18  Elasticity

In some contexts, it is useful to interpret the results of a regression model in terms of elasticity or semi-elasticity. One strategy to achieve that is to estimate a log-log or a semilog model, where the left and/or right-hand side variables are logged. Another approach is to note that $\frac{\partial ln(x)}{\partial x}=\frac{1}{x}$, and to post-process the marginal effects to transform them into elasticities or semi-elasticities.

For example, say we estimate a linear model of this form:

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\epsilon$$

Let $\hat{y}$ be the adjusted prediction made by the model for some combination of covariates $x_1$ and $x_2$. The slope with respect to $x_1$ (or “marginal effect”) is:

$$\frac{\partial \hat{y}}{\partial x_1}$$

We can estimate the “eyex”, “eydx”, and “dyex” (semi-)elasticities with respect to $x_1$ as follows:

$${\eta }_1=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{x_1}{\hat{y}}$$ $${\eta }_2=\frac{\partial \hat{y}}{\partial x_1}\cdot \frac{1}{\hat{y}}$$ $${\eta }_3=\frac{\partial \hat{y}}{\partial x_1}\cdot x_1$$

with interpretations roughly as follows:

1. A percentage point increase in $x_1$ is associated to a ${\eta }_1$ percentage points increase in $y$.
2. A unit increase in $x_1$ is associated to a ${\eta }_2$ percentage points increase in $y$.
3. A percentage point increase in $x_1$ is associated to a ${\eta }_3$ units increase in $y$.

For further intuition, consider the ratio of change in $y$ to change in $x$: $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$. We can turn this ratio into a ratio between relative changes by dividing both the numerator and the denominator: $\frac{\frac{\mathrm{\Delta }y}{y}}{\frac{\mathrm{\Delta }x}{x}}$. This is of course linked to the expression for the ${\eta }_1$ elasticity above.

With the marginaleffects package, these quantities are easy to compute:

library(marginaleffects)
mod <- lm(mpg ~ hp + wt, data = mtcars)

avg_slopes(mod)
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)    S   2.5 %  97.5 %
#>    hp  -0.0318    0.00903 -3.52   <0.001 11.2 -0.0495 -0.0141
#>    wt  -3.8778    0.63273 -6.13   <0.001 30.1 -5.1180 -2.6377
#> 
#> Type:  response 
#> Comparison: dY/dX

avg_slopes(mod, slope = "eyex")
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp   -0.285     0.0855 -3.34   <0.001 10.2 -0.453 -0.118
#>    wt   -0.746     0.1418 -5.26   <0.001 22.7 -1.024 -0.468
#> 
#> Type:  response 
#> Comparison: eY/eX

avg_slopes(mod, slope = "eydx")
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)    S    2.5 %    97.5 %
#>    hp -0.00173   0.000502 -3.46   <0.001 10.8 -0.00272 -0.000751
#>    wt -0.21165   0.037849 -5.59   <0.001 25.4 -0.28583 -0.137464
#> 
#> Type:  response 
#> Comparison: eY/dX

avg_slopes(mod, slope = "dyex")
#> 
#>  Term Estimate Std. Error     z Pr(>|z|)    S  2.5 % 97.5 %
#>    hp    -4.66       1.32 -3.52   <0.001 11.2  -7.26  -2.06
#>    wt   -12.48       2.04 -6.13   <0.001 30.1 -16.47  -8.49
#> 
#> Type:  response 
#> Comparison: dY/eX