Time Series Analysis: Forecasting and Control, Fourth Edition

CHAPTER NINE

Seasonal Models

In Chapters 3 to 8 we have considered the properties of a class of linear stochastic models, which are of value in representing stationary and nonstationary time series, and we have seen how these models may be used for forecasting. We then considered the practical problems of identification, fitting, and diagnostic checking that arise when relating these models to actual data. In the present chapter we apply these methods to analyzing and forecasting seasonal series and also provide an opportunity to show how the ideas of the previous chapters fit together.

9.1 PARSIMONIOUS MODELS FOR SEASONAL TIME SERIES

Figure 9.1 shows the totals of international airline passengers for 1952, 1953, and 1954. It is part of a longer series (12 years of data) quoted by Brown [79] and listed as Series G in the Collection of Time Series in Part Five. The series shows a marked seasonal pattern since travel is at its highest in the late summer months, while a secondary peak occurs in the spring. Many other series, particularly sales data, show similar seasonal characteristics.

In general, we say that a series exhibits periodic behavior with period s, when similarities in the series occur after s basic time intervals. In the example above, the basic time interval is 1 month and the period is s = 12 months. However, examples occur when s can take on other values. For example, s = 4 for quarterly data showing seasonal effects within years. It sometimes happens that there is ...