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.

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.

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  • $\begingroup$ Cheers, I'll use percentage point to make it clearer. $\endgroup$ – Aram Kocharyan Oct 9 '11 at 15:57
  • $\begingroup$ Is there a difference between log points and percentage points? $\endgroup$ – user1690130 May 20 '12 at 0:10
  • $\begingroup$ @JG A one unit increase in $\log(x)$ represents an approximate 100% increase in $x$ itself. The "approximate" part comes in because derivatives hold exactly only for infinitesimally small changes. $\endgroup$ – Charlie May 21 '12 at 20:47

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