Interpreting coefficient for regression with log dependent variable

Question

Say I estimate

$$ \ln y_i = \alpha + \beta x_i $$

I understand that my estimate , say it comes out, $\hat{\beta} = 0.01$ is approximately the percentage variation in $y_i$ since

$$ \ln y'_i - \ln y_i = \ln (\frac{y'_i - y_i}{y_i} + 1)\approx \frac{y'_i - y_i}{y_i} = \hat{\beta}. $$

Does that mean that a one unit increase of x leads to a 0.01% increase in y or does it mean it leads to a 1% increase in y? I think it should be the first one (so that unit match on both sides of the equation), but I am confused by a note in slides saying this is $100 \hat{\beta} \%$ increase in y.

Answer 1

I actually got it as I was writing it.

Indeed, the unit on the left hand side at the end are natural units (e.g. a 50% variation in y would read as 0.5). So does the left hand side. Hence, the need to multiply by one hundred to get it expressed as a percentage change.