Discussion
Let \(Y_{t}\) be the value of \(Y\) in year \(t\). The natural logarithmic value of \(Y_{t}\) is regressed on \(t\). A simple linear regression yields the statistically significant estimates of intercept and slope coefficient as 12 and 0.1512, respectively. The calculated compound annual growth rate of \(Y\) is
Answer: C
The compound annual growth rate is 16.32%.
The fitted semi-log trend is an exponential growth path:
\[ \begin{aligned} \ln Y_{t} &= 12 + 0.1512\,t \\ Y_{t} &= e^{12}\left(e^{0.1512}\right)^{t} \end{aligned} \]
Each year \(Y\) is multiplied by \(e^{0.1512}\), so the compound rate \(r\) satisfies:
\[ \begin{aligned} 1 + r &= e^{0.1512} \approx 1.1632 \\ r &\approx 0.1632 = 16.32\% \end{aligned} \]
15.12% is the slope itself, which is the instantaneous (continuously compounded) growth rate, not the compound annual rate. 116% treats the growth factor 1.16 as if it were the rate, and 1.16% misplaces the decimal point of that factor.
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