Interpretation of log-linear regression

Question

The regression is

$$\ln y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \varepsilon$$

I know there is an approximation, but what about the exact interpretation of $\beta_1$ in non-log terms of $y$?

1. This source says a one-unit in $x_1$ increase will give $(e^{\hat{\beta}} – 1) * 100$
2. This source says a one-unit in $x_1$ increase will give $e^{\hat{\beta}}$

Answer 1

Both sources give you a different interpretation of log-linear models.

1. The first source gives the interpretation in terms of the percentage change. That is, $ (e^\beta - 1) * 100 $ shows the percentage change in the dependent variable in response to one unit increase in the independent variable.

2. The second source shows the interpretation in different terms. In this case, a 1 unit increase in the independent variable multiplies the dependent variable by $ e^\beta $.