A time series is sequence of values recorded at successive intervals of time. The intervals can be days, weeks, months, years, or other time units. Examples include weekly sales of HP personal computers, quarterly earnings reports of Microsoft Corporation, daily shipments of Eveready batteries, and annual U.S. consumer price indices. A time series may consist of four possible components—trend, seasonal, cyclical, and random.

The trend (T) component is the general upward or downward movement of the data over a relatively long period of time. For example, total sales for a company may be increasing consistently over time. The consumer price index, one measure of inflation, is increasing over time. While consistent upward (or downward) movements are indicative of a trend, there may be a positive (or negative) trend present in a time series and yet the values do not increase (or decrease) in every time period. There may be an occasional movement that seems inconsistent with the trend due to random or other fluctuations.

Figure 5.2 shows scatter diagrams for several time series. The data for all the series are quarterly, and there are four years of data. Series 3 has both trend and random components present.

The seasonal (S) component is a pattern of fluctuations above or below an average value that repeats at regular intervals. For example, with monthly sales data on snowblowers, sales tend to be high in December and January and lower in summer months, and this pattern is expected to repeat every year. Quarterly sales for a consumer electronics store may be higher in the fourth quarter of every year and lower in other quarters. Daily sales in a retail store may be higher on Saturdays than on other days of the week. Hourly sales in a fast-food restaurant are usually expected to be higher around the lunch hour and the dinner hour, while other times are not as busy. Series 2 in Figure 5.2 illustrates seasonal variations for quarterly data.

If a seasonal component is present in a set of data, the number of seasons depends on the type of data. With quarterly data, there are four seasons because there are four quarters in a year. With monthly data, there would be 12 seasons because there are 12 months in a year. With daily sales data for a retail store that is open seven days a week, there would be seven seasons. With hourly data, there could be 24 seasons if the business is open 24 hours a day.

A cyclical (C) component of a time series is a pattern in annual data that tends to repeat every several years. The cyclical component of a time series is used only when making very long-range forecasts, and it is usually associated with the business cycle. Sales or economic activity might reach a peak and then begin to recede and contract, reaching a bottom or trough. At some point after the trough is reached, activity would pick up as recovery and expansion take place. A new peak would be reached, and the cycle would begin again. Figure 5.3 shows 16 years of annual data for a time series that has a cyclical as well as a random component.

While a time series with cyclical variations and a time series with seasonal variations may look similar on a graph, cyclical variations tend to be irregular in length (from a few years to 10 or more years) and magnitude, and they are very difficult to predict. Seasonal variations are very consistent in length and magnitude; are much shorter, as the seasons repeat in a year or less; and are much easier to predict. The forecasting models presented in this chapter will be for short-range or medium-range forecasts and the cyclical component will not be included in these models.

The random (R) component consists of irregular, unpredictable variations in a time series. Any variation in a times series that cannot be attributed to trend, seasonal, or cyclical variations would fall into this category. If data for a time series tend to be level with no discernible trend or seasonal pattern, random variations would be the cause for any changes from one time period to the next. Series 1 in Figure 5.2 is an example of a time series with only a random component.

There are two general forms of time-series models in statistics. The first is a multiplicative model, which assumes that demand is the product of the four components. It is stated as follows:

An additive model adds the components together to provide an estimate. Multiple regression is often used to develop additive models. This additive relationship is stated as follows:

There are other models that may be a combination of these. For example, one of the components (such as trend) might be additive, while another (such as seasonality) could be multiplicative. Understanding the components of a time series will help in selecting an appropriate forecasting technique to use. The methods presented in this chapter will be grouped according to the components considered when the forecast is developed.