic. The BSM forms the starting point for the model development process, and is the standard form employed in STAMP. The program then “tests down” to eliminate any components that are not required.