9781118041581

Forecasting 157

As an example, suppose that a cable television company has watched its

number of subscribers steadily increase over the last 10 quarters. With

500,000 subscribers in hand, the company wants to predict how many addi-

tional subscribers it will have one, two, and five years into the future. Three

crucial facts can help the company construct these forecasts. (1) Each quar-

ter, about 98 percent of current subscribers retain the service. (2) The size of

the potential market is about 1,000,000 households, so there are 500,000 poten-

tial customers not yet enlisted. (3) Each quarter, about 8 percent of unaffiliated

customers become new subscribers to the company. These facts imply the fol-

lowing equation for total subscribers in quarter t:

The first term on the right side of the equation is the number of retained

customers from last quarter; the second term is the number of new

subscribers.

Notice that this equation can be simplified to: Qt80,000 .90Qt 1 ,

showing that future subscriptions will grow at a decreasing rate. For instance,

starting from Q 0 500,000, next quarter’s subscriptions are predicted to

be: Q 1 80,000 (.9)(500,000) 530,000. Future quarterly forecasts are

found recursively. Having computed 530,000, the forecast for two quarters

in the future is: Q 2 80,000 (.9)(530,000) 557,000. The forecasts for

one year ahead (Q 4 ), two years ahead (Q 8 ), and five years ahead (Q 20 ) are

603,170, 670,860, and 763,527, respectively. These forecasts indicate that

subscriptions are expected to grow at a diminishing rate. Finally, if the cable

company did not have the specific facts in items (1) to (3), it could instead

use the record of its past quarterly subscriptions to fit the best regression

equation of the form Qta bQt 1 , estimate the coefficients a and b

directly, and then use these estimates for forecasting future numbers of sub-

scribers.

THE DEMAND FOR TOYS To illustrate some of the issues involved in time-

series modeling, consider the market for children’s toys. We have collected

sales data for 40 quarters over the period from 1995 to 2004. The tabular por-

tion of Figure 4.5 shows these hypothetical data. The data show an unmistak-

able upward trend and some obvious seasonal behavior. The pattern does not

seem completely regular, however, which indicates the presence of a random

element.

Let’s first estimate the long-term trend in toy sales. Assuming a linear trend,

we can use any standard computer program to estimate the OLS regression

equation

Qt141.161.998t.

Qt.98Qt 1 .08(1,000,000Qt 1 ).

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