In the history of economics, the analysis of economic fluctuations can reclaim a prominent part. Undoubtedly, the analysis of business cycle movements plays the dominant role in this field, but there are also different perspectives to look at the ups and downs of economic time series. Economic fluctuations are usually characterized with regard to their periodic recurrence. Variations that last several years and occur in more or less regular time intervals are called business cycles, whereas seasonality (originally) indicates regularly recurring fluctuations within a year, that appear due to the season. Such seasonal patterns can be observed for many macroeconomic time series like gross domestic product, unemployment, industrial production or construction.
The term seasonality is also used in a broader sense to characterize time series that show specific patterns that regularly recur within fixed time intervals (e.g. a year, a month or a week). Take as an example the demand for Christmas trees: the monthly demand in November and especially in December will be generally very high compared to the demand during the other months of the year. This pattern will be the same for every year--irrespective of the total demand for Christmas trees. Moreover, one can also detect seasonal patterns in financial time series like in the variance of stock market returns. The highest volatility is often observed on Monday, mainly because investors used the weekend to think carefully about their investments, to obtain new information and to come to a decision.
As we saw so far, seasonality has many different manifestations. Consequently, there are different approaches to model seasonality. If we focus on macroeconomic time series the class of seasonal models is confined to processes with dynamic properties at periods of a quarter or a month. However, when financial time series are studied, then our interest shifts to seasonal patterns at the daily level together with seasonal properties in higher moments. Therefore, it is no surprise, that a rich toolkit of econometric techniques has been developed to model seasonality.
In the following we are going to deal with seasonality in the mean only (for seasonality in higher moments see Ghysels and Osborn (2001)), but there are still different ways to do so. The choice of the appropriate technique depends on whether seasonality is viewed as deterministic or stochastic. The well-known deterministic approach is based on the assumption, that seasonal fluctuations are fix and shift solely the level of the time series. Therefore, deterministic seasonality can be modeled by means of seasonally varying intercepts using seasonal dummies. Stochastic seasonality however is a topic in recent time series analysis and is modeled using appropriate ARIMA models (Diebold; 1998, Chapter 5). Since these seasonal ARIMA models are just an extension of the usual ARIMA methodology, one often finds the acronym SARIMA for this class of models (Chatfield; 2001).
The topic of this chapter is modeling seasonal time series using SARIMA models. The outline of this chapter is as follows: the next Section 5.2.1 illustrates, how to develop an ARIMA model for a seasonal time series. Since these models tend to be quite large, we introduce in Section 5.2.2 a parsimonious model specification, that was developed by Box and Jenkins (1976)--the multiplicative SARIMA model. Section 5.3 deals with the identification these models in detail, using the famous airline data set of Box and Jenkins for illustrative purposes. Those, who already studied the Section ARIMA model building in Chapter 4 on Univariate Time Series Modeling, will recognize that we use the same tools to identify the underlying data generation process. Finally, in Section 5.4 we focus on the estimation of multiplicative SARIMA models and on the evaluation of the fitted models.
All quantlets for modeling multiplicative SARIMA models are collected in XploRe 's times library.