Time Series Analysis

Structural Models

Typically in a time series, four components can be distinguished: trend (T), cycle (C), seasonality (S), and irregular component (I). The trend is the longest-term behavior of the time series, and the cycle is a long-term cyclical component. Sometimes the time series is not long enough to distinguish between trend and cycle, and so a unique trend-cycle component can be considered. The seasonality is a short-term cyclical component often due to the seasons, and, finally, the irregular component is an erratic component obtained as residual once the other components are identified and removed. The nonsystematic behavior of the [Page 1006]irregular component is used to assess the goodness of the decomposition, and often it can be reasonably assumed that it follows a normal distribution.

Figure 1 Food Retail Sales in New Zealand

There are two main basic ways in which the components mix together: the additive model

where X is the original time series, and the multiplicative model

If the oscillations of the cyclical components increase with a higher level of the time series, a multiplicative model is more appropriate; if the oscillations remain constant, an additive model is preferable.

A possible way to find the different components is based on regression, and the time series of food retail sales provides an example to illustrate it.

Food retail sales have a tendency to increase over time, and this could be explained by the presence of a trend or by an ascending part of a cycle, although given the nature of the series, affected by price inflation, it is more likely to be a trend. We then consider three components in this decomposition: trend-cycle, seasonality, and irregular component. The oscillations of the seasonal appear independent from the level of the series, suggesting an additive model.

It is a good strategy to remove the longer-term components first, and in this case, the trend-cycle (T + C) is exponential and can be estimated by the equation

where t is a regular ascending sequence (0,1,2, …) representing a counter for the time and ∊ is the residual part. Consequently, the trend-cycle component is given by

where c, α, and β are the parameters to estimate. More specifically, the estimated equation by least squares is

The trend-cycle component estimated by the equation above is shown in Figure 2.

Figure 2 Trend-Cycle, Food Retail Sales, and Seasonal and Irregular Components

Figure 3 Irregular and Seasonal Components

The estimated component should then be removed from the original time series, and following Equation 1, we

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