12.3 Natural Logarithm
Transforming the dependent and/or independent variables using the natural logarithm has some important and useful interpretations. Consider the following simple consumption equation in which both variables are in logarithmic form: \[ln(consumption)=\beta_0+\beta_1 \cdot \ln(income)+\epsilon\] In this case, \(\beta_1\) is the elasticity of consumption with respect to income, i.e., a 1% increase in income leads to a \(\beta_1 \cdot 1\%\) increase in consumption. For example, if \(\beta_1=0.4\), then a 1% increase in income will rise consumption by 0.4%. Note that the percentage increase is only an approximation for small changes.
| Dep. Var. | Indep. Var | Interpretation |
|---|---|---|
| \(y\) | \(x\) | 1 dollar change in \(x\) changes y by \(\hat{\beta}\) dollars |
| \(\ln(y)\) | \(x\) | 1 dollar change in \(x\) changes y by 100 \(\times \hat{\beta}\) percent |
| \(\ln(y)\) | \(\ln(x)\) | 1 percent change in \(x\) changes y by \(\hat{\beta}\) percent |
| \(y\) | \(\ln(x)\) | 1 percent change in \(x\) changes y by \(\hat{\beta}/100\) dollars |
##
## Call:
## lm(formula = log(total) ~ yards + att + exp + draft1 + veteran +
## changeteam + pbowlever, data = nfl)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.4899 -0.4998 -0.0801 0.4554 3.1959
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.9289322 0.0767846 -12.098 < 2e-16 ***
## yards 0.0003566 0.0001411 2.527 0.011783 *
## att 0.0003927 0.0009657 0.407 0.684408
## exp 0.0108812 0.0160213 0.679 0.497312
## draft1 0.8876564 0.1132374 7.839 2.30e-14 ***
## veteran 0.6735244 0.1144567 5.885 6.88e-09 ***
## changeteam -0.3095919 0.0893125 -3.466 0.000568 ***
## pbowlever 0.4093324 0.0936078 4.373 1.46e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7909 on 560 degrees of freedom
## (441 observations deleted due to missingness)
## Multiple R-squared: 0.55, Adjusted R-squared: 0.5444
## F-statistic: 97.8 on 7 and 560 DF, p-value: < 2.2e-16