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Forecasting 153

business cycles. Although we do not show the random fluctuations, we can

describe their effect easily. If we took an even “finer” look at the data (plotted it

week by week, let’s say), the time series would look even more rough and jagged.

The relative importance of the components—trend, cycles, seasonal vari-

ations, and random fluctuations—will vary according to the time series in

question. Sales of men’s plain black socks creep smoothly upward (due to pop-

ulation increases) and probably show little cyclical or seasonal fluctuations.

By contrast, the number of lift tickets sold at a ski resort depends on cyclical,

seasonal, and random factors. The components’ relative importance also

depends on the length of the time period being considered. For instance, data

on day-to-day sales over a period of several months may show a great deal of

randomness. The short period precludes looking for seasonal, cyclical or trend

patterns. By contrast, if one looks at monthly sales over a three-year period, not

only will day-to-day randomness get averaged out, but we may see clear sea-

sonal patterns and even some evidence of the business cycle. Finally, annual

data over a 10-year horizon will let us observe cyclical movements and trends

but will average out, and thus mask, seasonal variation.

Fitting a Simple Trend

Figure 4.4 plots the level of annual sales for a product over a dozen years.

The time series displays a smooth upward trend. One of the simplest meth-

ods of time-series forecasting is fitting a trend to past data and then extrap-

olating the trend into the future to make a forecast. Let’s first estimate a

linear trend, that is, a straight line through the past data. We represent this

linear relationship by

[4.12]

where t denotes time and Qtdenotes sales at time t. As always, the coefficients a

and b must be estimated. We can use OLS regression to do this. To perform the

regression, we first number the periods. For the data in Figure 4.4, it is natural

to number the observations: year 1, year 2, and so on, through year 12. Fig-

ure 4.4a shows the estimated trend line superimposed next to the actual obser-

vations. According to the figure, the following linear equation best fits the data:

The figure shows that this trend line fits the past data quite closely.

A linear time trend assumes that sales increase by the same number of units

each period. Instead we could use the quadratic form

Qtabtct^2. [4.13]

Qt98.28.6t.

Qtabt,

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