Could some explain the calculation ((e0.0464 − 1 = 0.0299475) of the exponential growth in this example from the book?

EXAMPLE 3

A Log-Linear Regression for Quarterly Sales at Starbucks

Having rejected a linear trend model in Example 2, technology analyst Benedict

now tries a different model for the quarterly sales for Starbucks Corporation

from the second quarter of 2001 to the third quarter of 2019. The curvature in

the data plot shown in Exhibit 6 provides a hint that an exponential curve may

fit the data. Consequently, he estimates the following linear equation:

ln yt

= b0 + b1t + εt

, t = 1, 2, . . . , 74.

This equation seems to fit the sales data well. As Exhibit 10 shows, the R2

for this equation is 0.95. An R2 of 0.95 means that 95% of the variation in the

natural log of Starbucks’ sales is explained solely by a linear trend.

Exhibit 10 Estimating a Linear Trend in Lognormal Starbucks Sales

Regression Statistics

R2 0.9771

Standard error 0.1393

Observations 74

Durbin–Watson 0.26

© CFA Institute. For candidate use only. Not for distribution.

Log-Linear Trend Models 179

Coefficient Standard Error t-Statistic

Intercept 6.7617 0.0327 206.80

t (Trend) 0.0295 0.0008 36.875

Source: Compustat.

Although both Equations 1 and 3 have a high R2, Exhibit 11 shows how well

a linear trend fits the natural log of Starbucks’ sales (Equation 3). The natural

logs of the sales data lie very close to the linear trend during the sample period,

and log sales are not substantially above or below the trend for long periods of

time. Thus, a log-linear trend model seems better suited for modeling Starbucks’

sales than a linear trend model is.

1 Benedict wants to use the results of estimating Equation 3 to predict

Starbucks’ sales in the future. What is the predicted value of Starbucks’

sales for the fourth quarter of 2019?

Solution to 1:

The estimated value b

0 is 6.7617, and the estimated value b

1 is 0.0295. Therefore,

for fourth quarter of 2019 (t = 75), the estimated model predicts that ln y

75 =

6.7617 + 0.0295(75) = 8.9742 and that sales will be y e y

= ln 75 = e8.9742 =

$7,896.7 million. Note that a b

1 of 0.0295 implies that the exponential growth

rate per quarter in Starbucks’ sales will be 2.99475% (e0.0464 − 1 = 0.0299475).