interpreting level-log model that has a percentage variable

Question

For the level-log model

$$y = \beta_0+ \beta_1\log(x),$$

I know the interpretation is

If we increase $x$ by one percent, we expect $y$ to increase by $\beta_1/100$ units of $y$.

So, my questions are

1. If $y$ is a percentage variable, what is the correct interpretation of the resulting regression? Is it still the same but has a unit of percentage instead of a unit, so the interpretation becomes

If we increase $x$ by one percent, we expect $y$ to increase by $\beta_1/100$ percent of $y$?

Or we don’t have to divide $\beta_1$ by 100, so the interpretation becomes

If we increase $x$ by one percent, we expect $y$ to increase by $\beta_1$ percent of $y$?

2. Can I convert my variables (ex: interest rate and inflation rate) to logarithm if it has a unit of percentage? If it can be done, can I change my model into a log-log model, although my variable $y$ has a unit of percentage, so the model becomes $\log(y)=\beta_0+\beta_1 \log(x)$? What is the correct interpretation of this model?

Answer 1

General note: Distinguish between percent and percentage points.

Answer for Question 1: If y is a percentage your model will be: $$(\%)y=\beta_0+\beta_1 log(x)$$

$$\frac{d(\%)y}{d log(x)}=\beta_1 $$ $$\frac{d(\%)y}{d log(x)}=\frac{d(\%)y}{dx/x}=\beta_1$$ Dividing both sides by 100: $$\frac{d(\%)y}{d log(x)}=\frac{d(\%)y}{100 \times dx/x}=\frac{\beta_1}{100}$$

Your interpretation is: when X increases by 1 percent, y increases by $\beta_1$ over 100 percentage points.

Answer for Question 2: If you have any variable in percentage you can keep it in percentage. It is meaningless to transform a variable that is already in percentage into logarithm. You will be questioned with the zero problem: log (1)=0. I totally do not recommend this.