FORECASTING:TIME SERIES

TIME SERIES

Most statistical forecasting methods are based on using historical data from a time series.

A time series is a series of observations over time of some quantity of interest (a random variable). Thus, if Xi is the random variable of interest at time i, and if observations are taken at times1 i = 1, 2, . . . , t, then the observed values{X1 = x1, X2 = x2, . . . , Xt = xt} are a time series.

For example, the recent monthly sales figures for a product comprises a time series, as il- lustrated in Fig. 27.1.

1These times of observation sometimes are actually time periods (months, years, etc.), so we often will refer to the times as periods.

Because a time series is a description of the past, a logical procedure for forecasting the future is to make use of these historical data. If the past data are indicative of what we can expect in the future, we can postulate an underlying mathematical model that is representative of the process. The model can then be used to generate forecasts.

In most realistic situations, we do not have complete knowledge of the exact form of the model that generates the time series, so an approximate model must be chosen. Frequently, the choice is made by observing the pattern of the time series. Several typical time series patterns are shown in Fig. 27.2. Figure 27.2a displays a typical time series if the generating process were represented by a constant level superimposed with random fluctuations. Figure 27.2b displays a typical time series if the generating process were rep- resented by a linear trend superimposed with random fluctuations. Finally, Fig. 27.2c shows a time series that might be observed if the generating process were represented by a constant level superimposed with a seasonal effect together with random fluctuations. There are many other plausible representations, but these three are particularly useful in practice and so are considered in this chapter.

Once the form of the model is chosen, a mathematical representation of the generat- ing process of the time series can be given. For example, suppose that the generating process is identified as a constant-level model superimposed with random fluctuations, as illustrated in Fig. 27.2a. Such a representation can be given by

Xi = A + ei, for i = 1, 2, . . . ,

where Xi is the random variable observed at time i, A is the constant level of the model, and ei is the random error occurring at time i (assumed to have expected value equal to zero and constant variance). Let Ft+1 = forecast of the values of the time series at time t + 1, given the observed values, X1 = x1, X2 = x2, . . . , Xt = xt.

Because of the random error et+1, it is impossible for Ft+1 to predict the value Xt+1 = xt+1 precisely, but the goal is to have Ft+1 estimate the constant level A = E(Xt+1) as closely as possible. It is reasonable to expect that Ft+1 will be a function of at least some of the observed values of the time series.