Could some explain the calculation ((e0.0464 − 1 = 0.0299475) of the exponential growth in this example from the book?
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:
= 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
Standard error 0.1393
© 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
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
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).