9781118041581

Forecasting 155

A positive value of the coefficient c implies an increasing rate of growth in sales

over time; that is, sales tend to turn more steeply upward over time. Conversely,

if c is negative, sales grow more slowly over time. The quadratic form includes

the linear equation as a special case (when c equals zero). Thus, suppose the

manager runs a regression of sales versus the pair of explanatory variables, t

and t^2 , and arrives at the equation

where, according to t-tests, the constant term and both coefficients are statis-

tically significant. In such a case, the quadratic specification fits the past time-

series data better than the linear specification and has a higher adjusted R^2.

The bottom portion of Figure 4.4 shows the time series and the fitted quadratic

equation.

Besides the quadratic equation, forecasters often use the exponential form,

[4.14]

where the coefficients b and r are to be estimated. Here, the coefficient r is raised

to the power t. Thus, if r is greater than 1, then sales, Qt, grow proportionally as

time advances. For instance, if r equals 1.04, then sales grow by 4 percent each

year. Alternatively, if the estimated r falls short of 1, then sales decrease propor-

tionally. If r equals .94, then sales fall by 6 percent per period. By taking the nat-

ural log of both sides of the equation, we can convert this into a linear form so

that we can apply OLS:

[4.15]

To illustrate, suppose that the manager decides to fit an exponential equa-

tion to the time series in Figure 4.4. The resulting least-squares equation is

with both coefficients statistically significant. Here, 4.652 represents our best

estimate of log(b) and .0545 is our best estimate of log(r). To find the corre-

sponding estimates of b and r, we take the antilog of each coefficient: b 

antilog(4.652) 104.8 and r antilog(.0545) 1.056. (All standard regres-

sion programs, even most handheld calculators, have an antilog or exponen-

tial function.) Thus, the fitted exponential equation becomes

In other words, the exponential trend estimates annual growth of 5.6 percent

per year. Given only 12 observations in the time series, the quadratic and

Qt104.8(1.056)t.

log(Qt)4.652.0545t,

log(Qt)log(b)log(r)t,

Qtbrt,

Qt101.87.0t.12t^2 ,

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