Question

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

• A. \(15.12\%\)
• B. \(1.16\%\)
• C. \(16.32\%\)
• D. \(116\%\)

Hint:

For a log-linear trend equation:
\[ \ln Y_t = a + bt \]
the growth rate is:
\[ (e^b-1)\times100 \]
and not simply \(b\times100\).

Answer 1

The Correct Option is C

Step 1: Write the regression equation.
The regression model is
\[ \ln Y_t = 12 + 0.1512\,t \]
Here, the slope coefficient is
\[ b = 0.1512 \]

Step 2: Recall the CAGR formula from log-linear regression.
When the dependent variable is in natural logarithm form, the compound annual growth rate (CAGR) is calculated as
\[ \text{CAGR} = \left(e^b - 1\right)\times 100 \]

Step 3: Substitute the value of slope coefficient.
\[ \text{CAGR} = \left(e^{0.1512}-1\right)\times100 \]

Step 4: Evaluate the exponential term.
\[ e^{0.1512}\approx 1.1632 \]
Thus,
\[ \text{CAGR} = (1.1632-1)\times100 \]
\[ = 0.1632\times100 \]
\[ = 16.32\% \]

Step 5: Final conclusion.
Therefore, the compound annual growth rate is
\[ \boxed{16.32\%} \]
Hence, the correct option is (C).