# interpreting level-log model that has a percentage variable

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 $$y$$?

1. 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?

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.

• You are showing that (𝑑(%)𝑦)/(100×𝑑𝑥/𝑥)=𝛽1/100, which indicate that 𝛽1/100 represent the change in percentage point of y as a result on 1% change in x. How do you then interpret it as "when X increases by 1 percent, y increases by 𝛽1 percentage points"? Shouldn't is be when X increases by 1 percent, y increases by 𝛽1/100 percentage points?
– Lena
Feb 3, 2021 at 11:39