Datastories Hub: Quick Guide - Log linear models

Ralf Martin

What is the difference between the linear, log and log-log models? They describe different possible relationships between the dependent and explanatory variable. This affects how we can interpret estimated parameters. There are three cases

Linear: $y=a+bX$
Log: $\ln{y}=a+bX$
Log-log: $\ln{y}=a+b \ln{X}$

We can draw all three in the same y-x diagram by re-writing the equations (and recalling that the exponential function $\exp(\cdot)$ is the inverse of the natural logarithm function $\ln(\cdot)$

Linear: $y=a+bX$
Log: $y=\exp(a+bX)$
Log-log: $y=\exp(a+b\ln{X})$

Plotting this for a=1 and b=0.1 will look as follows:

i.e. log vs log-log are two different types of a non-linear relationship between x and y; e.g. in the log case a given change in x will have an increasingly bigger impact on y. The opposite is true in the log-log case. The interpretation of the parameters also changes in the three cases. To understand the differences consider the derivative (or gradient) in each case:

Linear: $\frac{\partial y}{\partial x} = b$
Log: $\frac{\partial y}{\partial x} = \exp(a+b x)b=y\times b$
Log-Log: $\frac{\partial y}{\partial x} = \exp(a+b \ln x)b=y\times b \times \frac{1}{x}$

For this we use the following rules for taking derivatives:

Derivative of $\ln z$ is $\frac{1}{z}$
Derivative of $exp(z)$ is $exp(z)$
Chain rule: $\frac{\partial(f(g(x)))}{\partial x} = \frac{\partial(f(g(x)))}{\partial g(x)} \times \frac{\partial g(x)}{\partial x}$

To understand how to interpret the b coefficient, we can re-write the derivative formulas so that we have b on one side of the equation. We can also use the fact that the derivative is approximately the small change in y you get in response to a small change in x; i.e. $\frac{\partial y}{\partial x} \approx \frac{dy}{dx}$

Linear: $b=\frac{\partial y}{\partial x} \approx \frac{dy}{dx}$
Log: $b=\frac{\partial y}{\partial x} \frac{1}{y}\approx \frac{dy}{y} \times \frac{1}{dx}$
Log-log: $b=\frac{\partial y}{\partial x} \frac{x}{y}\approx \frac{\frac{dy}{y}}{ \frac{dx}{x}}$

Hence, we get the following interpretations (recalling that $dz/z$ is the growth rate of a variable z):

Linear: Change in $y$ if we change $x$ by 1 unit
Log: Growth rate of $y$ if we change $x$ by 1 unit (the growth rate in percent will be b times 100)
Log-log: Growth rate of $y$ (in percent) in response to growth of $x$ by 1 percent. Note that if $x$ grows by 1 percent we have that $dx/x=0.01$

Examples:

Suppose that y is the wage in $ and $x$ is the years of experience of an individual

Linear: 1 year more of experience means b$ more wage
Log: 1 year more of experience means b×100 percent more wage
Log-log: If your experience goes up by 1% (e.g. if you current experience is 10 years, 1% is 0.1 years or about 1 months 1 one week) then your wage will go up by b%.

Suppose that y is the number of crimes and is the number of foreigners in an area

Linear: 1 foreigner more implies b more crimes
Log: 1 foreigner more implies $b\times 100$ percent more crimes
Log-log: If the number of foreigners goes up by 1% then the number of crimes goes up by b%.

Citation

For attribution, please cite this work as

Martin (2020, Nov. 14). Datastories Hub: Quick Guide - Log linear models. Retrieved from https://mondpanther.github.io/datastorieshub/posts/quickguides/quickguide_loglinear/

BibTeX citation

@misc{martin2020quick,
  author = {Martin, Ralf},
  title = {Datastories Hub: Quick Guide - Log linear models},
  url = {https://mondpanther.github.io/datastorieshub/posts/quickguides/quickguide_loglinear/},
  year = {2020}
}