Interpreting percentage units regression

Question

log(sales) = beta_0 + beta_1 * GDP

The usual process of transforming a variable such as price into log(price) to measure an approximate percentage change means that if you include an independent variable in your regression that is measured in units (e.g. GDP $) then you interpret it as:

• A one unit increase in GDP increases sales on average by (BETA_1 * 100) percent, holding all else constant.

This is because the beta coefficient on GDP is measuring changes in deminal percentages (0 - 1). What happens if your independent variable is not log transformed but already in percentages, except it ranges from 0 - 100?

sales_% = beta_0 + beta_1 * GDP

Would this mean we now interpret the beta coefficient as a proportional increase in sales_%? For example, if beta_1 = 10, then:

• A one unit increase in GDP increases sales on average by 10 percent, holding all else constant.

Answer 1

Not quite.

Your interpretation for the first model is correct, but your explanation isn't quite right. Note that $$ \begin{equation*} \beta_1 = \frac{\partial \log y}{\partial x}, \end{equation*}$$

but that isn't very easy to interpret. So, we recall the calculus result that $$ \begin{equation*} \frac{\partial \log y}{\partial y} = \frac{1}{y} \end{equation*}$$ or $$ \begin{equation*} \partial \log y = \frac{\partial y}{y}. \end{equation*}$$

Plugging this into the equation for $\beta_1$, we have

$$ \begin{equation*} \beta_1 = \frac{\partial y / y}{\partial x}. \end{equation*}$$

If we multiply both sides by 100, we have

$$ \begin{equation*} 100\beta_1 = \frac{100 \times \partial y / y}{\partial x}. \end{equation*}$$

We realize that $100 \times \partial y/y$ is just the percentage change in $y$, giving the interpretation that $100 \beta_1$ is the percentage change in the outcome for a one unit increase in $x$.

The correct interpretation for your second model would be that a 1 unit increase in GDP leads to a 10 percentage point increase in sales. It's easiest to understand this by thinking of your outcome as being measured in percentage points, rather than percent. Then, a 1 unit change in $x$ leads to a $\beta_1$ unit change in $y$, just as we normally get.

This is an important distinction. An increase in sales from 1% to 5% is a $5 - 1 = 4$ percentage point increase, but a $(5 - 1)/1 \times 100 =400$ percent change.